More than five-twelfths of the zeros of ζ are on the critical line

Kyle Pratt, Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler

Research output: Contribution to journalArticlepeer-review


The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form (μ⋆Λ1⋆k1⋆Λ2⋆k2⋆⋯⋆Λd⋆kd) is computed unconditionally by means of the autocorrelation of ratios of ζ techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of ζ(s)+λ1ζ′(s)logT+λ2ζ′′(s)log2T+⋯+λdζ(d)(s)logdT,where ζ( k ) stands for the kth derivative of the Riemann zeta-function and {λk}k=1d are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

Original languageEnglish (US)
Article number2
JournalResearch in Mathematical Sciences
Issue number2
StatePublished - 2019


  • Autocorrelation ratios
  • Bell diagrams
  • Convolution structure
  • Critical line
  • Generalized von Mangoldt functions
  • Incomplete Kloosterman sums
  • Mollifier
  • Riemann zeta-function
  • Zeros

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)
  • Computational Mathematics
  • Applied Mathematics


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