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Explicit sub-Weyl Bound for the Riemann Zeta Function

Patel, Dhir
http://rave.ohiolink.edu/etdc/view?acc_num=osu1626742085346834

Abstract Details

2021, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis we obtain the following explicit sub-Weyl bound for the Riemann zeta function: $|\zeta(1/2+it)|\leq 307.098t^{27/164}$ for $t\geq 3$. Applications of this bound include improvement to Turing's method as done by Trudgian, and improvement to explicit zero free regions as done by Kadiri--Lumley--Ng. Our main result advances explicit bounds for $|\zeta(1/2+it)|$ beyond the Weyl-exponent $1/6$. So we find a quadruple of numbers $(a, \kappa,C, t_0)$, with $a< 1/ 6$, such that $|\zeta(1/2+it)|\leq C t^a\log^\kappa t$ for $t\geq t_0$. Part of our motivation is to remove the $\log t$ term from the previous explicit bounds, that is to enable $\kappa=0$. This is significant in practice since for a long range of $t$ the benefit of removing the $\log t$ term is more substantial than lowering the exponent $a$ by a small amount.
Ghaith Hiary (Advisor)
102 p.
Mathematics
explicit bounds; riemann zeta function; sub-weyl estimate; exponential sums; exponent pairs; explicit exponential integrals; van der corput process

Recommended Citations

Citations

  • Patel, D. (2021). Explicit sub-Weyl Bound for the Riemann Zeta Function [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1626742085346834

    APA Style (7th edition)

  • Patel, Dhir. Explicit sub-Weyl Bound for the Riemann Zeta Function. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1626742085346834.

    MLA Style (8th edition)

  • Patel, Dhir. "Explicit sub-Weyl Bound for the Riemann Zeta Function." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1626742085346834

    Chicago Manual of Style (17th edition)

osu1626742085346834
311
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This open access ETD is published by The Ohio State University and OhioLINK.