Illinois Journal of Mathematics

Explicit bounds for primes in arithmetic progressions

Michael A. Bennett, Greg Martin, Kevin O’Bryant, and Andrew Rechnitzer

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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.

Article information

Source
Illinois J. Math., Volume 62, Number 1-4 (2018), 427-532.

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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1552442669

Digital Object Identifier
doi:10.1215/ijm/1552442669

Mathematical Reviews number (MathSciNet)
MR3922423

Zentralblatt MATH identifier
07036793

Subjects
Primary: 11N13: Primes in progressions [See also 11B25] 11N37: Asymptotic results on arithmetic functions 11M20: Real zeros of $L(s, \chi)$; results on $L(1, \chi)$ 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11Y35: Analytic computations 11Y40: Algebraic number theory computations

Citation

Bennett, Michael A.; Martin, Greg; O’Bryant, Kevin; Rechnitzer, Andrew. Explicit bounds for primes in arithmetic progressions. Illinois J. Math. 62 (2018), no. 1-4, 427--532. doi:10.1215/ijm/1552442669. https://projecteuclid.org/euclid.ijm/1552442669


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