Explicit bounds for primes in arithmetic progressions

TL;DR

This paper provides explicit upper bounds for counting functions of primes in arithmetic progressions, improving known constants and extending results to larger moduli and ranges of x, with applications to L-functions and zeros.

Contribution

It derives new explicit bounds for prime counting functions in arithmetic progressions, including for large moduli and ranges, and improves bounds for L(1,χ) and exceptional zeros.

Findings

Explicit bounds for θ(x;q,a) with sharp constants for q ≤ 10^5

Inequalities established for π(x;q,a) and ψ(x;q,a) functions

Improved bounds for L(1,χ) and exceptional zeros

Abstract

We derive explicit upper bounds for various functions counting primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\gcd(a,q)=1$ and $3 \leq q \leq 10^5$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p \equiv a \pmod{q}$ with $p \leq x$, we show that $$ \bigg| \theta (x; q, a) - \frac{x}{\phi (q)} \bigg| < \frac1{160} \frac{x}{\log x}, $$ for all $x \ge 8 \cdot 10^9$ (with sharper constants obtained for individual such moduli $q$). We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\pmod q$ when $q\le1200$. For moduli $q>10^5$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.