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Transactions of the American Mathematical Society

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Finite Euler products and the Riemann hypothesis

Author: S. M. Gonek
Journal: Trans. Amer. Math. Soc. 364 (2012), 2157-2191
MSC (2010): Primary 11M06, 11M26
Published electronically: December 1, 2011
MathSciNet review: 2869202
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Abstract: We investigate the approximation of the Riemann zeta-function by short truncations of its Euler product in the critical strip. We then construct a parameterized family of non-analytic functions that approximate the zeta-function to the right of the critical line. With the possible exception of finitely many zeros off the critical line, each function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter is not too large, the functions in the family have about the same number of zeros as the zeta-function, their zeros are all simple, and the zeros ``repel''. The structure of these functions makes the reason for the simplicity and repulsion of their zeros apparent. Computer calculations suggest that the zeros of functions in the family are remarkably close to those of the zeta-function, even for small values of the parameter. We show that if the Riemann Hypothesis holds for the Riemann zeta-function, then the zeros of these functions indeed converge to those of the zeta-function as the parameter increases and that, between consecutive zeros of the zeta-function, the functions tend to twice the zeta-function. Finally, we discuss analogues of the model for other L-functions and the insight they give into the distribution of zeros of linear combinations of L-functions.

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  • 1. E. Bogomolny, ``Spectral statistics'', Doc. Math. J.-Extra vol. ICM (1998), 99-108. MR 1648144 (99h:81053)
  • 2. E. Bombieri and D. A. Hejhal, ``On the zeros of Epstein zeta functions'', C. R. Acad. Sci. Paris Ser. I Math. 304 (1987), no. 9, 213-217. MR 883477 (89c:11136)
  • 3. E. Bombieri and D. A. Hejhal, ``On the distribution of zeros of linear combinations of Euler products'', Duke Math. J. 80 (1995), 821-862. MR 1370117 (96m:11071)
  • 4. E. B. Bogomolny and J. P. Keating, ``Gutzwiller's trace formula and spectral statistics: Beyond the diagonal approximation'', Phys. Rev. Lett. 77 (1996), no. 8, 1472-1475.
  • 5. R. Balasubramaian and K. Ramachandra, ``On the frequency of Titchmarsh's phenomenon for $ \zeta (s)$, III'', Proc. Indian Acad. of Sci. 86 A (1977), 341-351. MR 0506063 (58:21968)
  • 6. H. Davenport, Multiplicative Number Theory (2nd edition), Springer, New York, 1980. MR 606931 (82m:10001)
  • 7. D. Farmer, S. M. Gonek, C. P. Hughes, ``The maximum size of L-functions'', J. Reine Angew. Math. (Crelle's Journal) 609 (2007), 215-236. MR 2350784 (2009b:11140)
  • 8. D. A. Goldston and S. M. Gonek, ``A note on $ S(t)$ and the zeros of the Riemann zeta-function'', Bull. London Math. Soc. 39 (2007), 482-486. MR 2331578 (2008f:11097)
  • 9. S. M. Gonek, ``Mean values of the Riemann zeta-function and its derivatives'', Invent. Math. 75 (1984), 123-141. MR 728143 (85f:11063)
  • 10. S. M. Gonek, C. P. Hughes and J. P. Keating, ``A hybrid Euler-Hadamard product for the Riemann zeta function'', Duke Math. J. 136, no. 3 , 2007, 507-549. MR 2309173 (2008e:11100)
  • 11. A. P. Guinand, ``Some Fourier transforms in prime-number theory'', Quart. J. Math. Oxford Ser. (2) 18 (1947), 53-64. MR 0019677 (8:447a)
  • 12. A. P. Guinand, ``A summation formula in the theory of prime numbers'', Proc. London Math. Soc. (2) 50 (1948), 107-119. MR 0026086 (10:104g)
  • 13. 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 Arith. 41 (1918), 119-196. MR 1555148
  • 14. D. A. Hejhal, On a result of Selberg concerning zeros of linear combinations of L-functions, I.M.R.N., (2000), 551-577. MR 1763856 (2001m:11149)
  • 15. C. P. Hughes, J. P. Keating and N. O'Connell, ``Random matrix theory and the derivative of the Riemann zeta function'', Proc. R. Soc. Lond. A 456 (2000) 2611-2627. MR 1799857 (2002e:11117)
  • 16. A. E. Ingham, ``Mean-values theorems in the theory of the Riemann zeta-function'', Proc. Lond. Math. Soc. 27 (1926), 273-300.
  • 17. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications 53, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • 18. J. P. Keating, ``Quantum chaology and the Riemann zeta-function'', in Quantum Chaos, eds. G. Casati, I. Guarneri, and U. Smilansky, North-Holland, Amsterdam, 1993, 145-185. MR 1246830 (95d:81029)
  • 19. J. P. Keating and N. C. Snaith, ``Random matrix theory and $ \zeta (1/2+\mathrm {i} t)$'', Commun. Math. Phys. 214 (2000), 57-89. MR 1794265 (2002c:11107)
  • 20. H. L. Montgomery, ``Mean and large values of Dirichlet polynomials'', Invent. Math. 8 (1969), 334-345. MR 0268130 (42:3029)
  • 21. H. L. Montgomery, ``The pair correlation of zeros of the zeta function'', Analytic Number Theory, Proceedings of Symposia in Pure Mathematics 24 (1973), 181-193. MR 0337821 (49:2590)
  • 22. H. L. Montgomery and R. C. Vaughan, ``Hilbert's inequality'', J. London Math. Soc. (2) 8 (1974), 73-82. MR 0337775 (49:2544)
  • 23. A. Selberg, ``Old and new conjectures and results about a class of Dirichlet series'', in Collected Papers, vol. 2, Springer Verlag, 1991, 47-63. MR 1220477 (94f:11085)
  • 24. A. Selberg, ``On the remainder in the formula for $ N(T)$, the number of zeros of $ \zeta (s)$ in the strip $ 0<t<T$'', Avhandlinger Norske Vid. Akad. Oslo, no. 1, (1944), 1-27. MR 0015426 (7:417b)
  • 25. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function (2nd edition, revised by
    D. R. Heath-Brown), Oxford Science Publications, 1986. MR 882550 (88c:11049)

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Additional Information

S. M. Gonek
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Received by editor(s): September 17, 2010
Published electronically: December 1, 2011
Additional Notes: This work was supported in part by NSF grant DMS-0653809.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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