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2018 Explicit bounds for primes in arithmetic progressions
Michael A. Bennett, Greg Martin, Kevin O’Bryant, Andrew Rechnitzer
Illinois J. Math. 62(1-4): 427-532 (2018). DOI: 10.1215/ijm/1552442669

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.

Citation

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Michael A. Bennett. Greg Martin. Kevin O’Bryant. Andrew Rechnitzer. "Explicit bounds for primes in arithmetic progressions." Illinois J. Math. 62 (1-4) 427 - 532, 2018. https://doi.org/10.1215/ijm/1552442669

Information

Received: 21 December 2018; Revised: 21 December 2018; Published: 2018
First available in Project Euclid: 13 March 2019

zbMATH: 07036793
MathSciNet: MR3922423
Digital Object Identifier: 10.1215/ijm/1552442669

Subjects:
Primary: 11M20, 11M26, 11N13, 11N37
Secondary: 11Y35, 11Y40

Rights: Copyright © 2018 University of Illinois at Urbana-Champaign

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Vol.62 • No. 1-4 • 2018
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