### A hybrid Euler-Hadamard product for the Riemann zeta function

S. M. Gonek, C. P. Hughes, and J. P. Keating
Source: Duke Math. J. Volume 136, Number 3 (2007), 507-549.

#### Abstract

We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory

First Page:
Primary Subjects: 11M26
Secondary Subjects: 11M06, 15A52
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1170084897
Digital Object Identifier: doi:10.1215/S0012-7094-07-13634-2
Mathematical Reviews number (MathSciNet): MR2309173
Zentralblatt MATH identifier: 1171.11049

### References

E. L. Basor, Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc. 239 (1978), 33--65.
Mathematical Reviews (MathSciNet): MR0493480
Digital Object Identifier: doi:10.2307/1997847
Zentralblatt MATH: 0409.47018
E. Bombieri and D. A. Hejhal, On the distribution of zeros of linear combinations of Euler products, Duke Math. J. 80 (1995), 821--862.
Mathematical Reviews (MathSciNet): MR1370117
Digital Object Identifier: doi:10.1215/S0012-7094-95-08028-4
Project Euclid: euclid.dmj/1077246295
Zentralblatt MATH: 0853.11074
J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of L-functions, Proc. London. Math. Soc. (3) 91 (2005), 33--104.
Mathematical Reviews (MathSciNet): MR2149530
Digital Object Identifier: doi:10.1112/S0024611504015175
Zentralblatt MATH: 1075.11058
J. B. Conrey and A. Ghosh, A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices 1998, no. 15, 775--780.
Mathematical Reviews (MathSciNet): MR1639551
Digital Object Identifier: doi:10.1155/S1073792898000476
Zentralblatt MATH: 0920.11060
J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577--604.
Mathematical Reviews (MathSciNet): MR1828303
Digital Object Identifier: doi:10.1215/S0012-7094-01-10737-0
Project Euclid: euclid.dmj/1091737025
Zentralblatt MATH: 1006.11048
J. A. Gaggero Jara, Asymptotic mean square of the product of the second power of the Riemann zeta function and a Dirichlet polynomial, Ph.D. dissertation, University of Rochester, Rochester, N.Y., 1997.
S. M. Gonek, Mean values of the Riemann zeta function and its derivatives, Invent. Math. 75 (1984), 123--141.
Mathematical Reviews (MathSciNet): MR0728143
Digital Object Identifier: doi:10.1007/BF01403094
Zentralblatt MATH: 0531.10040
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., Academic Press, New York, 1980.
Mathematical Reviews (MathSciNet): MR0582453
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 (1917), 119--196.
Mathematical Reviews (MathSciNet): MR1555148
Digital Object Identifier: doi:10.1007/BF02422942
C. P. Hughes, J. P. Keating, and N. O'Connell, Random matrix theory and the derivative of the Riemann zeta function, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), 2611--2627.
Mathematical Reviews (MathSciNet): MR1799857
Digital Object Identifier: doi:10.1098/rspa.2000.0628
Zentralblatt MATH: 0996.11052
A. E. Ingham, Mean-values theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc. 27 (1926), 273--300.
H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, Amer. Math. Soc., Providence, 2004.
Mathematical Reviews (MathSciNet): MR2061214
J. P. Keating and N. C. Snaith, Random matrix theory and L-functions at $s\,=\,$1$/$2, Comm. Math. Phys. 214 (2000), 91--110.
Mathematical Reviews (MathSciNet): MR1794267
Digital Object Identifier: doi:10.1007/s002200000262
Zentralblatt MATH: 1051.11047
—, Random matrix theory and $\zeta(1/2+it)$, Comm. Math. Phys. 214 (2000), 57--89.
Mathematical Reviews (MathSciNet): MR1794265
Digital Object Identifier: doi:10.1007/s002200000261
—, Random matrices and $L$-functions, J. Phys. A 36 (2003), 2859--2881.
Mathematical Reviews (MathSciNet): MR1986396
Digital Object Identifier: doi:10.1088/0305-4470/36/12/301
Zentralblatt MATH: 1074.11053
F. Mezzadri and N. C. Snaith, eds., Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser. 322, Cambridge Univ. Press, Cambridge, 2005.
Mathematical Reviews (MathSciNet): MR2145172
H. L. Montgomery, The pair correlation of zeros of the zeta function'' in Analytic Number Theory (St. Louis, Mo., 1972), Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, 1973, 181--193.
Mathematical Reviews (MathSciNet): MR0337821
A. M. Odlyzko, The $10^20$-th zero of the Riemann zeta function and 175 million of its neighbors, preprint, 1992.
—, Zeros number $10^12+1$ through $10^12+10^4$ of the Riemann zeta function, table, http://www.dtc.umn.edu/$^\sim$odlyzko/zeta$\_$tables/index.html
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., New York, 1939.
Mathematical Reviews (MathSciNet): MR0000077