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 -