- PRZZ-RMS - Post Referee swap ab
Accepted author manuscript, 2.11 MB, PDF document

Available under license: CC BY: Creative Commons Attribution 4.0 International License

- https://link.springer.com/article/10.1007/s40687-019-0199-8
Final published version

Licence: CC BY: Creative Commons Attribution 4.0 International License

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

In: Research in the Mathematical Sciences., Vol. 7, No. 1, 2, 06.12.2019.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Zeindler, D, Robles, N, Zaharescu, A & Pratt, K 2019, 'More than five-twelfths of the zeros of ζ are on the critical line', *Research in the Mathematical Sciences.*, vol. 7, no. 1, 2. https://doi.org/10.1007/s40687-019-0199-8

Zeindler, D., Robles, N., Zaharescu, A., & Pratt, K. (2019). More than five-twelfths of the zeros of ζ are on the critical line. *Research in the Mathematical Sciences.*, *7*(1), Article 2. https://doi.org/10.1007/s40687-019-0199-8

Zeindler D, Robles N, Zaharescu A, Pratt K. More than five-twelfths of the zeros of ζ are on the critical line. Research in the Mathematical Sciences. 2019 Dec 6;7(1):2. doi: 10.1007/s40687-019-0199-8

@article{6ebe5b91ecef441e85ba5de794620bd8,

title = "More than five-twelfths of the zeros of ζ are on the critical line",

abstract = "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. ",

keywords = "Riemann zeta-function, Kloosterman sum, Zeors on the critical line",

author = "Dirk Zeindler and Nicolas Robles and Alexandru Zaharescu and Kyle Pratt",

year = "2019",

month = dec,

day = "6",

doi = "10.1007/s40687-019-0199-8",

language = "English",

volume = "7",

journal = "Research in the Mathematical Sciences.",

issn = "2522-0144",

publisher = "SPRINGER",

number = "1",

}

TY - JOUR

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

AU - Zeindler, Dirk

AU - Robles, Nicolas

AU - Zaharescu, Alexandru

AU - Pratt, Kyle

PY - 2019/12/6

Y1 - 2019/12/6

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 - Riemann zeta-function

KW - Kloosterman sum

KW - Zeors on the critical line

U2 - 10.1007/s40687-019-0199-8

DO - 10.1007/s40687-019-0199-8

M3 - Journal article

VL - 7

JO - Research in the Mathematical Sciences.

JF - Research in the Mathematical Sciences.

SN - 2522-0144

IS - 1

M1 - 2

ER -