## V Kumar Murty - The Barban-Vehov Theorem in Arithmetic Progressions

hrj:5118 - Hardy-Ramanujan Journal, January 23, 2019 - https://doi.org/10.46298/hrj.2019.5118
The Barban-Vehov Theorem in Arithmetic Progressions

Authors: V Kumar Murty

A result of Barban-Vehov (and independently Motohashi) gives an estimate for the mean square of a sequence related to Selberg's sieve. This upper bound was refined to an asymptotic formula by S. Graham in 1978. In 1992, I made the observation that Graham's method can be used to obtain an asymptotic formula when the sum is restricted to an arithmetic progression. This formula immediately gives a version of the Brun-Titchmarsh theorem. I am taking the occasion of a volume in honour of my friend S. Srinivasan to revisit and publish this observation in the hope that it might still be of interest.

Published on: January 23, 2019
Submitted on: January 23, 2019
Keywords: Selberg's sieve,Brun-Titchmarsh theorem,arithmetic progressions 2010 Mathematics Subject Classification Primary 11N37,11N13,Secondary 11B25,11N35,11N69, [ MATH ] Mathematics [math], [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT]