Source: Duke Math. J. Volume 107, Number 3
(2001), 577-604.
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.
References
R. Balasubramanian, On the frequency of Titchmarsh's phenomenon for $\zeta(s)$, IV, Hardy-Ramanujan J. 9 (1986), 1--10.
Mathematical Reviews (MathSciNet):
MR972558
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.
Mathematical Reviews (MathSciNet):
MR506063
J. B. Conrey and A. Ghosh, On mean values of the zeta-function, Mathematika 31 (1984), 159--161.
Mathematical Reviews (MathSciNet):
MR762188
--. --. --. --., ``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.
Mathematical Reviews (MathSciNet):
MR401651
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.
Mathematical Reviews (MathSciNet):
MR532980
--. --. --. --., Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), 65--78.
Mathematical Reviews (MathSciNet):
MR623671
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.
Mathematical Reviews (MathSciNet):
MR460255
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119--134.
Mathematical Reviews (MathSciNet):
MR374060
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.
Mathematical Reviews (MathSciNet):
MR577338
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.
Mathematical Reviews (MathSciNet):
MR337827
K. Soundararajan, Mean-values of the Riemann zeta-function, Mathematika 42 (1995), 158--174.
