Extreme values of zeta and L-functions

By Kannan Soundararajan. 2007

141 citations recorded by the source corpus.

We introduce a resonance method to produce large values of the Riemann zeta-function on the critical line, and large and small central values of L-functions.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d295474b4b8603cee91925ee135770ff46e52bf7

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In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Maxima of a randomized Riemann zeta function, and branching random walks (2015, background): “See the paper by Farmer, Gonek and Hughes [13] for more precise conjectures about this maximum size, and the paper of Soundararajan [26] and the recent work of Bondarenko and Seip [8] for rigorous lower bounds.”
Lower bounds for the maximum of the Riemann zeta function along vertical lines (2014, methodology): “To do so, the main idea of Soundararajan’s resonance method in [18] is (roughly speaking) to construct a function A(t) such that ∫ T”
Large greatest common divisor sums and extreme values of the Riemann zeta function (2015, methodology): “The best lower estimate for extreme values of | ζ (1/2 + it ) known previously was obtained in 2008 by Soundararajan [17] who proved that 1)) log T loglog T holds for some t , T ≤ t ≤ 2 T , if T is large enough.”
Extreme values of the Riemann zeta function on the 1-line (2017, methodology): “Improving earlier work of Littlewood, in 1972 Levinson [7] proved that (1) |ζ(1 + it)| ≥ eγ log2 t+O(1) for arbitrarily large t.1 This was further improved by Granville and Soundararajan [5], who in 2006 established the lower bound max t∈[0,T ] |ζ(1 + it)| ≥ eγ (log2 T + log3 T − log4 T +O(1)) for a”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, methodology): “in [30] under the Riemann hypothesis, and then made unconditional with improved constant c in [4] and [39].”
Extreme values of ζ′(ρ) (2007, methodology): “Z. Rudnick and K. Soundararajan, ‘Lower bounds for moments of L-functions’, Proc.”
Sommes de Gál et applications (2018, background): “Leur approche repose sur la méthode de résonance, développée indépendamment, dans ce cadre, par Soundararajan [23] et Hilberdink [14].”
Extreme values of the Riemann zeta function and its argument (2017, methodology): “The idea, following [8], is that |R(t)|2 should “resonate” with and pick out large values of the Dirichlet series in question, which will come from the right-hand side of either (9) or (10).”
An arithmetical mapping and applications to Ω-results for the Riemann zeta function (2009, methodology): “The author is grateful to the referee for pointing out the recent paper by Soundararajan [11], in which a “resonator” method was developed and used to find Ω-results for ζ(12 +it).”