Illinois Journal of Mathematics

Explicit bounds for primes in arithmetic progressions

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

Export citation


  • M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1965.
  • M. A. Bennett, Rational approximation to algebraic numbers of small height: The Diophantine equation $\vert ax\sp n-by\sp n\vert=1$, J. Reine Angew. Math. 535 (2001), 1–49.
  • D. Berkane and P. Dusart, On a constant related to the prime counting function, Mediterr. J. Math. 13 (2016), 929–938.
  • C. J. de la Vallée Poussin, Sur la fonction $\zeta(s)$ de Riemann et le nombre des nombres premieres inférieur a une limite donnée, Mém. Courronnés et autres Mém. Publ. Acad. Roy. Sci., des Letters Beaux-Arts Belgique 59 (1899/00).
  • P. Dusart, Estimates of $\theta(x;k,l)$ for large values of $x$, Math. Comp. 71 (2002), no. 239, 1137–1168.
  • P. Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2018), 227–251.
  • K. Ford, F. Luca and P. Moree, Values of the Euler $\varphi$-function not divisible by a given odd prime, and the distribution of Euler–Kronecker constants for cyclotomic fields, Math. Comp. 83 (2014), 1447–1476.
  • D. A. Frolenkov and K. Soundararajan, A generalization of the Pólya–Vinogradov inequality, Ramanujan J. 31 (2013), no. 3, 271–279.
  • D. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton–Dyer, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 3 (1976), 624–663.
  • X. Gourdon, The $10^{13}$ first zeros of the Riemann zeta function, and zeros computation at very large height, 2004. Accessed June 2018; available at
  • J. Hadamard, Sur la distribution des zéros de la fonction $\zeta(s)$ et ses consequences arithmétiques, Bull. Soc. Math. France 24 (1896), 199–220.
  • W. Haneke, Über die reellen Nullstellen der Dirichletschen $L$-Reihen, Acta Arith. 22 (1973), 391–421.
  • A. E. Ingham, The distribution of prime numbers, Cambridge tract, vol. 30, Cambridge University Press, Cambridge, 1932.
  • H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004.
  • H. Kadiri, Explicit zero-free regions for Dirichlet $L$-functions, Mathematika 64 (2018), no. 2, 445–474.
  • H. Kadiri and A. Lumley, Primes in arithmetic progression. In preparation.
  • K. S. McCurley, Explicit estimates for the error term in the prime number theorem for arithmetic progressions, Math. Comp. 42 (1984), no. 165, 265–285.
  • K. S. McCurley, Explicit estimates for $\theta(x;3,l)$ and $\psi(x;3,l)$, Math. Comp. 42 (1984), no. 165, 287–296.
  • K. S. McCurley, Explicit zero-free regions for Dirichlet L-functions, J. Number Theory 19 (1984), no. 1, 7–32.
  • H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), no. 40, 119–134.
  • H. L. Montgomery and R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007.
  • M. J. Mossinghoff and T. Trudgian, Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function, J. Number Theory 157 (2015), 329–349.
  • J. Oesterlé, Le problème de Gauss sur le nombre de classes, Enseign. Math. 34 (1988), 43–67.
  • F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds., NIST Digital Library of Mathematical Functions; available at, Release 1.0.20 of 2018-09-15.
  • D. J. Platt, Computing degree $1$ $L$-functions rigorously, Ph.D. Thesis, University of Bristol, 2011.
  • D. J. Platt, Numerical computations concerning the GRH, Math. Comp. 85 (2016), no. 302, 3009–3027.
  • D. J. Platt, Isolating some non-trivial zeros of zeta, Math. Comp. 86 (2017), no. 307, 2449–2467.
  • C. Pomerance, Remarks on the Pólya–Vinogradov inequality, Integers 11 (2011), A19.
  • O. Ramaré and R. Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), 397–425.
  • J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211–232.
  • J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.
  • J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday, Math. Comp. 29 (1975), 243–269.
  • M. O. Rubinstein, lcalc: The L-function calculator, a C++ class library and command line program. Available both through Sage and as Ubuntu linux package, 2008.
  • L. Schoenfeld, Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II, Math. Comp. 30 (1976), no. 134, 337–360.
  • The LMFDB Collaboration, The $L$-functions and Modular Forms Database; available at, 2013. Online; accessed January 2018.
  • The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.1); available at, 2018.
  • T. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line, Math. Comp. 81 (2012), 1053–1061.
  • T. Trudgian, An improved upper bound for the argument of the Riemann zeta-function on the critical line II, J. Number Theory 134 (2014), 280–292.
  • T. Trudgian, An improved upper bound for the error term in the zero-counting formulae for Dirichlet $L$-functions and Dedekind zeta-functions, Math. Comp. 84 (2015), 1439–1450.
  • T. Trudgian, Updating the error term in the prime number theorem, Ramanujan J. 39 (2016), 225–234.
  • K. Walisch, Fast C/C++ library for generating primes; available at
  • M. Watkins, Class numbers of imaginary quadratic fields, Math. Comp. 73 (2003), 907–938.
  • T. Yamada, Explicit formulae for primes in arithmetic progressions, I; available at \arxivurlarXiv:1306.5322v4.