## Duke Mathematical Journal

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

#### 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

#### Article information

Source
Duke Math. J., Volume 136, Number 3 (2007), 507-549.

Dates
First available in Project Euclid: 29 January 2007

https://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

#### Citation

Gonek, S. M.; Hughes, C. P.; Keating, J. P. A hybrid Euler-Hadamard product for the Riemann zeta function. Duke Math. J. 136 (2007), no. 3, 507--549. doi:10.1215/S0012-7094-07-13634-2. https://projecteuclid.org/euclid.dmj/1170084897

#### References

• E. L. Basor, Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc. 239 (1978), 33--65.
• E. Bombieri and D. A. Hejhal, On the distribution of zeros of linear combinations of Euler products, Duke Math. J. 80 (1995), 821--862.
• 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.
• 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.
• J. B. Conrey and S. M. Gonek, High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577--604.
• 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.
• I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged ed., Academic Press, New York, 1980.
• 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.
• 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.
• 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.
• J. P. Keating and N. C. Snaith, Random matrix theory and L-functions at $s\,=\,$1$/$2, Comm. Math. Phys. 214 (2000), 91--110.
• —, Random matrix theory and $\zeta(1/2+it)$, Comm. Math. Phys. 214 (2000), 57--89.
• —, Random matrices and $L$-functions, J. Phys. A 36 (2003), 2859--2881.
• 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.
• 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.
• 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.