Moments of the Riemann zeta function

By Kannan Soundararajan. 2006

294 citations recorded by the source corpus.

Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in other families of L-functions.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/59eb3c198122ed6d2510610fc6d78bc574e5b266

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Extrema of log-correlated random variables: Principles and Examples (2016, background): “This expansion can be made partially rigorous if the sum over primes is cut off at T , see Proposition 1 of [44] based on [56].”
SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION (2019, methodology): “Lower bounds of the form Ik(T ) ≫k T (log T )k 2 are established for all k > 1 in Radziwi l l and Soundararajan [9] unconditionally, and for all k > 0 conditionally on the Riemann Hypothesis in papers of Heath-Brown and Ramachandra, see [11, 12, 5].”
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line (2016, background): “The last lemma gives a useful bound for the 2kth moment of Dirichlet polynomials supported on primes; it may be deduced by a variant of our argument for the previous lemma, or see lemma 3 of [38].”
The fourth moment of quadratic Dirichlet L-functions over function fields (2016, methodology): “[7] Vorrapan Chandee and K. Soundararajan.”
MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS (2017, background): “See the introduction to Soundararajan’s paper [28], for example.”
Bounding ζ(s) in the critical strip (2010, background): “The advances have rather focused on reducing the value of the admissible constant C (see for instance the works by Ramachandra and Sankaranarayanan [12] and Soundararajan [13]) and extending the results to general Lfunctions (see the work of Chandee [3]).”
Lower bounds for moments of zeta and $L$-functions revisited (2020, background): “In [12] it was observed that the 2k-th moment of |ζ( 2 + it)| should be dominated by values of size (log T ), which should occur on a set of measure about T/(log T ) 2 .”
Mesoscopic fluctuations of the zeta zeros (2009, methodology): “In the proof of Proposition3.1, we showed that the remainder terms ( p > mt) converge to 0i n theL 2 -norm to simplify a problem of convergence of a sum over primes: this method seems to appear for the first time in Soundararajan [ 13 ].”
Sharp conditional bounds for moments of the Riemann zeta function (2013, methodology): “P As Soundararajan [17] discusses, if we knew that log | ζ (1 / 2 + it ) behaved in the same way as a Gaussian random variable with such mean and variance then we could obtain a moment bound as in Theorem 1, and his argument can be seen as an attempt to demonstrate as much Gaussian-like behaviour…”
The second moment of quadratic twists of modular L-functions (2009, background): “If we consider a smooth sum over the discriminants 8d in place of the “sharp cut-off” 0 8d ≤ X we would get an error term of O(X(logX) 34+ε), see Section 5 below.”
On the extreme values of the Riemann zeta function on random intervals of the critical line (2016, methodology): “Remark After the first submission of the present paper to arXiv.org, our result concerning the real part of zeta has been proven by Arguin, Belius, Bourgade, Radziwi l l and Soundararajan [4], without assuming the Riemann hypothesis.”
Upper bounds on L-functions at the edge of the critical strip (2009, background): “N T ] 1 9 Se p 20 09 UPPER BOUNDS ON L-FUNCTIONS AT THE EDGE OF THE CRITICAL STRIP XIANNAN LI”
Maxima of a randomized Riemann zeta function, and branching random walks (2015, methodology): “[25] K. Soundararajan.”
Moments and distribution of central L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-values of quadr (2014, background): “The connection betweenmoments and analogs of Selberg’s theorem for central values in families of L-functions is discussed in [32], where the possibility of establishing upper bounds as in Theorem 2 conditional on the Generalized Riemann Hypothesis is mentioned.”