## Duke Mathematical Journal

### High moments of the Riemann zeta-function

#### Abstract

In 1918 G. Hardy and J. Littlewood proved an asymptotic estimate for the Second moment of the modulus of the Riemann zeta-function on the segment [1/2,1/2+iT] in the complex plane, as T tends to infinity. In 1926 Ingham proved an asymptotic estimate for the fourth moment. However, since Ingham's result, nobody has proved an asymptotic formula for any higher moment. Recently J. Conrey and A. Ghosh conjectured a formula for the sixth moment. We develop a new heuristic method to conjecture the asymptotic size of both the sixth and eighth moments. Our estimate for the sixth moment agrees with and strongly supports, in a sense made clear in the paper, the one conjectured by Conrey and Ghosh. Moreover, both our sixth and eighth moment estimates agree with those conjectured recently by J. Keating and N. Snaith based on modeling the zeta-function by characteristic polynomials of random matrices from the Gaussian unitary ensemble. Our method uses a conjectural form of the approximate functional equation for the zeta-function, a conjecture on the behavior of additive divisor sums, and D. Goldston and S. Gonek's mean value theorem for long Dirichlet polynomials. We also consider the question of the maximal order of the zeta-function on the critical line.

#### Article information

Source
Duke Math. J. Volume 107, Number 3 (2001), 577-604.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1091737025

Digital Object Identifier
doi:10.1215/S0012-7094-01-10737-0

Mathematical Reviews number (MathSciNet)
MR1828303

Zentralblatt MATH identifier
1006.11048

#### Citation

Conrey, J. B.; Gonek, S. M. High moments of the Riemann zeta-function. Duke Math. J. 107 (2001), no. 3, 577--604. doi:10.1215/S0012-7094-01-10737-0. http://projecteuclid.org/euclid.dmj/1091737025.

#### References

• R. Balasubramanian, On the frequency of Titchmarsh's phenomenon for $\zeta(s)$, IV, Hardy-Ramanujan J. 9 (1986), 1--10.
• 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.
• J. B. Conrey and A. Ghosh, On mean values of the zeta-function, Mathematika 31 (1984), 159--161.
• --. --. --. --., 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.
• --. --. --. --., A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices 1998, 775--780.
• W. Duke, J. B. Friedlander, and H. Iwaniec, A quadratic divisor problem, Invent. Math. 115 (1994), 209--217.
• 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.
• S. M. Gonek, On negative moments of the Riemann zeta-function, Mathematika 36 (1989), 71--88.
• A. Good, Approximate Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die Spitzenformen assoziiert sind, Comment. Math. Helv. 50 (1975), 327--361.
• A. Granville and K. Soundararajan, The distribution of values of $L(1,\chi)$, preprint.
• 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.
• D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3) 38 (1979), 385--422.
• --. --. --. --., Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), 65--78.
• A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. (2) 27 (1926), 273--300.
• 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.
• J. Keating and N. Snaith, Random matrix theory and some zeta-function moments, lecture at Erwin Schrödinger Institute, Vienna, Sept. 1998.
• H. L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv. 52 (1977), 511--518.
• H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119--134.
• 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.
• 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.
• K. Soundararajan, Mean-values of the Riemann zeta-function, Mathematika 42 (1995), 158--174.