For coprime $a, n$, there are infinitely many primes $p \equiv a \pmod{n}$.

Proof via $L$-functions: Consider $L(s, \chi)$ for characters modulo $n$. The non-vanishing $L(1, \chi) \neq 0$ for non-principal $\chi$ implies: $\sum_{p \equiv a \pmod{n}} \frac{1}{p^s}$

diverges as $s \to 1^+$, proving infinitude of primes.

With more care, obtain density: primes $\equiv a \pmod{n}$ have natural density $1/\varphi(n)$.

For $K = \mathbb{Q}(\sqrt{d})$ with quadratic character $\chi_d$: $L(s, \chi_d) = \prod_p \frac{1}{1 - \chi_d(p)p^{-s}}$

Special values:

• $L(1, \chi_{-3}) = \frac{\pi}{3\sqrt{3}}$
• $L(1, \chi_{-4}) = \frac{\pi}{4}$
• $L(1, \chi_5) = \frac{\log(\phi)}{\sqrt{5}}$ where $\phi = (1+\sqrt{5})/2$

These relate to class numbers via: $h_K L(1, \chi_d) = \frac{2\pi}{w_K\sqrt{|d|}}$ for imaginary quadratic, and similar formulas for real quadratic (involving regulators).

For an elliptic curve $E/\mathbb{Q}$, the $L$-function is: $L(E, s) = \prod_p \frac{1}{1 - a_p p^{-s} + p^{1-2s}}$

where $a_p = p + 1 - \#E(\mathbb{F}_p)$. The Birch and Swinnerton-Dyer conjecture predicts: $\text{ord}_{s=1}L(E, s) = \text{rank}(E(\mathbb{Q}))$

and gives an exact formula for $\lim_{s \to 1}(s-1)^{-r}L(E, s)$ in terms of regulator, Shafarevich-Tate group, and Tamagawa numbers.

For $E: y^2 = x^3 - x$ (CM by $\mathbb{Z}[i]$): $L(E, 1) \neq 0$, so $\text{rank}(E(\mathbb{Q})) = 0$ (BSD predicts, proven for CM curves).

The Riemann zeta function satisfies: $-\frac{\zeta'(s)}{\zeta(s)} = \sum_n \frac{\Lambda(n)}{n^s}$

where $\Lambda$ is the von Mangoldt function. The zeros of $\zeta(s)$ control the distribution of primes via: $\pi(x) = \text{Li}(x) - \sum_\rho \text{Li}(x^\rho) + \text{lower order terms}$

where the sum is over nontrivial zeros $\rho$ of $\zeta(s)$. RH implies optimal error bounds for $\pi(x)$.

Modern algorithms compute $L(1, \chi)$ using:

1. Direct summation: Accelerate convergence with Euler-Maclaurin
2. Functional equation: Relate $L(1, \chi)$ to rapidly convergent series
3. Fast Fourier methods: For character sums and Gauss sums

For $\chi$ modulo $m$: Complexity $O(\sqrt{m})$ or better with precomputation.

Pari/GP example:

\\ L(1, chi) for chi of conductor 5
lfun(lfuncreate([5, 1, [0], 1, 0]), 1)
\\ Output: 0.728882...