Analytic Number Theory: Foundations¶

Job: set up Dirichlet series and $\zeta(s)$ as the basic analytic machine converting multiplicative arithmetic into meromorphic data (Euler products, zeros, growth).

Core¶

Dirichlet series¶

A Dirichlet series is $D(s)=\sum_{n=1}^\infty a_n n^{-s}$, $s=\sigma+it$. Absolute convergence in a half-plane $\sigma>\sigma_a$; conditional convergence possible for $\sigma>\sigma_c\le\sigma_a$. Uniqueness: $D\equiv 0$ on a half-plane $\Rightarrow$ all $a_n=0$.

Multiplicative $a_n$: if $a$ is completely multiplicative and $D$ converges absolutely for $\sigma>\sigma_a$, $$ D(s)=\prod_p\Bigl(1-a_p p^{-s}\Bigr)^{-1} $$ (Euler product). Partial summation links partial sums $A(x)=\sum_{n\le x}a_n$ to $D$ (and conversely via Perron).

Riemann zeta function¶

For $\sigma>1$, $$ \zeta(s)=\sum_{n\ge 1}n^{-s}=\prod_p(1-p^{-s})^{-1}. $$ Euler product $\Rightarrow$ $\zeta(s)\neq 0$ for $\sigma>1$. Harmonic series $\Rightarrow$ simple pole at $s=1$ with residue $1$.

Analytic continuation (sketch). For $\sigma>0$, $$ \zeta(s)=\frac{1}{s-1}+\sum_{n\ge 1}\int_n^{n+1}(n^{-s}-x^{-s})\,dx $$ or via $\xi$-function / $\theta$-functional equation. $\zeta$ extends meromorphically to $\mathbb{C}$ with only pole $s=1$.

Functional equation. With $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$, $$ \xi(s)=\xi(1-s). $$ Equivalently $\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is invariant under $s\mapsto 1-s$ up to the polynomial factor absorbed in $\xi$. Trivial zeros: $\zeta(-2k)=0$ for $k\in\mathbb{N}$ (from poles of $\Gamma$). Critical strip: $0<\sigma<1$; critical line: $\sigma=1/2$.

Prime number theorem¶

Chebyshev: $\theta(x)=\sum_{p\le x}\log p$, $\psi(x)=\sum_{n\le x}\Lambda(n)$ ($\Lambda=$ von Mangoldt). PNT equivalents: $$ \pi(x)\sim\frac{x}{\log x},\qquad \theta(x)\sim x,\qquad \psi(x)\sim x. $$

Theorem (PNT, Hadamard–de la Vallée Poussin, 1896). $\psi(x)\sim x$.

Mechanism. $-\zeta'/\zeta(s)=\sum\Lambda(n)n^{-s}$ ($\sigma>1$). Nonvanishing $\zeta(1+it)\neq 0$ ($t\neq 0$) + contour integration / tauberian theorems $\Rightarrow$ $\psi(x)\sim x$.

Zero-free regions (sketch)¶

Classical (de la Vallée Poussin type). There exists $c>0$ such that $$ \zeta(\sigma+it)\neq 0\quad\text{for}\quad \sigma\ge 1-\frac{c}{\log(|t|+2)}. $$ Width $\asymp 1/\log|t|$ near $\sigma=1$ yields $$ \psi(x)=x+O\bigl(x\exp(-c'\sqrt{\log x})\bigr) $$ (or better with Vinogradov–Korobov zero-free region). Full RH $\zeta(\rho)=0\Rightarrow\operatorname{Re}\rho=1/2$ would give $\psi(x)=x+O(x^{1/2}\log^2 x)$.

Proof idea of zero-free near $\sigma=1$: $3+4\cos\theta+\cos 2\theta\ge 0$ $\Rightarrow$ $$ \zeta(\sigma)^3\bigl|\zeta(\sigma+it)\bigr|^4\bigl|\zeta(\sigma+2it)\bigr|\ge 1 $$ for $\sigma>1$; a zero at $1+it$ would force a pole-like blow-up contradiction as $\sigma\downarrow 1$.

$L$-functions (pointer)¶

Dirichlet $L$-function for character $\chi\pmod q$: $$ L(s,\chi)=\sum_n\chi(n)n^{-s}=\prod_p(1-\chi(p)p^{-s})^{-1}\quad(\sigma>1). $$ Non-principal $\chi$: entire after continuation; principal $\chi_0$: pole from $\zeta$. Dirichlet’s theorem on primes in arithmetic progressions uses $L(1,\chi)\neq 0$. General automorphic / Hecke / Artin $L$: same package — Euler product, analytic continuation, functional equation, zeros control primes in thicker sets. See L-functions.

See it¶

Mertens and Euler product at $s=1$. $\sum_{p\le x}p^{-1}=\log\log x+B+o(1)$. The divergence of $\sum p^{-1}$ is the “real” face of the pole of $\zeta$ at $s=1$; the Euler product cannot be continued through $s=1$ as a holomorphic nonzero function.

Explicit formula (one line). Under RH-friendly growth, $$ \psi(x)=x-\sum_{|\operatorname{Im}\rho|\le T}\frac{x^\rho}{\rho}+O\Bigl(\frac{x\log^2(xT)}{T}+\log x\Bigr); $$ oscillations in the prime-counting error are dual to ordinates of zeros.

Check¶

Show that $\zeta(s)\neq 0$ for $\sigma>1$ from the Euler product, and that $\zeta$ has a pole at $s=1$.

For $\sigma>1$ each factor $(1-p^{-s})^{-1}\neq 0$ and the product converges absolutely and nonvanishing. As $s\downarrow 1^+$, $\zeta(s)\ge\sum_{n\le N}n^{-s}\to\sum_{n\le N}n^{-1}$ for any $N$, so $\zeta(s)\to+\infty$; combined with the Laurent expansion $\zeta(s)=1/(s-1)+\gamma+O(s-1)$, simple pole residue $1$.

Why does a zero-free region $\sigma\ge 1-c/\log(|t|+2)$ give a power-saving-free but subpolynomial error in PNT?

Truncated Perron + horizontal/vertical growth: shifting to $\sigma=1-c'/\log T$ with $T$ optimized against $x$ produces error $x\exp(-c''\sqrt{\log x})$ (classical), not $O(x^{1-\delta})$ for fixed $\delta>0$ (which needs a zero-free strip of fixed width).

Pitfalls¶

$\sigma_c$ vs $\sigma_a$: abscissae of conditional vs absolute convergence need not coincide (e.g. $\sum(-1)^{n}n^{-s}$).
Landau’s theorem: for $a_n\ge 0$, singularity on the real axis at $\sigma=\sigma_c$ — cannot continue past that point along the real line.
RH is about nontrivial zeros in the critical strip; trivial zeros are understood and do not obstruct PNT-quality estimates the same way.
$\psi$ vs $\pi$: elementary comparisons $\psi(x)\sim x\Leftrightarrow\pi(x)\sim x/\log x$; analytic proofs usually target $\psi$.

Refs¶

Davenport, Multiplicative Number Theory
Titchmarsh, The Theory of the Riemann Zeta-Function
Iwaniec–Kowalski, Analytic Number Theory
Edwards, Riemann’s Zeta Function (historical + classical)

Next¶

Analytic NT overview · L-functions · Number theory foundations · Complex analysis · Residue theory · Algebraic NT