L-functions¶

Job: encode arithmetic of characters, Galois representations, and automorphic forms in Euler products with analytic continuation and functional equations; zeros control refined prime laws.

Core¶

Dirichlet $L$-functions¶

For a Dirichlet character $\chi\pmod q$, $$ L(s,\chi)=\sum_{n=1}^\infty\chi(n)n^{-s}=\prod_p\bigl(1-\chi(p)p^{-s}\bigr)^{-1},\qquad\sigma>1. $$ - $\chi$ principal $\chi_0$: $L(s,\chi_0)=\zeta(s)\prod_{p\mid q}(1-p^{-s})$ — same pole at $s=1$ as $\zeta$. - $\chi$ non-principal: $L(s,\chi)$ continues to an entire function; $L(1,\chi)\neq 0$ (Dirichlet) $\Rightarrow$ primes in coprime residue classes $a\pmod q$ have Dirichlet density $1/\varphi(q)$.

Functional equation (primitive $\chi$). Completed $\Lambda(s,\chi)= (q/\pi)^{(s+a)/2}\Gamma\bigl((s+a)/2\bigr)L(s,\chi)$ ($a=0$ or $1$ by parity of $\chi$) satisfies $$ \Lambda(s,\chi)=\frac{\tau(\chi)}{i^a\sqrt{q}}\,\Lambda(1-s,\bar\chi), $$ $\tau(\chi)=$ Gauss sum, $|\tau(\chi)|=\sqrt{q}$ for primitive $\chi$.

Hecke $L$-functions (number fields)¶

Let $K/\mathbb{Q}$ be a number field, $\mathfrak{m}$ a modulus, $\chi$ a Hecke character of the ray class group modulo $\mathfrak{m}$ (idèle-class character of finite order in the simplest case). The Hecke $L$-function $$ L(s,\chi)=\sum_{\mathfrak{a}}\chi(\mathfrak{a})\,N(\mathfrak{a})^{-s}=\prod_{\mathfrak{p}}\bigl(1-\chi(\mathfrak{p})N(\mathfrak{p})^{-s}\bigr)^{-1} $$ (sum over integral ideals coprime to the conductor) continues meromorphically (entire if $\chi$ nontrivial) and satisfies a functional equation relating $s\leftrightarrow 1-s$ with $\Gamma$-factors of degree $n=[K:\mathbb{Q}]$. Dedekind $\zeta_K(s)=L(s,\mathbf{1})$ has a simple pole at $s=1$ with residue related to $h_K R_K$ (class number formula).

Analytic continuation package (general shape)¶

A standard $L$-function (Selberg class / automorphic $L$) is expected to have:

Axiom	Content
Dirichlet series	$L(s)=\sum a_n n^{-s}$, $a_n\ll n^{\varepsilon}$ (Ramanujan on average)
Euler product	local polynomials of fixed degree $d$
Analytic continuation	entire or meromorphic of controlled poles
Functional equation	$\Lambda(s)=\varepsilon \overline{\Lambda}(1-s)$, $
Growth	finite order in strips

Artin $L$-functions $L(s,\rho)$ for Galois representations $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_n(\mathbb{C})$: Euler factors from $\det(1-\rho(\mathrm{Frob}_p)p^{-s}|V^{I_p})$; Artin conjecture predicts entirety for nontrivial irreducible $\rho$ (known in many cases via modularity / Langlands).

Grand Riemann Hypothesis (statement)¶

GRH (Dirichlet form). Every nontrivial zero of every Dirichlet $L(s,\chi)$ lies on $\operatorname{Re}s=1/2$.

Broader GRH. Nontrivial zeros of automorphic / arithmetic $L$-functions of interest lie on the critical line $\operatorname{Re}s=1/2$ (after standard normalization of the functional equation center).

Consequences (sample). Under GRH for Dirichlet $L$: $\pi(x;q,a)=\mathrm{li}(x)/\varphi(q)+O(x^{1/2}\log(qx))$ uniformly in wide ranges of $q$; effective Chebotarev error terms; improved bounds on class numbers and least quadratic non-residues.

Explicit formulae (one-liner)¶

For $-\frac{L'}{L}(s,\chi)=\sum\Lambda(n)\chi(n)n^{-s}$ ($\sigma>1$), contour shift yields schematic identity $$ \sum_{n\le x}\chi(n)\Lambda(n)= \mathbf{1}{\chi=\chi_0}\,x -\sum, $$ sum over nontrivial zeros }\frac{x^\rho}{\rho}+\text{archimedean / trivial-zero terms$\rho$ of $L(s,\chi)$. Moral: prime-power sums weighted by $\chi$ are dual to zeros; GRH optimizes the dual sum.

Weil’s explicit formula packages test functions on primes vs zeros via Mellin transforms — the spectral form used in Guinand–Weil and modern pair correlation.

See it¶

Dirichlet’s theorem via $L(1,\chi)$. Orthogonality of characters: $$ \sum_{p\equiv a\pmod q}\frac{1}{p}=\frac{1}{\varphi(q)}\sum_{\chi\bmod q}\bar\chi(a)\log L(\sigma,\chi)+O(1)\quad(\sigma\downarrow 1^+). $$ Principal character contributes $\log 1/(\sigma-1)$; non-principal $L(\sigma,\chi)\to L(1,\chi)\neq 0$ stays bounded — hence infinitely many primes in the progression.

Class number formula (real quadratic hint). $L(1,\chi_d)\asymp h_d\sqrt{|d|}^{-1}$ (Dirichlet): low-lying zeros of $L(s,\chi_d)$ constrain $h_d$ (Siegel’s theorem: ineffective $L(1,\chi)\gg_\varepsilon |d|^{-\varepsilon}$).

Check¶

Why is $L(s,\chi)$ entire for non-principal $\chi$, while $\zeta$ has a pole?

For non-principal $\chi$, partial sums $\sum_{n\le x}\chi(n)$ are bounded (character orthogonality / complete sums over periods), so partial summation gives continuation at least to $\sigma>0$; the functional equation / contour methods complete the job. The pole of $\zeta$ reflects $a_n=1$ (non-oscillatory); oscillation of $\chi$ cancels the harmonic divergence at $s=1$.

State GRH for a fixed non-principal $\chi$ and one arithmetic corollary for $\psi(x,\chi)$.

All zeros of $L(s,\chi)$ in $0<\operatorname{Re}s<1$ satisfy $\operatorname{Re}s=1/2$. Then $\psi(x,\chi)=\sum_{n\le x}\chi(n)\Lambda(n)=O(x^{1/2}\log^2(qx))$ (standard form), hence primes equidistribute in APs with square-root cancellation quality.

Pitfalls¶

Imprimitive characters: $L(s,\chi)$ factors through a primitive inducing character; functional equation is cleanest for primitive $\chi$.
Siegel zeros: hypothetical real zeros of $L(s,\chi_d)$ extremely close to $s=1$ wreck uniformity in $q$; known zero-free regions have exceptional-zero caveats.
Normalization: automorphic $L$-functions may be centered so critical line is $\operatorname{Re}s=1/2$ after completing; check the conductor and $\Gamma$-factors before quoting GRH.
Euler product degree: degree $d$ functional equations imply $d$-dimensional local roots — mixing “$\zeta$-like” intuition with higher-degree $L$ needs care (e.g. Sato–Tate vs prime races).

Refs¶

Davenport, Multiplicative Number Theory
Iwaniec–Kowalski, Analytic Number Theory
Neukirch, Algebraic Number Theory (Hecke $L$, class number formula)
Goldfeld, Automorphic Forms and L-Functions for the Group $\mathrm{GL}(n,\mathbb{R})$ (higher rank pointer)

Next¶

Analytic NT overview · Foundations · Algebraic NT foundations · Class field theory · Complex analysis · AG / Mordell–Weil & BSD — the BSD / the $L$-function / the modularity · Number theory