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

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

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More than five-twelfths of the zeros of ζ are on the critical line. / Zeindler, Dirk; Robles, Nicolas; Zaharescu, Alexandru et al.
In: Research in the Mathematical Sciences., Vol. 7, No. 1, 2, 06.12.2019.

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

Harvard

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

APA

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

Vancouver

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

Author

Zeindler, Dirk ; Robles, Nicolas ; Zaharescu, Alexandru et al. / More than five-twelfths of the zeros of ζ are on the critical line. In: Research in the Mathematical Sciences. 2019 ; Vol. 7, No. 1.

Bibtex

@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",

}

RIS

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 -