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Published September 18, 2013 | Submitted
Journal Article Open

The Riemann-zeta function on vertical arithmetic progressions


We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression 1/2 + i(an + b) with a>0, b real, exhibits a remarkable correspondence with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression an+b are not the ordinates of some zero of ζ(s) lying on the critical line. This improves on an earlier work of Martin and Ng. We then complement this result by producing large values of ζ(s) on arithmetic progressions which are of the same quality as the best Ω results currently known for ζ(1/2 + it) with t real.

Additional Information

© The Author(s) 2013. Published by Oxford University Press. Received February 5, 2013; Revised August 21, 2013; Accepted August 22, 2013. Advance Access Publication September 18, 2013. We would like to thank Professor Soundararajan for a number of useful remarks on a draft of this paper. This work was done while both authors were visiting the Centre de Recherches Mathématiques. We are grateful for their kind hospitality. This work was partially supported by a NSERC PGS-D award (to M.R.).

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August 19, 2023
October 18, 2023