These notes synthesize material from Michael Rubinstein’s PMATH 440 lectures at the University of Waterloo with supplementary material drawn from Tom Apostol’s Introduction to Analytic Number Theory*, Harold Davenport’s* Multiplicative Number Theory*, Andrew Granville’s lecture notes, K. Soundararajan’s Stanford lecture notes, MIT OCW 18.785, and Terry Tao’s blog. The goal is to present analytic number theory not merely as a list of theorems but as a living mathematical narrative — tracing how the deep questions about prime distribution forced mathematicians to build an entirely new kind of analysis.*

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Sources and References

Primary textbook — None required.

Supplementary texts — Tom Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976; Harold Davenport, Multiplicative Number Theory; M. Ram Murty, Problems in Analytic Number Theory; G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford, 1979.

Online resources — MIT OCW 18.785 (Analytic Number Theory); Andrew Granville’s lecture notes; Terry Tao’s blog posts on analytic number theory; K. Soundararajan’s lecture notes (Stanford).

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Prologue: What Is Analytic Number Theory?

Number theory asks questions about integers: which numbers are prime? how many primes are there below a trillion? can every even number be written as a sum of two primes? These seem like purely combinatorial questions, and yet — remarkably — the most powerful tools for answering them come from complex analysis, Fourier theory, and the study of special functions. This collision of the discrete and the continuous is what makes analytic number theory so beautiful and so surprising.

\[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \frac{1}{1-p^{-s}}, \]

which simultaneously encodes all positive integers (through the sum) and all prime numbers (through the Euler product). Riemann’s profound 1859 insight was that by treating \(s\) as a complex variable and studying \(\zeta(s)\) as an analytic function on the complex plane, one could extract precise information about how the primes are distributed. The zeros of \(\zeta(s)\) — particularly the nontrivial zeros lying in the strip \(0 < \mathrm{Re}(s) < 1\) — directly control the error in the approximation \(\pi(x) \approx x/\log x\). Understanding these zeros is the central unsolved problem in mathematics: the Riemann Hypothesis.

The course follows a logical arc: we begin with elementary but sharp estimates for primes using Chebyshev’s method, then develop the analytical machinery (summation formulas, Dirichlet series, the zeta function), prove the Prime Number Theorem via complex analysis, extend everything to Dirichlet L-functions for the study of primes in arithmetic progressions, and close with a discussion of the Riemann Hypothesis and what is known about zeros.

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Chapter 1: Primes and Elementary Estimates

1.1 The Prime Numbers

The prime numbers are the atoms of multiplicative arithmetic. Recall that a positive integer \( p > 1 \) is prime if its only positive divisors are \( 1 \) and \( p \) itself. Every integer \( n > 1 \) factors uniquely as a product of primes (the Fundamental Theorem of Arithmetic). This uniqueness is why primes are so fundamental: they are the building blocks, and any multiplicative structure must be understood prime by prime.

The first several primes are \( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, \ldots \). Looking at this list, a natural pattern-seeking instinct kicks in: how are the primes distributed? Do they eventually stop (Euclid tells us no)? Do they thin out (yes — the gaps between primes grow on average)? How rapidly?

The first few values are \( \pi(10) = 4 \), \( \pi(100) = 25 \), \( \pi(1000) = 168 \), \( \pi(10^6) = 78498 \), \( \pi(10^9) = 50{,}847{,}534 \). Notice the pattern: primes become sparser as numbers grow larger, but they never disappear entirely. The ratio \(\pi(x)/x\) declines toward zero — roughly like \(1/\log x\). The central theorem governing the distribution of primes is:

The logarithmic integral \( \mathrm{li}(x) \) is a significantly better approximation. For instance, \( \mathrm{li}(10^6) \approx 78628 \) while \( 10^6/\log(10^6) \approx 72382 \). The true count \(\pi(10^6) = 78498\) lies much closer to \(\mathrm{li}(10^6)\). Heuristically, one can think of each integer \(n\) near \(x\) as being “prime with probability \(1/\log x\)” — this is made precise by the PNT. The proof of PNT, given in Chapters 4–5, requires understanding the analytic continuation and zero-free regions of the Riemann zeta function. This motivates everything that follows.

1.2 Euclid’s Theorem

This proof is a logical gem, but it is quantitatively weak — it shows \(\pi(x) \to \infty\) but gives no rate. Note that the number \(N = p_1 \cdots p_k + 1\) constructed in the proof is not necessarily prime itself (e.g., \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 = 30031 = 59 \times 509\)), but it has a prime factor not in our list. With more work one can extract a quantitative lower bound: \(\pi(x) \geq \log_2 \log_2 x\) follows from iterating Euclid’s argument. But we can do much better.

1.3 Bertrand’s Postulate and Chebyshev Bounds

Bertrand’s Postulate (proved by Chebyshev in 1852) states that for every integer \( n \geq 1 \), there exists a prime \( p \) with \( n < p \leq 2n \). Chebyshev proved this and simultaneously established the right order of magnitude for \( \pi(x) \). Bertrand had conjectured this after numerical verification for \(n \leq 3{,}000{,}000\). Chebyshev’s analytic proof was a landmark.

The proof relies on properties of the central binomial coefficient \( \binom{2n}{n} \). Observe that \( 4^n = (1+1)^{2n} \geq \binom{2n}{n} \), and all prime powers \( p^k \) with \( p \leq 2n \) and \( p^k > n \) divide \( \binom{2n}{n} \). These estimates give both bounds after careful bookkeeping. We do not reproduce the full proof here, but instead develop the sharper Chebyshev functions that lead more naturally to PNT.

1.3a Full Proof of Chebyshev’s Upper and Lower Bounds via Central Binomial Coefficients

\[ \binom{2n}{n} = \frac{(2n)!}{(n!)^2}. \]

This innocent object carries a surprising amount of information about primes. We work through the proof in enough detail that every step is traceable.

Setup. We need two facts:

1. \(\binom{2n}{n} \leq 4^n\) (since it is one of the \(2^{2n}\) terms in \((1+1)^{2n}\)).
2. \(\binom{2n}{n} \geq 4^n / (2n+1)\) (since the sum of all \(2n+1\) terms is \(4^n\) and \(\binom{2n}{n}\) is the largest).

Step 1: Upper bound.

\[ \prod_{n < p \leq 2n} p \;\Big|\; \binom{2n}{n} \leq 4^n. \]

Taking logarithms: \(\theta(2n) - \theta(n) \leq n \log 4 = 2n \log 2\).

\[ \theta(x) = \sum_{k=0}^{\lfloor \log_2 x \rfloor} [\theta(x/2^k) - \theta(x/2^{k+1})] \leq \sum_{k=0}^{\lfloor \log_2 x \rfloor} 2 \cdot \frac{x}{2^{k+1}} \log 2 \leq 2x \log 2. \]

Since \(\psi(x) = \theta(x) + O(\sqrt{x}\log x)\), we get \(\psi(x) \leq 2x\log 2 + O(\sqrt{x}\log x)\), giving the upper bound \(\psi(x) < c_2 x\) for all large \(x\), with \(c_2 = 2\log 2 + \varepsilon\). The explicit bound \(\psi(x) \leq 2x\log 2\) requires a slightly more careful argument for small \(x\).

Step 2: Lower bound.

\[ \binom{2n}{n} = \prod_{p \leq 2n} p^{v_p(\binom{2n}{n})} \leq \prod_{p \leq 2n} (2n)^{v_p(\binom{2n}{n})}. \]\[ \frac{4^n}{2n+1} \leq \binom{2n}{n} \leq (2n)^{\pi(2n)} \cdot 4^{n/2}, \]\[ \pi(2n) \geq \frac{n\log 4 - (1/2)\log(2n) \cdot \pi(\sqrt{2n})}{\log(2n)} \geq c_1 \frac{n}{\log n}. \]

The cleanest explicit constants come from Chebyshev’s original analysis: \(c_1 = \log 2 \cdot (30^{1/30}) \approx 0.9212\log 2 \approx 0.638\) and \(c_2 = 6\log 2 \approx 4.16\), which were improved by later work. The key point is that the proof is purely elementary — no complex analysis, no integrals over half-planes, just binomial coefficients and prime factorizations.

Why this proof works. The central binomial coefficient \(\binom{2n}{n}\) is large (exponential in \(n\)), and yet its prime factorization only involves primes up to \(2n\). So the product of (powers of) primes up to \(2n\) must be large — at least \(4^n/(2n+1)\) — which forces there to be many primes, each contributing a logarithmic amount. This is Chebyshev’s beautiful insight.

1.3b A Table Comparing π(x), x/log x, and li(x)

One of the most illuminating facts about the PNT is that \(\mathrm{li}(x)\) approximates \(\pi(x)\) far better than \(x/\log x\) does. The following table makes this precise:

\(x\)	\(\pi(x)\)	\(\lfloor x/\log x \rfloor\)	\(\lfloor\mathrm{li}(x)\rfloor\)	Error \(\pi(x) - \lfloor x/\log x\rfloor\)	Error \(\pi(x) - \lfloor\mathrm{li}(x)\rfloor\)
\(10\)	4	4	5	0	\(-1\)
\(100\)	25	21	29	\(+4\)	\(-4\)
\(10^3\)	168	144	177	\(+24\)	\(-9\)
\(10^4\)	1229	1085	1245	\(+144\)	\(-16\)
\(10^6\)	78498	72382	78627	\(+6116\)	\(-129\)
\(10^9\)	50847534	48254942	50849234	\(+2592592\)	\(-1700\)
\(10^{12}\)	37607912018	36191206825	37607950280	\(+1416705193\)	\(-38262\)

The numbers speak for themselves: the error \(\pi(x) - x/\log x\) grows like \(x/(\log x)^2\) (a genuine secondary term), while \(\pi(x) - \mathrm{li}(x)\) grows much more slowly and oscillates. The PNT says both errors are \(o(x/\log x)\); the Riemann Hypothesis would say the li-error is \(O(\sqrt{x}\log x)\), which the table is entirely consistent with.

Note on the sign. Empirically \(\pi(x) < \mathrm{li}(x)\) for all \(x\) up to roughly \(10^{316}\) (Skewes’s number), but Littlewood proved in 1914 that \(\pi(x) - \mathrm{li}(x)\) changes sign infinitely often. The persistent negative sign for small \(x\) is related to the “Chebyshev bias” discussed in Remark 4.12.

1.4 The Chebyshev Functions

Why introduce \(\theta(x)\) and \(\psi(x)\)? The function \(\pi(x)\) has an inconvenient jump of size 1 at each prime, and the primes have varying sizes (logarithms). The Chebyshev functions package the information more smoothly by weighting each prime by its logarithm, making them better adapted to Dirichlet series and Abel summation techniques.

\[ \log n = \sum_{d \mid n} \Lambda(d), \]

which follows from writing \( n = \prod_p p^{v_p(n)} \) and summing: the divisors \(d\) of \(n\) that are prime powers \(p^j\) (with \(j \geq 1\)) contribute \(\log p\) each, and summing \(\log p\) over all prime power divisors \(p^j \mid n\) gives \(\sum_p v_p(n)\log p = \log n\).

This identity is crucial because it connects \(\Lambda\) (a function on prime powers) to the more tractable function \(\log\) (defined on all integers) via Dirichlet convolution with the constant function \(\mathbf{1}\). Möbius inversion then gives \(\Lambda = \mu * \log\), i.e., \(\Lambda(n) = -\sum_{d \mid n}\mu(d)\log d\).

1.5 The Equivalence with PNT

\[ \pi(x) \sim \frac{x}{\log x}, \qquad \theta(x) \sim x, \qquad \psi(x) \sim x. \]

1.5a Why ψ(x) Is Easier Than π(x): The Role of the von Mangoldt Function

One might wonder: why not work directly with \(\pi(x)\)? The answer lies in the analytic structure of Dirichlet series.

\[ -\frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}. \]\[ \frac{\zeta'(s)}{\zeta(s)} = \frac{d}{ds}\log\zeta(s) = -\sum_p\frac{-\log p \cdot p^{-s}}{1-p^{-s}} \cdot (-1) = \sum_p \frac{\log p}{p^s - 1} = -\sum_{p,k \geq 1}\frac{\log p}{p^{ks}} = -\sum_{n \geq 1}\frac{\Lambda(n)}{n^s}. \]

Why this matters. The Dirichlet series \(-\zeta'(s)/\zeta(s)\) has:

• A simple pole at \(s=1\) (because \(\zeta(s)\) has a simple pole there, so \(\log\zeta(s)\) has a logarithmic singularity, and its derivative has a simple pole with residue 1).
• Simple poles at each zero \(\rho\) of \(\zeta(s)\) (where the denominator vanishes), each with residue \(-1\) (from the zero of first order).
• Simple poles at \(s=0\) and the trivial zeros \(s = -2, -4, \ldots\).

\[ \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2}\log(1-x^{-2}) \]

is the direct consequence. Each zero \(\rho\) of \(\zeta\) produces a term; the dominant term \(x\) comes from the pole at \(s=1\).

\[ \pi(x) = \sum_{k=1}^\infty \frac{\mu(k)}{k}\cdot\mathrm{li}(x^{1/k}) + O(\text{smaller}). \]

The leading term is \(\mathrm{li}(x)\), and the correction terms involve \(\mathrm{li}(\sqrt{x})\), \(\mathrm{li}(x^{1/3})\), etc. This is Riemann’s formula \(\pi(x) = R(x) - \sum_\rho R(x^\rho) + \cdots\) where \(R(x) = \sum_k \mu(k)/k \cdot \mathrm{li}(x^{1/k})\) is the Riemann \(R\)-function. But deriving this from scratch requires understanding \(\psi(x)\) first.

Summary. \(\psi(x)\) is analytically more tractable than \(\pi(x)\) because: (1) it is the partial sum of a naturally occurring Dirichlet series \(-\zeta'/\zeta\); (2) its generating function has a simple pole structure (simple poles at zeros and at \(s=1\)) amenable to residue calculus; (3) the passage from \(\psi(x) \sim x\) to \(\pi(x) \sim x/\log x\) is a routine application of partial summation (Abel’s formula). This is why analysts “always” prove \(\psi(x) \sim x\) first and deduce the PNT from it.

Why is \(\psi(x) \sim x\) the most convenient form to prove? Because \(\psi(x) = \sum_{n \leq x}\Lambda(n)\) is the partial sum of the Dirichlet series \(-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n)n^{-s}\). Via Perron’s formula (Chapter 10), proving \(\psi(x) \sim x\) reduces to showing that the contour integral of \(-\zeta'(s)/\zeta(s) \cdot x^s/s\) over a vertical line captures only the simple pole at \(s=1\) — which in turn requires \(\zeta(s) \neq 0\) on the line \(\mathrm{Re}(s) = 1\).

1.6 An Elementary Upper Bound for ψ(x)

From the divisibility properties of binomial coefficients, one can show:

Combined with the lower bounds, this places \( \psi(x) \) in the range \( [c_1 x, c_2 x] \) for explicit constants. The PNT sharpens this to \( \psi(x) \sim x \), requiring the analytic theory of \( \zeta(s) \).

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Chapter 2: Summation Methods

Analytic number theory often requires converting sums involving arithmetic functions into integrals or estimating them asymptotically. In analysis one routinely passes from integrals to sums; here we need to go in the opposite direction. The key tools are the Abel summation formula (summation by parts, the discrete analogue of integration by parts) and the Euler–Maclaurin formula (which gives an asymptotic expansion of the error between sum and integral in terms of Bernoulli numbers). These techniques appear throughout the subject and are worth mastering thoroughly before proceeding.

2.1 Abel’s Summation Formula

\[ A(x) = \sum_{n \leq x} a_n. \]

Fully Worked Example: ∑_{p≤x} 1/p ~ log log x via Abel Summation

One of the most striking applications of Abel summation in prime number theory is the derivation of the divergence rate of \(\sum_{p \leq x} 1/p\). This sum grows extremely slowly — slower than any power of \(\log x\) — but it does diverge, and its rate is exactly \(\log\log x\).

Corollaries of Abel Summation

Abel summation is the workhorse for converting sums to integrals. Key applications include:

• Estimating partial sums of Dirichlet series: if \( f(x) = x^{-s} \), Abel’s formula relates \( \sum_{n \leq x} a_n n^{-s} \) to \( \sum_{n \leq x} a_n \).
• Proving convergence of Dirichlet series on vertical lines given growth bounds on \( A(x) \).
• Deriving asymptotics for \( \sum_{n \leq x} \frac{\log n}{n} \), \( \sum_{p \leq x} \frac{1}{p} \), etc.

2.2 Euler–Maclaurin Summation Formula

The Euler–Maclaurin formula gives a precise relationship between a sum and an integral, with an explicit error in terms of Bernoulli numbers. It is more powerful than Abel summation when one wants a complete asymptotic expansion rather than just a leading term.

Fully Worked Example: ∑_{n≤x} log n = x log x - x + O(log x) via Euler–Maclaurin

2.3 Poisson Summation Formula

The Poisson summation formula connects a sum over the integers with a sum over the integers of the Fourier transform. It is fundamental in the analytic theory of L-functions, particularly for proving functional equations. The key idea is that periodizing a function and expanding in Fourier series gives a relation between the function values at integers and the Fourier transform values at integers.

2.4 Voronoi Summation

The Voronoi summation formula is a more refined tool for sums involving the divisor function \( d(n) \). It generalizes Poisson summation to arithmetic settings.

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Chapter 3: Arithmetic Functions and Dirichlet Series

Arithmetic functions — functions from the positive integers to the complex numbers — are the basic vocabulary of number theory. The Fundamental Theorem of Arithmetic gives the positive integers a rich multiplicative structure, and functions that respect this structure (multiplicative functions) are especially tractable. The formalism of Dirichlet series converts arithmetic functions into complex analytic objects, and Dirichlet convolution becomes ordinary multiplication of these series. This dictionary between arithmetic and analysis is the engine of the subject.

3.0a Motivating Dirichlet Series: Linearizing Multiplicative Structure

Before we define things formally, let us explain why Dirichlet series are the right tool.

Suppose you want to study an arithmetic function \(f : \mathbb{Z}_{>0} \to \mathbb{C}\) — say, \(f(n) = \mu(n)\) or \(f(n) = d(n)\) — by “generating function” methods. The natural first guess for a generating function is the power series \(\sum_{n \geq 1}f(n)x^n\). But this is poorly adapted to multiplicative structure: the product \((\sum f(n)x^n)(\sum g(n)x^n)\) gives Cauchy convolution \(\sum_{n}(\sum_{k}f(k)g(n-k))x^n\), which involves additive structure \(k + (n-k) = n\). Multiplicative arithmetic is about divisors, not additive decompositions.

\[ \left(\sum_{n \geq 1}\frac{f(n)}{n^s}\right)\!\left(\sum_{m \geq 1}\frac{g(m)}{m^s}\right) = \sum_{n \geq 1}\frac{(f*g)(n)}{n^s}, \]

where \((f*g)(n) = \sum_{d \mid n}f(d)g(n/d)\) is Dirichlet convolution. This works because \(n^{-s}m^{-s} = (nm)^{-s}\), and \(nm\) runs over products — exactly what divisor sums are about. Multiplication of Dirichlet series corresponds exactly to Dirichlet convolution of arithmetic functions.

\[ \sum_{n \geq 1}\frac{f(n)}{n^s} = \prod_p \left(\sum_{k \geq 0}\frac{f(p^k)}{p^{ks}}\right). \]\[ \zeta(s) = \prod_p \frac{1}{1-p^{-s}}. \]

The key dictionary. If you know the Dirichlet series of \(f\) and \(g\), then:

• \(f*g\) has Dirichlet series \(F(s)G(s)\).
• \(f*\mathbf{1}\) (summing \(f\) over divisors) has Dirichlet series \(F(s)\zeta(s)\).
• Möbius inversion \(f = g*\mu\) corresponds to \(F(s) = G(s)/\zeta(s)\).
• Differentiation \(-F'(s) = \sum f(n)\log(n)/n^s\) corresponds to weighting by \(\log\).

Every identity between arithmetic functions via Dirichlet convolution translates into an algebraic identity between Dirichlet series — and those identities can then be leveraged analytically (studying singularities, zero-free regions, contour integrals) to extract asymptotics.

3.1 Multiplicative Functions

Key Examples

\[ \mu(n) = \begin{cases} 1 & n = 1, \\ (-1)^k & n = p_1 p_2 \cdots p_k \text{ (distinct primes)}, \\ 0 & p^2 \mid n \text{ for some prime } p. \end{cases} \]

This is multiplicative. Its defining property: \( \sum_{d \mid n} \mu(d) = [n=1] \) (the Iverson bracket). Intuitively, \(\mu\) is an inclusion-exclusion coefficient: summing \(\mu(d)\) over divisors of \(n\) counts all \(n\) with a “plus” and cancels everything with a “minus.” The fact that this sum is zero for \(n > 1\) is equivalent to the Möbius inversion formula.

\[ \varphi(p^k) = p^k - p^{k-1} = p^k\left(1 - \frac{1}{p}\right), \]

and hence \( \varphi(n) = n \prod_{p \mid n} (1 - 1/p) \). Also \( \sum_{d \mid n} \varphi(d) = n \). For example: \(\varphi(12) = \varphi(4)\varphi(3) = 2 \cdot 2 = 4\), and indeed \(\{1, 5, 7, 11\}\) are the units mod 12.

The divisor functions \( \sigma_k(n) = \sum_{d \mid n} d^k \). In particular, \( d(n) = \sigma_0(n) \) counts the number of divisors and \( \sigma(n) = \sigma_1(n) \) sums them. Both are multiplicative. For a prime power: \(\sigma_k(p^e) = 1 + p^k + p^{2k} + \cdots + p^{ek} = (p^{k(e+1)}-1)/(p^k-1)\).

The von Mangoldt function \( \Lambda(n) \) (defined above) is not multiplicative but satisfies \( \log n = \sum_{d \mid n} \Lambda(d) \).

The Liouville function \( \lambda(n) = (-1)^{\Omega(n)} \) where \( \Omega(n) = \sum_p v_p(n) \) is the total number of prime factors with multiplicity. This is completely multiplicative. The Dirichlet series \(\sum_n \lambda(n)n^{-s} = \zeta(2s)/\zeta(s)\). The statement \(\sum_{n \leq x}\lambda(n) = o(x)\) is equivalent to PNT.

3.1a A Zoo of Arithmetic Functions

It is useful to have a systematic taxonomy before proceeding. The landscape of arithmetic functions can be organized along several axes.

By multiplicativity:

Class	Definition	Examples
Completely multiplicative	\(f(mn) = f(m)f(n)\) for all \(m,n\)	\(\mathrm{id}(n) = n\), \(n^s\), \(\chi(n)\) (Dirichlet char.), \(\lambda(n)\) (Liouville)
Multiplicative	\(f(mn) = f(m)f(n)\) for \(\gcd(m,n)=1\)	\(\mu(n)\), \(\varphi(n)\), \(d(n)\), \(\sigma_k(n)\)
Additive	\(f(mn) = f(m)+f(n)\) for \(\gcd(m,n)=1\)	\(\omega(n)\), \(\log n\) on prime powers…
Completely additive	\(f(mn) = f(m)+f(n)\) for all \(m,n\)	\(\Omega(n) = \sum_p v_p(n)\), \(\log n\)
Neither	—	\(\Lambda(n)\), \(r_2(n)\) (as a function of \(n\))

The distinction matters because:

• Multiplicative functions are determined by their values at prime powers. So to specify \(f\) completely, you only need to say: “at prime power \(p^k\), \(f(p^k) = \ldots\).”
• Completely multiplicative functions are determined by their values at primes alone: if you know \(f(p)\) for all primes \(p\), then \(f(p^k) = f(p)^k\), and \(f(n) = \prod_p f(p)^{v_p(n)}\).
• The product of two multiplicative functions (under Dirichlet convolution) is multiplicative. This means the class of multiplicative arithmetic functions is closed under the natural operations, making it a well-structured abelian group.

\[ \mu(1) = 1, \quad \mu(p) = -1, \quad \mu(p^2) = 0, \quad \mu(pq) = 1, \quad \mu(p^2q) = 0, \quad \mu(pqr) = -1, \ldots \]

The pattern: \(\mu(n) = (-1)^k\) if \(n\) is squarefree with \(k\) prime factors, and \(\mu(n)=0\) otherwise. The Dirichlet series is \(\sum \mu(n)n^{-s} = 1/\zeta(s)\), which encodes the “inverse” of \(\zeta\) in the Dirichlet series ring. This is the number-theoretic content of the identity \(\zeta(s) \cdot (1/\zeta(s)) = 1\).

The von Mangoldt function \(\Lambda(n)\). Not multiplicative, not additive. It is supported on prime powers and satisfies \(\log n = \sum_{d \mid n}\Lambda(d)\) — a factored form of the logarithm. Its Dirichlet series \(-\zeta'/\zeta = \sum \Lambda(n)n^{-s}\) is the logarithmic derivative of \(\zeta\), which naturally arises when you differentiate \(\log\zeta(s) = \sum_p\sum_k p^{-ks}/k\). The non-multiplicativity of \(\Lambda\) is not a defect: it reflects the fact that \(-\zeta'/\zeta\) is not an Euler product, but a ratio \(\zeta'/\zeta\).

Additive functions. An additive function satisfies \(f(mn) = f(m)+f(n)\) when \(\gcd(m,n)=1\). The key examples:

• \(\omega(n) = \#\{p : p \mid n\}\): number of distinct prime factors. \(\omega(12) = \omega(2^2\cdot 3) = 2\).
• \(\Omega(n) = \sum_p v_p(n)\): total number of prime factors with multiplicity. \(\Omega(12) = \Omega(2^2 \cdot 3) = 3\).
• \(\Omega(n) - \omega(n)\) counts the “repeat prime factors.”

Additive functions are less common in analytic number theory but appear in probabilistic number theory (Erdős–Kac theorem: \(\omega(n)\) is approximately Gaussian with mean and variance \(\log\log n\)).

Summary table of key arithmetic functions and their Dirichlet series:

Function	Formula	Multiplicative?	Dirichlet series
\(\mathbf{1}(n) = 1\)	constant 1	completely	\(\zeta(s)\)
\(\mathrm{id}(n) = n\)	identity	completely	\(\zeta(s-1)\)
\(\mu(n)\)	Möbius	multiplicative	\(1/\zeta(s)\)
\(\varphi(n)\)	Euler totient	multiplicative	\(\zeta(s-1)/\zeta(s)\)
\(d(n)\)	divisor count	multiplicative	\(\zeta(s)^2\)
\(\sigma_k(n)\)	divisor power sum	multiplicative	\(\zeta(s)\zeta(s-k)\)
\(\Lambda(n)\)	von Mangoldt	neither	\(-\zeta'(s)/\zeta(s)\)
\(\lambda(n)\)	Liouville	completely	\(\zeta(2s)/\zeta(s)\)
\(\chi(n)\)	Dirichlet char.	completely	\(L(s,\chi)\)

3.2 Dirichlet Convolution and Möbius Inversion

Dirichlet convolution is commutative and associative. The identity element is \( \varepsilon(n) = [n=1] \). The convolution of two multiplicative functions is multiplicative. This makes the set of multiplicative functions (with Dirichlet convolution) an abelian group.

Möbius inversion is the number-theoretic analogue of the inclusion-exclusion principle. Just as inclusion-exclusion counts elements with or without certain properties, Möbius inversion recovers \(f\) from \(g = f * \mathbf{1}\) by reversing the summation over divisors.

Applications of Möbius Inversion

• \( \varphi = \mu * \mathrm{id} \) (where \( \mathrm{id}(n) = n \)), so \( n = \sum_{d \mid n} \varphi(d) \) inverts to \( \varphi(n) = \sum_{d \mid n} \mu(d)(n/d) \).
• \( \log = \Lambda * \mathbf{1} \), so \( \Lambda = \log * \mu \), giving \( \Lambda(n) = -\sum_{d \mid n} \mu(d)\log d \).
• \( d = \mathbf{1} * \mathbf{1} \), \( \sigma_k = \mathrm{id}_k * \mathbf{1} \).

3.3 Dirichlet Series

The reason Dirichlet series are so natural here is that the substitution \(n^{-s} = e^{-s\log n}\) converts the multiplicative structure of the integers (via unique factorization) into additive structure (via \(\log(mn) = \log m + \log n\)), which complex analysis handles with ease. The Dirichlet series is essentially the Laplace transform of the arithmetic function \(a(n)\) with the discrete measure \(\sum_n a(n)\delta_{\log n}\).

Convergence of Dirichlet Series

More precisely, there is an abscissa of absolute convergence \( \sigma_a \) and an abscissa of conditional convergence \( \sigma_c \leq \sigma_a \leq \sigma_c + 1 \), such that \( F(s) \) converges absolutely for \( \mathrm{Re}(s) > \sigma_a \) and conditionally for \( \mathrm{Re}(s) > \sigma_c \).

Landau’s theorem is powerful: it says that for Dirichlet series with non-negative coefficients, the analytic properties of the function on the real axis completely determine where the series converges. We will apply it in Chapter 8 to prove \(L(1,\chi) \neq 0\) for real characters.

Multiplication of Dirichlet Series

This is the fundamental reason Dirichlet series are adapted to multiplicative problems. The algebraic identity \(F(s)G(s) = H(s)\) at the level of complex functions corresponds to \(a*b = c\) at the level of arithmetic functions — a purely multiplicative statement becomes a statement about analytic functions, and vice versa.

Euler Products

3.4 Euler’s Constant and Key Asymptotic Sums

The existence of this limit is not obvious. It says that the harmonic series \(\sum 1/n\) grows like \(\log N\), and the difference \(\sum_{n \leq N} 1/n - \log N\) stabilizes. The constant \(\gamma\) appears throughout analytic number theory: in the Laurent expansion of \(\zeta(s)\) near \(s=1\), in Mertens’ theorems, in the asymptotic for the divisor sum, and in many other places.

3.5 The Divisor Sum Formula

This is the Dirichlet divisor problem; see Chapter 6 for the proof. The error term \( O(\sqrt{x}) \) has been improved significantly (by van der Corput and others), but the final answer (Dirichlet’s conjecture gives \( O(x^{1/4+\varepsilon}) \)) remains open.

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Chapter 4: The Riemann Zeta Function

The Riemann zeta function is the centerpiece of analytic number theory. Euler studied it for real \(s > 1\) and proved the Euler product, relating it to primes. Riemann’s 1859 paper extended \(\zeta\) to a meromorphic function on all of \(\mathbb{C}\), discovered the functional equation \(\xi(s) = \xi(1-s)\), identified the nontrivial zeros as lying in the critical strip, and made the Riemann Hypothesis. Hadamard and de la Vallée Poussin used the non-vanishing on \(\mathrm{Re}(s) = 1\) to prove PNT in 1896.

In this chapter we build the theory of \(\zeta(s)\) step by step: definition and Euler product for \(\mathrm{Re}(s) > 1\), analytic continuation to \(\mathrm{Re}(s) > 0\) (and then to all of \(\mathbb{C}\)), the functional equation, non-vanishing on \(\mathrm{Re}(s) = 1\), and zero-free regions.

4.1 Definition and Basic Properties

\[ \zeta(s) = \prod_p \frac{1}{1 - p^{-s}}, \quad \mathrm{Re}(s) > 1. \]

This product immediately implies \( \zeta(s) \neq 0 \) for \( \mathrm{Re}(s) > 1 \) (each factor is nonzero and the product converges absolutely). The connection between the Dirichlet series and the Euler product is the analytic manifestation of the Fundamental Theorem of Arithmetic.

4.2 Analytic Continuation to Re(s) > 0

One can continue further:

This is proved via the functional equation (Section 4.4) or via the Euler–Maclaurin formula.

The Laurent expansion near s = 1

\[ \zeta(s) = \frac{1}{s-1} + \gamma + \gamma_1(s-1) + \gamma_2(s-1)^2 + \cdots, \]\[ \gamma_k = \frac{(-1)^k}{k!}\lim_{N\to\infty}\left(\sum_{n=1}^N \frac{(\log n)^k}{n} - \frac{(\log N)^{k+1}}{k+1}\right). \]

The fact that the constant term in the Laurent expansion is \(\gamma\) (Euler’s constant) — the same constant that appears in \(\sum_{n \leq x} 1/n = \log x + \gamma + O(1/x)\) — is not a coincidence: it follows directly from writing \(\zeta(s) = \sum n^{-s}\) and expanding near \(s=1\).

4.3 Non-Vanishing on the Line Re(s) = 1

The non-vanishing of \( \zeta(s) \) on the line \( \mathrm{Re}(s) = 1 \) is the key analytic input for PNT. Without it, we cannot rule out that the sum over zeros in the explicit formula contributes a term of order \(x\), swamping the main term.

The proof uses a classical trigonometric inequality:

4.4 The Functional Equation

4.5 The Riemann Hypothesis

The Riemann Hypothesis (RH) is the statement that all nontrivial zeros \(\rho\) of \(\zeta(s)\) satisfy \(\mathrm{Re}(\rho) = 1/2\). It is one of the Millennium Prize Problems (with a $1 million prize for a proof), and it has been verified numerically for the first \(10^{13}\) zeros (all lying on the critical line, none lying off it). But no proof is known.

\[ \psi(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2}\log(1-x^{-2}). \]

Each zero \(\rho = \beta + i\gamma\) contributes a term \(x^\rho/\rho = x^\beta e^{i\gamma\log x}/\rho\) of absolute value \(x^\beta/|\rho|\). If all zeros have \(\beta = 1/2\), these terms are bounded by \(x^{1/2}/|\rho|\), and their sum converges to give an error of size \(O(x^{1/2}\log^2 x)\) in \(\psi(x) - x\). This is the best possible error one could hope for from the explicit formula.

If a zero existed with \(\beta > 1/2\), it would contribute a term growing like \(x^\beta\), degrading the error estimate. The known zero-free region \(\sigma \geq 1 - c/\log t\) gives an error of \(O(x\exp(-c'\sqrt{\log x}))\) — much weaker than RH but still sufficient for many applications.

4.6 Newman’s Tauberian Theorem

A key tool connecting analytic properties of \( \zeta(s) \) to number-theoretic asymptotics is the following Tauberian theorem:

4.7 The Riemann Hypothesis and Zero-Free Regions

While the Riemann Hypothesis (RH) remains unproved, one can establish explicit zero-free regions:

\[ \pi(x) = \mathrm{li}(x) + O\!\left(x \exp\!\left(-c\sqrt{\log x}\right)\right). \]

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Chapter 5: The Prime Number Theorem

5.1 Statement and Equivalences

The proof proceeds via the analytic properties of \( \zeta(s) \) established in Chapter 4.

5.2 Strategy: Möbius Inversion Approach

One approach (using the non-vanishing of \( \zeta(s) \) on \( \mathrm{Re}(s)=1 \)) proceeds through the following function.

5.3 Key Lemma: Cancellation in the Möbius Sum

5.4 Completing the Proof of PNT

5.5 Explicit Formula for ψ(x)

The Riemann explicit formula connects \( \psi(x) \) directly to the zeros of \( \zeta(s) \). It is one of the most beautiful and surprising results in all of mathematics: the distribution of primes — a completely multiplicative, discrete phenomenon — is governed by the zeros of a complex-analytic function.

PNT then follows from the explicit formula if one can show \( \zeta(1+it) \neq 0 \) for all \( t \neq 0 \) (which we proved in Chapter 4), ensuring the sum over zeros does not contribute a term of size \( x \).

5.6 The PNT via Contour Integration: A Bird’s Eye View

Let us outline the contour integration proof of PNT more explicitly, as it illustrates the general strategy used throughout the subject.

\[ \psi(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \left(-\frac{\zeta'(s)}{\zeta(s)}\right)\frac{x^s}{s}\,ds. \]

Step 2 (Truncate). Replace \(i\infty\) by \(iT\) for a large parameter \(T = T(x)\), with an error of \(O(x\log^2 x / T)\).

Step 3 (Move the contour). Shift the line of integration from \(\mathrm{Re}(s) = c\) to the left: specifically to the contour \(L\) consisting of the left-hand boundary of the zero-free region, \(\sigma = 1 - c_0/\log(|t|+2)\). The horizontal segments contribute \(O(\cdots)\).

Step 4 (Pick up residues). The residue at \(s = 1\) (the only pole of \(-\zeta'/\zeta\) we cross) gives \(x^1/1 = x\). All other poles — the zeros of \(\zeta\) — are to the left of our contour and do not contribute.

Step 5 (Bound the remaining integral). On the contour \(L\), one needs bounds on \(\zeta(s)\) and \(\zeta'(s)/\zeta(s)\). Using \(|\zeta(\sigma+it)| = O(\log t)\) in the critical strip, and the bounds on \(\zeta'/\zeta\), one shows the integral over \(L\) contributes \(O(x\exp(-c_1\sqrt{\log x}))\).

Conclusion. \(\psi(x) = x + O(x\exp(-c_1\sqrt{\log x}))\), giving \(\psi(x) \sim x\) and hence PNT.

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Chapter 6: Lattice Point Problems

6.1 The Divisor Function and Its Averages

The exact formula for the partial sum:

6.2 Dirichlet’s Hyperbola Method

6.3 Gauss Circle Problem

The Gauss circle problem asks: how many lattice points \( (m,n) \in \mathbb{Z}^2 \) lie in the disk \( m^2 + n^2 \leq r^2 \)?

6.4 Van der Corput’s Method

Van der Corput’s method improves the error terms in both divisor and circle problems by bounding exponential sums using the geometry of the phase function.

The method transforms a sum over an interval into a shorter sum (the B-process uses the Poisson formula), and iterating gives better and better bounds. For the divisor problem, one sets \( f(n) = (t/2\pi)\log n \) and estimates the resulting exponential sum to improve on \( O(x^{1/4}) \) toward \( O(x^{1/4 + \varepsilon}) \).

6.5 Connection to Dirichlet Series

\[ \sum_{n \leq x} d(n) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \zeta(s)^2 \frac{x^s}{s}\,ds. \]\[ \zeta(s)^2 = \frac{1}{(s-1)^2} + \frac{2\gamma}{s-1} + O(1), \]\[ \mathrm{Res}_{s=1}\left(\zeta(s)^2 \frac{x^s}{s}\right) = \frac{d}{ds}\left[(s-1)^2 \zeta(s)^2 \frac{x^s}{s}\right]_{s=1} = x\log x + (2\gamma - 1)x. \]

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Chapter 7: Quadratic Residues and Primitive Roots

7.1 The Legendre Symbol

Elements \( a \not\equiv 0 \pmod{p} \) with \( \left(\frac{a}{p}\right) = 1 \) are quadratic residues mod \( p \); those with \( \left(\frac{a}{p}\right) = -1 \) are quadratic non-residues. There are exactly \( (p-1)/2 \) of each in \( (\mathbb{Z}/p\mathbb{Z})^* \).

The Legendre symbol is essentially a Dirichlet character mod \(p\): it is completely multiplicative (by Corollary 7.3 below), periodic mod \(p\), and takes values in \(\{0, \pm 1\}\). It is the unique nontrivial real character mod \(p\).

7.2 Euler’s Criterion and Basic Properties

7.3 Gauss’s Lemma

7.4 The Law of Quadratic Reciprocity

Quadratic reciprocity has over 200 proofs. One elegant proof uses the Gauss sum \( \tau(\chi) = \sum_{a=0}^{p-1} \left(\frac{a}{p}\right) e^{2\pi i a/p} \), for which \( \tau(\chi)^2 = (-1)^{(p-1)/2} p \).

7.5 Structure of (Z/nZ)*

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Chapter 8: Dirichlet Characters and L-Functions

Characters are the key tool for studying primes in arithmetic progressions. The idea is beautifully simple: to study primes in the residue class \(\ell \pmod k\), we use the fact that we can detect the condition \(n \equiv \ell \pmod k\) by averaging over characters — exactly as Fourier analysis detects the frequency content of a signal by averaging against exponentials. Characters are the “harmonics” of the group \((\mathbb{Z}/k\mathbb{Z})^*\).

8.1 Characters of Finite Abelian Groups

8.2 Dirichlet Characters

A Dirichlet character mod \( k \) is essentially a character of \( (\mathbb{Z}/k\mathbb{Z})^* \), extended by zero. There are exactly \( \varphi(k) \) distinct Dirichlet characters mod \( k \).

The principal character \( \chi_0 \) mod \( k \): \( \chi_0(n) = 1 \) if \( \gcd(n,k)=1 \), \( 0 \) otherwise.

This orthogonality is the key that allows us to detect primes in specific residue classes.

8.3 Primitive Characters and the Conductor

Every character is induced from a unique primitive character; the modulus of that primitive character is the conductor of \( \chi \).

8.4 Gauss Sums

Gauss sums are essential for the functional equation of \( L(s,\chi) \) and also appear in the evaluation of Gauss’s proof of quadratic reciprocity.

8.5 Dirichlet L-Functions

\[ L(s,\chi_0) = \zeta(s) \prod_{p \mid k} (1 - p^{-s}), \]

so \( L(s,\chi_0) \) has a simple pole at \( s=1 \) with residue \( \varphi(k)/k \).

For non-principal \( \chi \), since \( \sum_{n=1}^k \chi(n) = 0 \) (orthogonality), Abel summation shows \( L(s,\chi) \) converges for \( \mathrm{Re}(s) > 0 \) and is entire.

8.6 Analytic Continuation and Functional Equation of L(s,χ)

8.7 Non-Vanishing of L(1,χ)

This is the key analytic step for Dirichlet’s theorem on primes in arithmetic progressions.

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Chapter 9: Primes in Arithmetic Progressions

9.1 Dirichlet’s Theorem

The proof combines orthogonality of characters (to detect the condition \( p \equiv \ell \pmod k \)) with non-vanishing of L-functions.

9.2 Detecting Congruences with Characters

\[ \mathbf{1}[n \equiv \ell \pmod k] = \frac{1}{\varphi(k)} \sum_{\chi \bmod k} \chi(\ell)^{-1} \chi(n), \quad \gcd(n,k)=1. \]\[ \sum_{\substack{n \leq x \\ n \equiv \ell \pmod k}} f(n) = \frac{1}{\varphi(k)} \sum_\chi \overline{\chi(\ell)} \sum_{n \leq x} f(n)\chi(n). \]\[ \sum_{\substack{n \leq x \\ n \equiv \ell \pmod k}} \frac{\Lambda(n)}{n} = \frac{1}{\varphi(k)}\sum_\chi \overline{\chi(\ell)} \sum_{n \leq x} \frac{\Lambda(n)\chi(n)}{n}. \]

9.3 Analysis of L-Function Logarithmic Derivatives

\[ \frac{L'(s,\chi)}{L(s,\chi)} = -\sum_n \frac{\Lambda(n)\chi(n)}{n^s}. \]\[ -\frac{L'(s,\chi_0)}{L(s,\chi_0)} \sim \frac{1}{s-1} \]

near \( s=1 \). Non-principal characters have \( L(1,\chi) \neq 0 \), so \( -L'(s,\chi)/L(s,\chi) \) is holomorphic at \( s=1 \) and bounded.

\[ \sum_{n \leq x} \frac{\Lambda(n)\chi_0(n)}{n} = \log x + O(1), \quad \sum_{n \leq x} \frac{\Lambda(n)\chi(n)}{n} = O(1) \text{ for } \chi \neq \chi_0. \]\[ \sum_{\substack{p \leq x \\ p \equiv \ell \pmod k}} \frac{\log p}{p} = \frac{1}{\varphi(k)}\log x + O(1). \]

9.4 The Proof of Dirichlet’s Theorem

9.5 Density of Primes in Arithmetic Progressions

9.6 Siegel Zeros and Effective Bounds

The ineffectivity stems from an argument that either Siegel zeros don’t exist (giving a good bound via an explicit constant), or if two different real characters have zeros very close to 1, they “repel” each other (Deuring–Heilbronn phenomenon), also giving a good bound. But extracting an explicit constant from this argument requires knowing which case holds.

9.7 The Generalized Riemann Hypothesis

Unconditionally, the best known error term is \( O(x \exp(-c\sqrt{\log x})) \) uniformly for \( k \leq (\log x)^A \), and the Bombieri–Vinogradov theorem (an average form of GRH) gives a near-optimal result on average.

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Chapter 10: Additional Topics

10.1 Dirichlet Series: Deeper Convergence Theory

This theorem (sometimes called the uniqueness theorem for Dirichlet series) also implies: if a Dirichlet series converges for \( \mathrm{Re}(s) > \sigma_0 \) and equals zero there, then all coefficients are zero.

10.2 Perron’s Formula

Perron’s formula converts partial sums of Dirichlet series into contour integrals. It is the fundamental tool connecting the analytic properties of a Dirichlet series (poles, zeros, growth) to the arithmetic properties of its coefficients (partial sums).

Perron’s formula is the bridge between analytic properties of \( F(s) \) (poles, zeros) and arithmetic asymptotics (partial sums of \( a(n) \)).

10.3 The Selberg Sieve

The Selberg sieve is a powerful combinatorial method for upper bounds on primes and related quantities.

10.4 Linnik’s Theorem and the Least Prime in an AP

The best known value is \( L \leq 5 \) (unconditional); under GRH, \( L = 2 \) suffices.

10.5 Zero-Free Regions for L-Functions and PNT in AP

The prime number theorem in arithmetic progressions (PNTAP) takes the form:

The Bombieri–Vinogradov theorem gives a stronger average statement:

This says GRH holds “on average” for \( k \leq x^{1/2-\varepsilon} \).

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Chapter 11: The Explicit Formula and its Consequences

11.1 Counting Zeros: The Riemann–von Mangoldt Formula

Before using the explicit formula, we need to know how many zeros there are. Let \(N(T)\) denote the number of nontrivial zeros \(\rho = \beta + i\gamma\) of \(\zeta(s)\) with \(0 < \beta < 1\) and \(0 < \gamma \leq T\).

This formula tells us that zeros have average density \(\sim \log T/(2\pi)\) at height \(T\), growing logarithmically. The sum \(\sum_\rho x^\rho/\rho\) in the explicit formula therefore converges (conditionally) since \(\sum_{|\gamma| \leq T} 1/|\rho| \sim (\log T)^2 / (2\pi)\).

11.2 The Explicit Formula in Detail

11.3 PNT and the Explicit Formula

From the explicit formula, PNT is equivalent to showing that \(\sum_{|\gamma| \leq T} x^\rho/\rho = o(x)\) as \(x \to \infty\) (for an appropriate \(T = T(x)\)). This requires two things:

1. No zeros on \(\mathrm{Re}(s) = 1\): otherwise a zero \(\rho = 1+i\gamma\) would contribute \(x^1/(1+i\gamma)\) to the sum, giving a term of order \(x\).
2. Zero-free region to the left of \(\mathrm{Re}(s)=1\): zeros with \(\mathrm{Re}(\rho) = 1 - c/\log T\) contribute \(x^{1-c/\log T}/|\rho|\), and summing over \(O(N(T))\) such zeros gives a total of \(O(xe^{-c\sqrt{\log x}})\) with the optimal choice \(T = e^{\sqrt{\log x}}\).

This is the quantitative PNT with error term \(O(xe^{-c\sqrt{\log x}})\).

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Chapter 12: The Gamma Function and Special Values

12.1 The Gamma Function Revisited

The Gamma function \(\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt\) (for \(\mathrm{Re}(s) > 0\)) is the unique meromorphic continuation of the factorial, and its role in the functional equation of \(\zeta(s)\) is fundamental.

\[ \psi(s) = -\gamma - \frac{1}{s} + \sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+s}\right). \]

12.2 Special Values of ζ(s)

12.3 The Connection to Bernoulli Numbers

The Bernoulli numbers \(B_k\) defined by \(\sum_k B_k t^k/k! = t/(e^t-1)\) appear throughout number theory:

• In the Euler–Maclaurin formula as correction terms.
• In the values of \(\zeta\) at negative integers: \(\zeta(-n) = -B_{n+1}/(n+1)\).
• In the theory of cyclotomic fields (Kummer’s criterion for regular primes).
• In the formula for \(\zeta(2k) = (-1)^{k+1}(2\pi)^{2k}B_{2k}/(2(2k)!)\).

The von Staudt–Clausen theorem characterizes the denominators of Bernoulli numbers: \(B_{2k} + \sum_{(p-1)\mid 2k} 1/p \in \mathbb{Z}\), where the sum is over primes \(p\) such that \((p-1) \mid 2k\).

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Appendix A: Complex Analysis Background

A.1 Holomorphic and Meromorphic Functions

A function \( f : U \to \mathbb{C} \) (where \( U \subseteq \mathbb{C} \) is open) is holomorphic if it is complex differentiable at every point of \( U \). Holomorphic functions are infinitely differentiable and equal their Taylor series locally. A meromorphic function is holomorphic except at isolated poles.

Key facts used in the course:

• A holomorphic function determined on a sequence with an accumulation point is zero everywhere (identity theorem).
• If \( f \) is nonzero on a connected open set, then \( \log f \) can be defined locally and \( f'/f \) is holomorphic.
• The Argument Principle: \( \frac{1}{2\pi i} \oint \frac{f'(s)}{f(s)}\,ds \) counts zeros minus poles (with multiplicity) of \( f \) inside the contour.
• Phragmén–Lindelöf principle: If \(f\) is holomorphic on a strip and bounded on the boundary, then the bound holds throughout the strip (under mild growth conditions). This is used to bound \(\zeta(s)\) in the critical strip.

A.2 The Gamma Function

A.3 Cauchy’s Integral Formula and Residues

The residue of \( f \) at a simple pole \( z_0 \) is \( \lim_{z \to z_0}(z-z_0)f(z) \). At a double pole: \( \mathrm{Res}(f,z_0) = \lim_{z \to z_0} \frac{d}{dz}[(z-z_0)^2 f(z)] \).

A.4 The Phragmén–Lindelöf Principle

This is used to show that if \(\zeta(s)\) is bounded on the boundaries of the critical strip (\(\sigma = 0\) and \(\sigma = 1\)), it is bounded throughout. Together with the convexity of \(\log|\zeta(\sigma+it)|\) as a function of \(\sigma\), this gives the “convexity bound” \(|\zeta(1/2+it)| = O(t^{1/4+\varepsilon})\) — the Lindelöf Hypothesis states \(\varepsilon = 0\), which would follow from RH.

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Appendix B: Summary of Key Formulas

B.1 The Fundamental Identities

Identity	Formula
Divisor sum	\( \sum_{d \mid n} \mu(d) = [n=1] \)
Log identity	\( \log n = \sum_{d \mid n} \Lambda(d) \)
Totient sum	\( n = \sum_{d \mid n} \varphi(d) \)
Euler product	\( \zeta(s) = \prod_p (1-p^{-s})^{-1} \)
Log derivative	\( -\zeta'(s)/\zeta(s) = \sum_n \Lambda(n)/n^s \)

B.2 Asymptotic Formulas

Sum	Asymptotic
\( \sum_{n \leq x} 1/n \)	\( \log x + \gamma + O(1/x) \)
\( \sum_{n \leq x} d(n) \)	\( x\log x + (2\gamma-1)x + O(\sqrt{x}) \)
\( \sum_{p \leq x} \log p / p \)	\( \log x + O(1) \)
\( \sum_{p \leq x} 1/p \)	\( \log\log x + M + O(1/\log x) \)
\( \pi(x) \)	\( x/\log x \) (PNT)
\( \psi(x) \)	\( x \) (PNT)
\( R(x) \) (circle)	\( \pi x + O(\sqrt{x}) \)

B.3 Dirichlet Series Dictionary

Function \( f(n) \)	Dirichlet series \( \sum f(n)/n^s \)	Euler product
\( 1 \)	\( \zeta(s) \)	\( \prod_p (1-p^{-s})^{-1} \)
\( \mu(n) \)	\( 1/\zeta(s) \)	\( \prod_p (1-p^{-s}) \)
\( \Lambda(n) \)	\( -\zeta'(s)/\zeta(s) \)	—
\( d(n) \)	\( \zeta(s)^2 \)	\( \prod_p (1-p^{-s})^{-2} \)
\( \varphi(n) \)	\( \zeta(s-1)/\zeta(s) \)	—
\( \chi(n) \)	\( L(s,\chi) \)	\( \prod_p (1-\chi(p)p^{-s})^{-1} \)
\( \lambda(n) \)	\( \zeta(2s)/\zeta(s) \)	\( \prod_p (1+p^{-s})^{-1} \)

B.4 L-Function Properties Summary

Property	\( \zeta(s) \)	\( L(s,\chi) \), \( \chi \neq \chi_0 \)
Convergence	\( \mathrm{Re}(s) > 1 \) absolute; \( \mathrm{Re}(s) > 0 \) conditional	\( \mathrm{Re}(s) > 0 \) conditional
Pole	Simple at \( s=1 \), residue 1	Entire
Zeros \( \mathrm{Re}(s) = 1 \)	None	None (same proof)
Trivial zeros	\( s = -2,-4,-6,\ldots \)	Related to \( \chi(-1) \)
Functional eq.	\( \xi(s) = \xi(1-s) \)	\( \Lambda(s,\chi) = \frac{\tau(\chi)}{i^a\sqrt{k}}\Lambda(1-s,\bar\chi) \)

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Appendix C: Worked Examples and Exercises

C.1 Worked Example: Evaluating ζ(2)

C.2 Worked Example: Abel Summation for log(n!)

C.3 Worked Example: Non-vanishing of L(1,χ) for χ a quadratic character

C.4 Worked Example: The Characters mod 8

C.5 Exercises

Exercise C.1. Prove using Abel summation that \( \sum_{n \leq x} \frac{\log n}{n} = \frac{(\log x)^2}{2} + C + O(\log x / x) \) for an explicit constant \( C \).

Exercise C.2. Show that the series \( L(1, \chi) = \sum_{n=1}^\infty \chi(n)/n \) converges for any non-principal character \( \chi \) mod \( k \), by bounding the partial sums \( |\sum_{n \leq N} \chi(n)| \leq \varphi(k) \).

Exercise C.3. Prove that \( \sum_{d \mid n} \mu(d)/d = \varphi(n)/n \) using multiplicativity and the formula at prime powers.

Exercise C.4. Use the functional equation to show \( \zeta(0) = -1/2 \) and \( \zeta(-2k) = 0 \) for \( k \geq 1 \).

Exercise C.5. Prove the inequality \( |\zeta(\sigma)|^3 |\zeta(\sigma+it)|^4 |\zeta(\sigma+2it)| \geq 1 \) for \( \sigma > 1 \), \( t \in \mathbb{R} \), directly from the Euler product representation.

Exercise C.6. Let \( r_4(n) \) denote the number of representations of \( n \) as a sum of 4 squares. Using the identity \( \left(\sum_{n} r(n) q^n\right)^2 = \sum_n r_4(n)q^n \) and Dirichlet series techniques, show \( r_4(n) = 8\sum_{4 \nmid d \mid n} d \).

Exercise C.7. Prove Mertens’ theorem: \( \prod_{p \leq x}(1 - 1/p) \sim e^{-\gamma}/\log x \), using the asymptotic \( \sum_{p \leq x} 1/p = \log\log x + M + o(1) \) and properties of the exponential.

Exercise C.8. Define \( \Lambda_k(n) \) by \( (-1)^k \frac{d^k}{ds^k}\log\zeta(s) = \sum_n \Lambda_k(n)/n^s \). Express \( \Lambda_2(n) \) explicitly in terms of \( \Lambda \) and show that \( \sum_{n \leq x} \Lambda_2(n)/n \sim 2\log x \).

Exercise C.9. Let \(\chi_1\) be the non-principal character mod 4. Compute \(L(1,\chi_1) = 1 - 1/3 + 1/5 - 1/7 + \cdots = \pi/4\) by evaluating the Fourier series of \(f(x) = 1\) on \([0,\pi]\) at \(x = \pi/2\).

Exercise C.10. Using the Perron formula heuristic, explain why the main term in the asymptotic \(\sum_{n \leq x} d(n) \sim x\log x\) arises from the double pole of \(\zeta(s)^2\) at \(s=1\). Compute the residue at the double pole explicitly.

Exercise C.11. Show that if RH is true, then \(\pi(x) = \mathrm{li}(x) + O(\sqrt{x}\log x)\). What does the explicit formula tell you about the contribution of the zeros to the error \(\pi(x) - \mathrm{li}(x)\)?

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Appendix D: Historical Notes

D.1 Riemann’s 1859 Paper

Bernhard Riemann’s eight-page memoir “Über die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the Number of Primes Less Than a Given Magnitude, 1859) introduced nearly all the tools used in modern analytic number theory: the Euler product for \( \zeta(s) \) extended to complex \( s \), the analytic continuation, the functional equation, the explicit formula for \( \pi(x) \) in terms of zeros, and the Riemann Hypothesis. The prime number theorem was proved by Hadamard and de la Vallée Poussin (independently) in 1896, using the non-vanishing on \( \mathrm{Re}(s)=1 \).

What makes Riemann’s paper extraordinary is its density of new ideas. In 8 pages, Riemann:

• Extended \(\zeta(s)\) to a meromorphic function on all of \(\mathbb{C}\) (using both the integral formula and the theta function approach).
• Derived the functional equation \(\xi(s) = \xi(1-s)\).
• Found the product formula for \(\xi\) in terms of its zeros.
• Stated (without proof) the asymptotic \(\pi(x) \sim \mathrm{li}(x)\).
• Gave the explicit formula relating \(\psi(x)\) to the zeros.
• Computed the first few zeros numerically and conjectured they all lie on \(\mathrm{Re}(s)=1/2\).

Most of the claims in the paper were not proved rigorously until much later. The explicit formula was proved by von Mangoldt. The asymptotic for \(N(T)\) was proved by Riemann–von Mangoldt. The PNT was proved by Hadamard and de la Vallée Poussin. The RH remains open.

D.2 Dirichlet’s 1837 Paper

Peter Gustav Lejeune Dirichlet’s 1837 paper “Über die Primzahlen in arithmetischen Progressionen” (On Primes in Arithmetic Progressions) introduced Dirichlet characters, L-functions, and proved that every arithmetic progression \( a, a+d, a+2d, \ldots \) with \( \gcd(a,d)=1 \) contains infinitely many primes. This was the first systematic use of analysis to solve a purely number-theoretic problem.

The paper was motivated by Gauss’s empirical observations about the distribution of primes in residue classes. Gauss had verified the equidistribution of primes numerically but could not prove it. Dirichlet’s invention of characters and L-functions to solve this problem is considered one of the founding acts of analytic number theory.

D.3 Chebyshev’s Contributions (1848–1852)

Pafnuty Chebyshev proved that \( \pi(x)/(x/\log x) \) has lim inf at most \( 1 \) and lim sup at least \( 1 \) (getting within a factor of \( 6\log 2 \approx 4.16 \) of the correct constant), and proved Bertrand’s postulate. His introduction of \( \psi(x) \) and \( \theta(x) \) and the reduction of PNT to \( \psi(x)/x \to 1 \) remain fundamental.

D.4 Mertens (1874) and the Constants

Franz Mertens proved the three Mertens theorems: \( \sum_{p \leq x}(\log p)/p = \log x + O(1) \), \( \sum_{p \leq x} 1/p = \log\log x + M + O(1/\log x) \), and \( \prod_{p \leq x}(1-1/p) = (e^{-\gamma}/\log x)(1+O(1/\log x)) \). His constant \( M \approx 0.2615 \) (the Meissel–Mertens constant) was the first successful asymptotic for the sum of reciprocals of primes.

D.5 Hardy, Ramanujan, and Circle Method

G. H. Hardy and S. Ramanujan (1917) pioneered the circle method (Hardy–Littlewood method), which, combined with exponential sums, led to Vinogradov’s proof that every sufficiently large odd integer is a sum of three primes (1937) and other breakthroughs. Hardy’s collaboration with Littlewood is also responsible for conjectures about the distribution of prime constellations (the Hardy–Littlewood conjecture).

D.6 Selberg and Erdős: The Elementary Proof of PNT (1949)

\[ \sum_{p \leq x} \log^2 p + \sum_{p q \leq x} \log p \log q = 2x\log x + O(x), \]

where the second sum is over products of two (not necessarily distinct) primes. From this, one can derive \(\psi(x) \sim x\) using elementary inequalities. The elementary proof is in some ways more difficult to follow than the analytic proof, despite avoiding complex analysis.

D.7 Recent Developments

The Riemann Hypothesis remains open after over 160 years. Major recent developments include:

• Green–Tao theorem (2004): The primes contain arbitrarily long arithmetic progressions, proved using Fourier analysis and ergodic-theoretic methods.
• Zhang’s bounded prime gaps (2013): There are infinitely many primes \(p\) with \(p_{n+1}-p_n \leq 70{,}000{,}000\), improved to 246 by the Polymath project.
• Maynard (2015): Infinitely many intervals \([n, n+C_k]\) containing at least \(k\) primes, for any \(k\).
• Helfgott (2013): Every odd integer \(>5\) is a sum of three primes (Vinogradov’s theorem made effective).

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Appendix E: Notation Index

Symbol	Meaning
\( \pi(x) \)	Number of primes \( \leq x \)
\( \theta(x) \)	First Chebyshev function: \( \sum_{p \leq x}\log p \)
\( \psi(x) \)	Second Chebyshev function: \( \sum_{p^k \leq x}\log p \)
\( \Lambda(n) \)	Von Mangoldt function
\( \mu(n) \)	Möbius function
\( \varphi(n) \)	Euler totient function
\( d(n) \) or \( \tau(n) \)	Number of divisors of \( n \)
\( \sigma_k(n) \)	Sum of \( k \)-th powers of divisors
\( \gamma \)	Euler–Mascheroni constant \( \approx 0.5772 \)
\( \mathrm{li}(x) \)	Logarithmic integral \( \int_2^x dt/\log t \)
\( \zeta(s) \)	Riemann zeta function
\( \chi \)	Dirichlet character
\( \chi_0 \)	Principal Dirichlet character
\( L(s,\chi) \)	Dirichlet L-function
\( \tau(\chi) \)	Gauss sum
\( \varphi(k) \)	Euler totient of modulus \( k \)
\( f * g \)	Dirichlet convolution of \( f \) and \( g \)
\( \Gamma(s) \)	Gamma function
\( \xi(s) \)	Completed zeta function: \( \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s) \)
\( M(x) \)	Mertens function: \( \sum_{n \leq x}\mu(n) \)
\( \pi(x;k,\ell) \)	Number of primes \( \leq x \) congruent to \( \ell \pmod k \)
\( \{x\} \)	Fractional part of \( x \)
\( \lfloor x \rfloor \)	Floor function: largest integer \( \leq x \)
\( [P] \)	Iverson bracket: 1 if \( P \) is true, 0 otherwise
\( N(T) \)	Number of nontrivial zeros \(\rho\) of \(\zeta\) with \(0 < \mathrm{Im}(\rho) \leq T\)
\( B_k \)	Bernoulli numbers
\( \lambda(n) \)	Liouville function \((-1)^{\Omega(n)}\)
\( \Omega(n) \)	Total number of prime factors of \(n\) with multiplicity
\( \omega(n) \)	Number of distinct prime factors of \(n\)
\( \sigma_a, \sigma_c \)	Abscissae of absolute and conditional convergence

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Chapter 13: Deep Theory of the Riemann Zeta Function

The material in this chapter goes beyond what is strictly needed for the proof of PNT, but it is the heart of the subject — the structures that every research mathematician working in analytic number theory must understand thoroughly. We give complete derivations of the functional equation via the theta function, the Hadamard product, the explicit formula, the trivial zeros, the first nontrivial zeros, and the zero-free region with its quantitative consequence.

13.1 The Theta Function and Functional Equation: Complete Derivation

The most elegant approach to the functional equation of \(\zeta(s)\) uses the Jacobi theta function and the Poisson summation formula. We present it in full detail.

\[ \theta(t) = \sum_{n=-\infty}^{\infty} e^{-\pi n^2 t}, \qquad t > 0. \]

This is the Jacobi theta function already introduced in Section 2.3. Note \(\theta(t) = 1 + 2\sum_{n=1}^{\infty}e^{-\pi n^2 t} = 1 + 2\omega(t)\) where \(\omega(t) = \sum_{n=1}^\infty e^{-\pi n^2 t}\).

The Mellin transform computation. We now compute the Mellin transform of \(\omega(t) = (\theta(t)-1)/2\) in two ways to derive the functional equation of \(\zeta(s)\).

\[ \pi^{-s/2}\Gamma(s/2)\zeta(s) = \int_0^\infty t^{s/2-1}\omega(t)\,dt. \]\[ \Lambda(s) = \int_1^\infty t^{s/2-1}\omega(t)\,dt + \int_0^1 t^{s/2-1}\omega(t)\,dt. \]\[ \int_0^1 t^{s/2-1}\omega(t)\,dt = \int_1^\infty t^{-(s/2-1)}t^{-2}\omega(1/t)\,dt = \int_1^\infty t^{-s/2-1}\omega(1/t)\,dt. \]\[ \omega(1/t) = \frac{t^{1/2}(1+2\omega(t))-1}{2} = \frac{t^{1/2}-1}{2} + t^{1/2}\omega(t). \]\[ \int_0^1 t^{s/2-1}\omega(t)\,dt = \int_1^\infty t^{-s/2-1}\left(\frac{t^{1/2}-1}{2} + t^{1/2}\omega(t)\right)dt. \]\[ = \int_1^\infty \frac{t^{-s/2-1/2}-t^{-s/2-1}}{2}\,dt + \int_1^\infty t^{-s/2-1/2}\omega(t)\,dt. \]\[ \int_1^\infty \frac{t^{-s/2-1/2} - t^{-s/2-1}}{2}\,dt = \frac{1}{2}\left(\frac{1}{s/2-1/2} - \frac{1}{s/2}\right) = \frac{1}{s-1} - \frac{1}{s} = \frac{1}{s(s-1)}. \]\[ \Lambda(s) = \frac{1}{s(s-1)} + \int_1^\infty \left(t^{s/2-1} + t^{(1-s)/2-1}\right)\omega(t)\,dt. \]

This expression is symmetric under \(s \mapsto 1-s\): swapping \(s\) and \(1-s\) leaves the integral unchanged (since \(s/2-1 \leftrightarrow (1-s)/2-1 = -s/2-1/2\) but the pair \(\{t^{s/2-1}, t^{(1-s)/2-1}\}\) is exchanged), and \(1/(s(s-1)) = 1/((1-s)(1-(1-s)))\). Hence \(\Lambda(s) = \Lambda(1-s)\). \(\square\)

13.2 Table of the First Nontrivial Zeros

The nontrivial zeros \(\rho = \beta + i\gamma\) of \(\zeta(s)\) lie in the critical strip \(0 < \beta < 1\) and come in conjugate pairs: if \(\rho\) is a zero then so are \(\bar\rho\), \(1-\rho\), and \(1-\bar\rho\). The Riemann Hypothesis asserts \(\beta = 1/2\) for all nontrivial zeros.

The first ten nontrivial zeros (listed by positive imaginary part, using the notation \(\rho_n = 1/2 + i\gamma_n\) which would hold under RH) are:

\(n\)	\(\gamma_n\) (imaginary part)	Verified real part
1	14.134725141734693…	\(1/2\) (numerical)
2	21.022039638771554…	\(1/2\)
3	25.010857580145688…	\(1/2\)
4	30.424876125859513…	\(1/2\)
5	32.935061587739189…	\(1/2\)
6	37.586176070968646…	\(1/2\)
7	40.918719012147495…	\(1/2\)
8	43.327073280914999…	\(1/2\)
9	48.005150881167159…	\(1/2\)
10	49.773832477672302…	\(1/2\)

All of these have been verified to have real part exactly \(1/2\) to many decimal places of precision. As of 2025, over \(10^{13}\) zeros have been verified on the critical line, all with real part \(1/2\). No zero off the critical line has ever been found.

13.3 The Hadamard Product for ζ(s)

Just as a polynomial of degree \(n\) can be written as a product of linear factors \(\prod_j (s - s_j)\) over its roots \(s_j\), a general entire function of finite order can be expressed as an infinite product over its zeros. For \(\zeta(s)\), the relevant object is the entire function \(\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\).

13.4 The Explicit Formula: Full Statement and Proof Sketch

The explicit formula is the precise quantitative version of the statement “the zeros of \(\zeta\) control the primes.” It is one of Riemann’s greatest contributions, made rigorous by von Mangoldt.

\[ \psi(x) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\left(-\frac{\zeta'(s)}{\zeta(s)}\right)\frac{x^s}{s}\,ds, \qquad c > 1. \]

Using the Hadamard product representation of \(\zeta'/\zeta\), the integrand \(-(\zeta'/\zeta)(s) \cdot x^s/s\) is a meromorphic function whose poles are:

• A simple pole at \(s = 1\) with residue \(x^1/1 = x\) (from the simple pole of \(\zeta\) at \(s=1\));
• Simple poles at each nontrivial zero \(\rho\) with residue \(-x^\rho/\rho\) (from the zeros of \(\zeta\) in the denominator of \(-\zeta'/\zeta\));
• A simple pole at \(s=0\) with residue \(-\zeta'(0)/\zeta(0) = \log(2\pi)\) (since \(\zeta(0) = -1/2\) and \(\zeta'(0)/\zeta(0) = -\log(2\pi)\));
• Simple poles at the trivial zeros \(s = -2,-4,-6,\ldots\), each contributing \(x^{-2k}/(-2k)\), whose sum telescopes to \(-\frac{1}{2}\log(1-x^{-2})\).

Shifting the contour to the left and collecting residues gives the explicit formula. The crucial analytic issue — proving the contour shift is valid, i.e., that the contributions from the horizontal segments vanish — requires the estimate \(|\zeta'/\zeta(\sigma+iT)| = O(\log^2 T)\) uniformly in \(\sigma \in [-1, 2]\) (away from zeros), which follows from the zero-density estimates.

\[ -\frac{x^\rho}{\rho} = -\frac{x^{1/2}e^{i\gamma\log x}}{1/2 + i\gamma} = -\frac{x^{1/2}}{|\rho|}e^{i(\gamma\log x - \arg\rho)}. \]\[ -\frac{x^\rho}{\rho} - \frac{x^{\bar\rho}}{\bar\rho} = -\frac{2x^{1/2}}{|\rho|}\cos(\gamma\log x - \arg\rho). \]

This is a pure oscillation with amplitude \(2x^{1/2}/|\rho|\) and period \(2\pi/\gamma\) on the \(\log x\) scale. Since \(\sum_\rho 1/|\rho|^2 < \infty\) but \(\sum_\rho 1/|\rho| = \infty\) (the zeros are too numerous), the sum converges only conditionally — but it converges in the symmetric sense and represents \(\psi(x) - x\) to high precision.

Numerical verification. For \(x = 100\): \(\psi(100) = \sum_{p^k \leq 100}\log p = \log 2\cdot 6 + \log 3\cdot 4 + \log 5 \cdot 2 + \log 7^2 + \log 11 + \cdots \approx 94.0\). The main term is \(x = 100\). The correction from the first zero \(\rho_1 = 1/2 + 14.134i\) is approximately \(-2\cdot 100^{1/2}/|\rho_1|\cdot\cos(14.134\log 100 - \arg\rho_1) \approx -2\cdot 10/14.14 \cdot \cos(64.9^\circ) \approx -6\), bringing the estimate to \(\approx 94\), consistent with the actual value.

13.5 Zero-Free Regions: The Classical Result and Proof

We now establish the classical zero-free region due to de la Vallée Poussin (1899) in full detail.

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