TY - JOUR

T1 - More than five-twelfths of the zeros of ζ are on the critical line

AU - Pratt, Kyle

AU - Robles, Nicolas

AU - Zaharescu, Alexandru

AU - Zeindler, Dirk

N1 - Funding Information:
The authors would like to acknowledge Matthew Faust of the Illinois Geometry Lab project for helping them write the code that produced their results as well as Siegfried Baluyot for proofreading an earlier version of this manuscript. Moreover, the authors would like to thank Hung Bui and Arindam Roy for useful discussions. For part of this work, the first author was supported by NSF Grant DMS-1501982. The authors are extremely grateful to the anonymous referees for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and more readable. 1 At one point in Levinson’s original paper, there are twenty-four cancellations going on simultaneously! See [ 51 , p. 308] for further details. 2 It is not exactly an improvement but rather a different problem altogether.
Funding Information:
The authors would like to acknowledge Matthew Faust of the Illinois Geometry Lab project for helping them write the code that produced their results as well as Siegfried Baluyot for proofreading an earlier version of this manuscript. Moreover, the authors would like to thank Hung Bui and Arindam Roy for useful discussions. For part of this work, the first author was supported by NSF Grant DMS-1501982. The authors are extremely grateful to the anonymous referees for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and more readable.
Publisher Copyright:
© 2019, The Author(s).

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Autocorrelation ratios

KW - Bell diagrams

KW - Convolution structure

KW - Critical line

KW - Generalized von Mangoldt functions

KW - Incomplete Kloosterman sums

KW - Mollifier

KW - Riemann zeta-function

KW - Zeros

UR - http://www.scopus.com/inward/record.url?scp=85076348083&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076348083&partnerID=8YFLogxK

U2 - 10.1007/s40687-019-0199-8

DO - 10.1007/s40687-019-0199-8

M3 - Article

AN - SCOPUS:85076348083

SN - 2522-0144

VL - 7

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

IS - 2

M1 - 2

ER -