Lower bounds for moments of the derivative of the Riemann zeta function

The study on Mk(T ) dates back to the work of G.H. Hardy and J. E. Littlewood [13], who established an asymptotic formula for M1(T ). In [18], A. E. Ingham established an asymptotic formula for M2(T ). No other asymptotic formulas are known for Mk(T ) except for the trivial case k = 0. Despite of this, J. P. Keating and N. C. Snaith [21] made precise conjectured formulas for Mk(T ) for all real k ≥ 0 by drawing analogues with the random matrix theory. Using multiple Dirichlet series, A. Diaconu, D. Goldfeld and J. Hoffstein [10] also obtained the same conjectured formulas. More precise asymptotic formulas with lower order terms were conjectured by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith in [5]. Owing much to the work in [1, 2, 6–8, 14–17, 26–31, 34, 35], we now have sharp upper and lower bounds for Mk(T ) of the conjectured order of magnitude for all k ≥ 0 with some of them being valid under the truth of the Riemann hypothesis (RH). Among the many methods applied in the above work, we point out notably a simple and powerful method developed by Z. Rudnick and K. Soundararajan [31, 32] towards establishing sharp lower bounds for moments of families of Lfunctions, a method of K. Soundararajan [35] and its refinement by A. J. Harper [14] to derive sharp upper bounds for moments of families of L-functions under the generalized Riemann hypothesis (GRH). We note also an upper bounds principle developed by M. Radziwi l l and K. Soundararajan in [27] for establishing upper bounds for moments of families of L-functions as well as its dual lower bounds principle developed by W. Heap and K. Soundararajan in [16]. Similar to Mk(T ), it is also interesting to study moments of the derivatives of ζ(s) on the critical line. For integers l ≥ 1, let

[1]  J. Littlewood,et al.  Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes , 1916 .

[2]  A. E. Ingham.  Mean-Value Theorems in the Theory of the Riemann Zeta-Function , 1928 .

[3]  H. Davenport.  Multiplicative Number Theory , 1967 .

[4]  K. Ramachandra.  Some remarks on the mean value of the Riemann zetafunction and other Dirichlet series. III , 1978 .

[5]  S. Gonek.  Mean values of the Riemann zeta-function and its derivatives , 1984 .

[6]  J. Conrey,et al.  On mean values of the zeta-function , 1984 .

[7]  THE FOURTH MOMENT OF DERIVATIVES OF THE RIEMANN ZETA-FUNCTION , 1988 .

[8]  K. Ramachandra.  Proof of some conjectures on the mean-value of titchmarsh series with applications to Titchmarsh's phenomenon , 1990 .

[9]  Kannan Soundararajan,et al.  Mean-values of the Riemann zeta-function , 1995 .

[10]  Fractional moments of the Riemann zeta-function , 1997 .

[11]  High moments of the Riemann zeta-function , 1999, math/9902162.

[12]  Nina C Snaith,et al.  Random Matrix Theory and L-Functions at s= 1/2 , 2000 .

[13]  J. P. Keating,et al.  Integral Moments of L‐Functions , 2002, math/0206018.

[14]  Dorian Goldfeld,et al.  Multiple Dirichlet Series and Moments of Zeta and L-Functions , 2003, Compositio Mathematica.

[15]  H. Iwaniec,et al.  Analytic Number Theory , 2004 .

[16]  Lower bounds for moments of L-functions. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Lower bounds for moments of L-functions: symplectic and orthogonal examples , 2006, math/0601498.

[18]  Moments of the Riemann zeta function , 2006, math/0612106.

[19]  Moments of the Derivative of Characteristic Polynomials with an Application to the Riemann Zeta Function , 2006 .

[20]  A note on a conjecture of Gonek , 2011, 1106.1160.

[21]  Moments of the Riemann zeta‐function at its relative extrema on the critical line , 2011, 1106.1154.

[22]  The 4.36th Moment of the Riemann Zeta-Function , 2012 .

[23]  Lower bounds for the moments of the derivatives of the Riemann zeta-function and Dirichlet L-functions , 2012 .

[24]  J. Kalpokas,et al.  Upper bounds for discrete moments of the derivatives of the riemann zeta-function on the critical line* , 2012, Lithuanian Mathematical Journal.

[25]  M. Milinovich,et al.  Lower Bounds for Moments of ζ′(ρ) , 2014 .

[26]  K. Soundararajan,et al.  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, Inventiones mathematicae.

[27]  The mean square of the product of the Riemann zeta-function with Dirichlet polynomials , 2014, 1411.7764.

[28]  SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION , 2019, The Quarterly Journal of Mathematics.

[29]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[30]  K. Soundararajan,et al.  Lower bounds for moments of zeta and L$L$ ‐functions revisited , 2020, Mathematika.