Explicit bounds for primes in arithmetic progressions

Abstract

We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\mathop{\mathrm{gcd}}\nolimits (a,q)=1$ and $3\leq q\leq10^{5}$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p\equiv a\ (\operatorname{mod}q)$ with $p\leq x$, we show that

\[\vert \theta(x;q,a)-{x}/{\varphi(q)}\vert <\frac{1}{160}\frac{x}{\log x}\] for all $x\geq8\cdot10^{9}$, with significantly sharper constants obtained for individual 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 $a\ (\operatorname{mod}q)$ 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.