Zero-free regions for Dirichlet series

Delaunay, C.; Fricain, E.; Mosaki, E.; Robert, O.

doi:10.1090/S0002-9947-2012-05735-7

Zero-free regions for Dirichlet series
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by C. Delaunay, E. Fricain, E. Mosaki and O. Robert PDF

Trans. Amer. Math. Soc. 365 (2013), 3227-3253 Request permission

Abstract:

In this paper, we are interested in explicit zero-free discs for some Dirichlet series and we also study a general Beurling-Nyman criterion for $L$-functions. Our results generalize and improve previous results obtained by N. Nikolski and by A. de Roton. As a concrete application, we get, for example, a Beurling-Nyman type criterion for the Siegel zero problem.

References

Luis Báez-Duarte, A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), no. 1, 5–11 (English, with English and Italian summaries). MR 2057270
Luis Báez-Duarte, Michel Balazard, Bernard Landreau, and Eric Saias, Notes sur la fonction $\zeta$ de Riemann. III, Adv. Math. 149 (2000), no. 1, 130–144 (French, with English summary). MR 1742356, DOI 10.1006/aima.1999.1861
Arne Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 312–314. MR 70655, DOI 10.1073/pnas.41.5.312
H. Bercovici and C. Foias, A real variable restatement of Riemann’s hypothesis, Israel J. Math. 48 (1984), no. 1, 57–68. MR 768266, DOI 10.1007/BF02760524
Michel Balazard and Eric Saias, Notes sur la fonction $\zeta$ de Riemann. 1, Adv. Math. 139 (1998), no. 2, 310–321 (French). MR 1654169, DOI 10.1006/aima.1998.1760
Jean-François Burnol, A lower bound in an approximation problem involving the zeros of the Riemann zeta function, Adv. Math. 170 (2002), no. 1, 56–70. MR 1929303, DOI 10.1006/aima.2001.2066
Anne de Roton, Une approche hilbertienne de l’hypothèse de Riemann généralisée, Bull. Soc. Math. France 134 (2006), no. 3, 417–445 (French, with English and French summaries). MR 2246000, DOI 10.24033/bsmf.2516
Anne de Roton, Généralisation du critère de Beurling-Nyman pour l’hypothèse de Riemann, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6111–6126 (French, with English and French summaries). MR 2336318, DOI 10.1090/S0002-9947-07-04261-4
Anne de Roton, On the mean square of the error term for an extended Selberg class, Acta Arith. 126 (2007), no. 1, 27–55. MR 2284311, DOI 10.4064/aa126-1-2
Andrew Granville and K. Soundararajan, Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 357–384. MR 2276774, DOI 10.1090/S0894-0347-06-00536-4
G. Hardy, A. Ingham, and G. Pólya. Theorems concerning mean values of analytic functions. Proc. of the Royal Math. Soc., 113:542–569, 1927.
Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
M. N. Huxley, Exponential sums and the Riemann zeta function. V, Proc. London Math. Soc. (3) 90 (2005), no. 1, 1–41. MR 2107036, DOI 10.1112/S0024611504014959
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Nikolai Nikolski, Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann $\zeta$-function, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 1, 143–159 (English, with English and French summaries). MR 1324128, DOI 10.5802/aif.1451
Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
B. Nyman. On some groups and semi-groups of translations. Ph.D, thesis, Upsala, 1950.
G. Tenenbaum. Introduction à la théorie analytique et probabiliste des nombres. 3ème édition. Belin, 2008.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550