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2009

Bounding |\zeta(1/2 + it)| on the Riemann hypothesis

Chandee, Vorrapan, Soundararajan, Kannan

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In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |\zeta(1/2+it)| \ll \exp(C\log t/\log \log t).

In this note we show how the problem of bounding |\zeta(1/2+it)| may be framed in terms of minorizing the function \log ((4+x^2)/x^2) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant.
Thus we establish that any C> (\log 2)/2 is permissible in Littlewood's result.

Bounding |\zeta(1/2 + it)| on the Riemann hypothesis · Around

Built on

J.E. Littlewood, On the zeros of the Riemann zeta-function
1924
Earlier work this paper cites.
S. Graham and J. Vaaler, A class of extremal functions for the Fourier transform
1981
Earlier work this paper cites.
J. Vaaler, Some extremal functions in Fourier analysis
1985
Earlier work this paper cites.

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K. Ramachandra and A. Sankaranarayanan, On some theorems of Littlewood and Selberg. I
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H. Davenport, Multiplicative number theory
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H. Iwaniec and E. Kowalski, Analytic Number Theory
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E. Carneiro and J. D. Vaaler, Some Extremal Functions in Fourier Analysis II
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V. Chandee, Explicit upper bound for L L -functions on the critical line
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K. Soundararajan, Moments of the Riemann zeta-function
Cited in the paper.

Then

D.W. Farmer, S.M. Gonek, and C.P. Hughes, The maximum size of L L -functions
2007
Later among the works it cites.
D.A. Goldston and S.M. Gonek, A note on S ⁡ ( t ) S(t) and the zeros of the Riemann zeta-function
2007
Later among the works it cites.
K. Soundararajan, Extreme values of zeta and L L -functions
2008
Later among the works it cites.

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