The mean square of the product of the Riemann zeta-function with Dirichlet polynomials

Sandro Bettin, Vorrapan Chandee, Maksym Radziwill

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

Improving earlier work of Balasubramanian, Conrey and Heath-Brown [1], we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length T1/2+δ , with δ = 0:01515 ⋯. As an application we obtain an upper bound of the correct order of magnitude for the third moment of the Riemann zeta-function. We also refine previous work of Deshouillers and Iwaniec [8], obtaining asymptotic estimates in place of bounds. Using the work ofWatt [19], we compute the mean-square of the Riemann zeta-function times a Dirichlet polynomial of length going up to T3/4 provided that the Dirichlet polynomial assumes a special shape. Finally, we exhibit a conjectural estimate for trilinear sums of Kloosterman fractions which implies the Lindelöf Hypothesis.

Original languageEnglish (US)
Pages (from-to)51-79
Number of pages29
JournalJournal fur die Reine und Angewandte Mathematik
Volume2017
Issue number729
DOIs
StatePublished - Aug 2017

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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