The maximum size of L-functions

By Hugh Montgomery. 2005

122 citations recorded by the source corpus.

Abstract We conjecture the true rate of growth of the maximum size of the Riemann zeta-function and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/499cad61ba4801ba107dc32486a4b0d42ab950ff

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, background): “It has been conjectured by Farmer, Gonek, and Hughes [15], that the supremum of |ζ(1/2+ it)| on [0, T ] is exp((1 + o(1)) √ (1/2) log T log log T ) for T → ∞: their heuristics is related to the fact that the large values of |ζ| are expected to be roughly independent when they occur at points which a”
Bounding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S(t)$$\end{document} and \documentclass[12pt]{minimal} \usepa (2012, background): “For instance, a heuristic argument of Farmer, Gonek and Hughes [9] suggests that S ( t ) = O ( √ log t log log t ) .”
Partial sums of the Möbius function (2007, result): “ 2. Namely, perhaps the bound R ≪ T exp(−CV 2/loglogT) holds for some absolute positive constant C, uniformly in V. This is in keeping with the recent conjecture of D.W. Farmer, Gonek and C.P. Hughes [2] that S(t) ≪ √ logT loglogT. By adapting the ideas in [12] it would be possible to establish the ”
Extreme values of the Riemann zeta function and its argument (2017, background): “Thirty years later, however, Farmer, Gonek, and Hughes [5] offered alternate heuristic arguments suggesting that one should expect S(t) to grow like √ log t log log t .”
Lower bounds for the maximum of the Riemann zeta function along vertical lines (2014, unknown): “[8] conjectured that maxt∈[0,T ] |ζ(1/2 + i t)| = exp ( (1 + o(1)) √ 1 2 log T log log T ) .”
Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “ for T large. The Lindelof hypothesis, which is implied by the Riemann hypothesis, states that max 0tTj(1=2 + it)jgrows slower than any small power of T. See the paper of Farmer, Gonek and Hughes [12] for more precise conjectures about this maximum size, and the paper of Soundararajan [25] for a rig”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, background): “In [14], Farmer, Gonek and Hughes have conjectured that max t∈[0,T ] log |ζ(1/2 + it)| ∼ √ 1 2 log T log log T , but at the end of their paper they also point to dissenting views, advocating that (1) is closer to the maximal size.”
The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line (2016, background): “In the setting of random matrices, we provide a rigorous proof of a connection between the statistical behavior of the zeta function and random characteristic polynomials, while in the setting of the theory of log-correlated fields, our results can be seen as further evidence for log-correlated fiel”
Extreme values of zeta and L-functions (2007, background): “Hughes [4] have suggested, based on several interesting heuristic considerations, that the maximum size of |ζ( 1 2 + it)| is about exp(C √ log t log log t) with C = 1/ √ 2 + o(1).”