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A NEW UPPER BOUND FOR $\vert \zeta (1+ it)\vert $

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  13 June 2013

TIMOTHY TRUDGIAN
TIMOTHY TRUDGIAN*
Affiliation:
Mathematical Sciences Institute, The Australian National University, ACT 0200, Australia email [email protected]
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Abstract

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It is known that $\zeta (1+ it)\ll \mathop{(\log t)}\nolimits ^{2/ 3} $ when $t\gg 1$. This paper provides a new explicit estimate $\vert \zeta (1+ it)\vert \leq \frac{3}{4} \log t$, for $t\geq 3$. This gives the best upper bound on $\vert \zeta (1+ it)\vert $ for $t\leq 1{0}^{2\cdot 1{0}^{5} } $.

Keywords

Riemann zeta-functioncritical stripexponential sums

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 259 - 264
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Backlund, R. J., ‘Sur les zéros de la fonction $\zeta (s)$ de Riemann’, C. R. Math. Acad. Sci. Paris 158 (1914), 1979–1982.Google Scholar
Cheng, Y. F. and Graham, S. W., ‘Explicit estimates for the Riemann zeta function’, Rocky Mountain J. Math. 34 (4) (2004), 1261–1280.Google Scholar
Ford, K., ‘Vinogradov’s integral and bounds for the Riemann zeta function’, Proc. Lond. Math. Soc. 85 (3) (2002), 565–633.Google Scholar
Granville, A. and Ramaré, O., ‘Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients’, Mathematika 43 (1) (1996), 73–197.Google Scholar
Mellin, Hj., ‘Eine Formel für den Logarithmus transcendenter Funktionen von endlichem Geschlecht’, Acta Soc. Sci. Fenn. 24 (4) (1902), 1–50.Google Scholar
Ram Murty, M., Problems in Analytic Number Theory, Graduate Texts in Mathematics, 206 (Springer, New York, 2008).Google Scholar
Titchmarsh, E. C., The Theory of the Riemann Zeta-function, 2nd edn. (Oxford University Press, Oxford, 1986).Google Scholar
Trudgian, T. S., ‘An improved upper bound for the argument of the Riemann zeta-function on the critical line II’. Preprint.Google Scholar