Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Alisa Sedunova
A partial Bombieri–Vinogradov theorem with explicit constants
(Aspects explicites d’un théorème de Bombieri–Vinogradov)
Publications mathématiques de Besançon (2018), p. 101-110, doi: 10.5802/pmb.24
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Class. Math.: 11N13, 11N05, 11N36
Keywords: Primes in arithmetic progressions, Bombieri–Vinogradov theorem, large sieve


In this paper we improve the result of [1] with getting $(\log x)^{\frac{7}{2}}$ instead of $(\log x)^{\frac{9}{2}}$. In particular we obtain a better version of Vaughan’s inequality by applying the explicit variant of an inequality connected to the Möbius function from [5].


[1] Amir Akbary & Kyle Hambrook, “A variant of the Bombieri-Vinogradov theorem with explicit constants and applications”, Math. Comput. 84 (2015) no. 294, p. 1901-1932 Article
[2] Alina Carmen Cojocaru & M. Ram Murty, An introduction to sieve methods and their applications, London Mathematical Society Student Texts 66, Cambridge University Press, 2006
[3] François Dress, Henryk Iwaniec & Gérald Tenenbaum, “Sur une somme liée à la fonction de Möbius”, J. Reine Angew. Math. 340 (1983), p. 53-58 Article
[4] Patrick X. Gallagher, “A large sieve density estimate near $\sigma =1$”, Invent. Math. 11 (1970), p. 329-339 Article
[5] Harald Andres Helfgott, “The ternary Goldbach problem”,, 2015
[6] Carl Pomerance, “Remarks on the Pólya-Vinogradov inequality”, Integers 11 (2011) no. 4, p. 531-542 Article