High moments of the Riemann zeta-function. (English) Zbl 1006.11048

The asymptotic formula for the integral \[ I_k(T) = \int_0^T|\zeta({1\over 2}+it)|^{2k} dt\qquad(k = 1,2,\ldots) \] represents a central problem in the theory of the Riemann zeta-function \(\zeta(s)\) (for an extensive account see the reviewer’s monographs [John Wiley & Sons (1985; Zbl 0556.10026)] and [Springer Verlag (1991; Zbl 0758.11036)]). Except for the classical cases \(k =1,2\) (op. cit.) no asymptotic formula of the form \[ I_k(T) \sim c_kT(\log T)^{k^2}\qquad(T\to\infty)\tag{1} \] is known when \(k > 2\). In fact, it is difficult even to guess what the value of the constant \(c_k\) should be, in case the above formula holds. J. B. Conrey and A. Ghosh [Int. Math. Res. Not. 1998, 775-780 (1998; Zbl 0920.11060)] conjectured that \[ c_3 = {42\over 9!}\prod_p {\left(1 - {1\over p}\right)}^4 \left(1 + {4\over p} + {1\over p^2}\right),\tag{2} \] where \(p\) denotes primes. In the present paper the authors propose that, when \(k=4\), (1) should hold with \[ c_4 = {24024\over 14!}\prod_p {\left(1 - {1\over p}\right)}^9 \left(1 + {9\over p} + {9\over p^2} + {1\over p^3}\right). \tag{3} \] This is achieved by considerations involving Dirichlet polynomials of the form \(D_{k,N}(s) = \sum_{n\leq N}d_k(n)n^{-s}\), where the divisor function \(d_k(n)\) denotes the number of ways \(n\) may be written as a product of \(k\) factors. Such polynomials appear in various approximate functional equations, used to approximate \(\zeta^k(s)\). After squaring and integration, the problem of the asymptotic evaluation of \(I_k(T)\) is reduced to problems involving the so-called general binary additive divisor problem, namely the study of the function \(D_k(x,h) = \sum_{n\leq x}d_k(n)d_k(n+h)\) where the ‘shift parameter’ \(h\) may not be fixed. This condition makes the problem one of extra difficulty, and the authors make a conjecture involving the function \(m_k(x,h)\), the main term approximating \(D_k(x,h)\). This conjecture, too complicated to be reproduced here, is similar in nature to the one previously made by the reviewer [in New Trends Probab. Stat. 4, 69-89 (1997; Zbl 0924.11070)]. By using this conjecture the authors make a detailed analysis which leads to the conjectural value (3).
It is worth remarking that recently J. P. Keating and N. C. Snaith [Commun. Math. Phys. 214, 57-89 (2000; Zbl 1051.11048)] made the general conjecture \[ c_k = {g_k a_k\over \Gamma(1+k^2)}, \] where \[ a_k = \prod_p \left\{{\left(1 - {1\over p}\right)}^{k^2} \sum_{j=0}^\infty d_k^2(p^j)p^{-j}\right\}, \]
\[ g_k = \prod_{j=0}^{k-1}{j!\over(j+k)!}. \] It is remarkable that the values provided by the Keating-Snaith conjecture (an asymptotic formula should hold also when \(k \geq 0\) is not necessarily an integer, in which case they also produce the explicit value of the conjectural \(c_k\)) coincide with the known values when \(k=1,2\) and with the values furnished by (2) and (3), respectively. It is even more remarkable that the Keating-Snaith conjecture follows from considerations totally different from those employed by the authors, namely by modelling the behaviour of \(\zeta({1\over 2}+it)\) by analogies from random matrix theory. Further ramifications in this exciting field of research have been lately obtained by various authors, most notably by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith [Integral moments of \(L\)-functions, preprint, 58 pp, arXiv:mat.NT/0206018].


11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
Full Text: DOI arXiv


[1] R. Balasubramanian, On the frequency of Titchmarsh’s phenomenon for \(\zeta(s)\), IV , Hardy-Ramanujan J. 9 (1986), 1–10. · Zbl 0662.10030
[2] R. Balasubramanian and K. Ramachandra, On the frequency of Titchmarsh’s phenomenon for \(\zeta(s)\), III , Proc. Indian Acad. Sci. Sect. A 86 (1977), 341–351. · Zbl 0373.10026
[3] J. B. Conrey and A. Ghosh, On mean values of the zeta-function , Mathematika 31 (1984), 159–161. · Zbl 0528.10026
[4] –. –. –. –., “Mean values of the Riemann zeta-function, III” in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, Italy, 1989) , Univ. Salerno, Salerno, Italy, 1992, 35–59.
[5] –. –. –. –., A conjecture for the sixth power moment of the Riemann zeta-function , Internat. Math. Res. Notices 1998 , 775–780. · Zbl 0920.11060
[6] W. Duke, J. B. Friedlander, and H. Iwaniec, A quadratic divisor problem , Invent. Math. 115 (1994), 209–217. · Zbl 0791.11049
[7] D. A. Goldston and S. M. Gonek, Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series , Acta Arith. 84 (1998), 155–192. · Zbl 0902.11033
[8] S. M. Gonek, On negative moments of the Riemann zeta-function , Mathematika 36 (1989), 71–88. · Zbl 0673.10032
[9] A. Good, Approximate Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die Spitzenformen assoziiert sind , Comment. Math. Helv. 50 (1975), 327–361. · Zbl 0315.10038
[10] A. Granville and K. Soundararajan, The distribution of values of \(L(1,\chi)\) , · Zbl 1044.11080
[11] G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes , Acta Math. 41 (1918), 119–196. · JFM 46.0498.01
[12] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function , Proc. London Math. Soc. (3) 38 (1979), 385–422. · Zbl 0403.10018
[13] –. –. –. –., Fractional moments of the Riemann zeta function , J. London Math. Soc. (2) 24 (1981), 65–78. · Zbl 0431.10024
[14] A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function , Proc. London Math. Soc. (2) 27 (1926), 273–300. · JFM 53.0313.01
[15] A. Ivić, “The general additive divisor problem and moments of the zeta-function” in New Trends in Probability and Statistics, Vol. 4: Analytic and Probabalistic Methods in Number Theory (Palanga, Lithuania, 1996) , VSP, Utrecht, Netherlands, 1997, 69–89. · Zbl 0924.11070
[16] J. Keating and N. Snaith, Random matrix theory and some zeta-function moments , lecture at Erwin Schrödinger Institute, Vienna, Sept. 1998.
[17] H. L. Montgomery, Extreme values of the Riemann zeta function , Comment. Math. Helv. 52 (1977), 511–518. · Zbl 0373.10024
[18] H. L. Montgomery and R. C. Vaughan, The large sieve , Mathematika 20 (1973), 119–134. · Zbl 0296.10023
[19] K. Ramachandra, Some remarks on the mean value of the Riemann zeta function and other Dirichlet series, II , Hardy-Ramanujan J. 3 (1980), 1–24. · Zbl 0426.10046
[20] D. Shanks, “Systematic examination of Littlewood’s bounds on \(L(1,\chi)\)” in Analytic Number Theory (St. Louis, 1972) , Proc. Sympos. Pure Math. 24 , Amer. Math. Soc., Providence, 1973, 267–283. · Zbl 0281.12011
[21] K. Soundararajan, Mean-values of the Riemann zeta-function , Mathematika 42 (1995), 158–174. · Zbl 0830.11032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.