zbMATH — the first resource for mathematics

Zero-free regions for Dirichlet series. (English) Zbl 1322.11091
The authors study zero-free regions of general Dirichlet series (\(L\)-functions) of the form \[ L(s) = \sum_{n=1}^\infty a_n n^{-s}, \] where some mild conditions, such as \(a_n \ll_\varepsilon n^\varepsilon\), are assumed. They study a general Beurling-Nyman criterion for the zeros of \(L\)-functions. The well-known criterion of B. Nyman [On some groups and semi-groups of translations. (PhD Thesis) (1950)] states that the Riemann hypothesis (all complex zeros of the Riemann zeta-function \(\zeta(s)\) have real parts 1/2) is equivalent to the statement that the characteristic function of the interval (0,1) belongs to the closure of \(\mathcal N\) in \(L^2(0,1)\), the set of functions \[ f(x) = \sum_{j=1}^n c_j\left\{\frac{\theta_j}{x}\right\}\quad(0 < \theta_j \leq 1, c_j \in \mathbb C, \,{y} = y - [y]), \] and \(\sum_{j=1}^nc_j\theta_j =0\).
In their investigations the authors primarily use methods from functional analysis, and they are also interested in sharpening and generalizing the work of N. Nikolski [Ann. Inst. Fourier 45, No. 1, 143–159 (1995; Zbl 0816.30026)] and A. de Roton [Acta Arith. 126, No. 1, 27–55 (2007; Zbl 1131.11062)]. In particular, the authors generalize Nikolski’s work to the well-known class \(\mathcal S\) of the Selberg class of \(L\)-functions. In section 2 four theorems of quite a general nature are presented, and since their formulations are technical, they will not be reproduced here. Applications to the zero-free regions for, \(L(s,\chi)\) and functions of \(\mathcal S\) are given in section 7.

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
30H10 Hardy spaces
Full Text: DOI
[1] 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). · Zbl 1097.11041
[2] Luis Báez-Duarte, Michel Balazard, Bernard Landreau, and Eric Saias, Notes sur la fonction \? de Riemann. III, Adv. Math. 149 (2000), no. 1, 130 – 144 (French, with English summary). · Zbl 1008.11032
[3] Arne Beurling, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 312 – 314. · Zbl 0065.30303
[4] H. Bercovici and C. Foias, A real variable restatement of Riemann’s hypothesis, Israel J. Math. 48 (1984), no. 1, 57 – 68. · Zbl 0569.46011
[5] Michel Balazard and Eric Saias, Notes sur la fonction \? de Riemann. 1, Adv. Math. 139 (1998), no. 2, 310 – 321 (French). · Zbl 0920.11062
[6] 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. · Zbl 1029.11045
[7] 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). · Zbl 1204.11145
[8] 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). · Zbl 1136.11054
[9] Anne de Roton, On the mean square of the error term for an extended Selberg class, Acta Arith. 126 (2007), no. 1, 27 – 55. · Zbl 1131.11062
[10] 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. · Zbl 1210.11090
[11] G. Hardy, A. Ingham, and G. Pólya. Theorems concerning mean values of analytic functions. Proc. of the Royal Math. Soc., 113:542-569, 1927. · JFM 53.0304.05
[12] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. · Zbl 0734.46033
[13] M. N. Huxley, Exponential sums and the Riemann zeta function. V, Proc. London Math. Soc. (3) 90 (2005), no. 1, 1 – 41. · Zbl 1083.11052
[14] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. · Zbl 1059.11001
[15] Nikolai Nikolski, Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann \?-function, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 1, 143 – 159 (English, with English and French summaries). · Zbl 0816.30026
[16] 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. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. · Zbl 1007.47001
[17] B. Nyman. On some groups and semi-groups of translations. Ph.D, thesis, Upsala, 1950. · Zbl 0037.35401
[18] G. Tenenbaum. Introduction à la théorie analytique et probabiliste des nombres. 3ème édition. Belin, 2008. · Zbl 0788.11001
[19] 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. · Zbl 0601.10026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.