Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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 Home 1975-2014 archives All online articles Advanced Search Latest articles Search for an article Table of contents for this volume | Previous article | Next article Alisa SedunovaA partial Bombieri–Vinogradov theorem with explicit constants(Aspects explicites d’un théorème de Bombieri–Vinogradov)Publications mathématiques de Besançon (2018), p. 101-110 Article PDF Class. Math.: 11N13, 11N05, 11N36Keywords: Primes in arithmetic progressions, Bombieri–Vinogradov theorem, large sieve AbstractIn 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]. Bibliography[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”, https://arxiv.org/abs/1501.05438, 2015 [6] Carl Pomerance, “Remarks on the Pólya-Vinogradov inequality”, Integers 11 (2011) no. 4, p. 531-542 Article