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An Explicit Upper Bound for $|\zeta(1+it)|$

Number Theory 2020-09-03 v1

Abstract

In this paper we provide an explicit bound for ζ(1+it)|\zeta(1+it)| in the form of ζ(1+it)min(logt,12logt+1.93,15logt+44.02)|\zeta(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right). This improves on the current best-known explicit bound of ζ(1+it)62.6(logt)2/3|\zeta(1+it)|\leq 62.6(\log t)^{2/3} up until tt of the magnitude 1010710^{10^7}.

Keywords

Cite

@article{arxiv.2009.00769,
  title  = {An Explicit Upper Bound for $|\zeta(1+it)|$},
  author = {Dhir Patel},
  journal= {arXiv preprint arXiv:2009.00769},
  year   = {2020}
}

Comments

18 pages

R2 v1 2026-06-23T18:15:18.097Z