×

Zeros of Dirichlet polynomials via a density criterion. (English) Zbl 1421.30003

Summary: We obtain a necessary and sufficient condition in order that a semi-plane of the form \(\mathfrak{R}(s) > r\), \(r \in \mathbb{R}\), is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalisation of a well-known criterion for the truth of the Riemann hypothesis due to Báez-Duarte. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte’s one is also established.

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
11R45 Density theorems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Báez-Duarte, L., On Beurling’s real variable reformulation of the Riemann hypothesis, Adv. Math., 101, 10-30 (1993) · Zbl 0795.11035
[2] Báez-Duarte, L., A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 14, 5-11 (2003) · Zbl 1097.11041
[3] Báez-Duarte, L., A general strong Nyman-Beurling criterion for the Riemann hypothesis, Publ. Inst. Math., 78, 92, 117-125 (2005) · Zbl 1119.11048
[4] Báez-Duarte, L.; Balazard, M.; Landreau, B.; Saias, E., Notes sur la fonction \(ζ\) de Riemann, 3, Adv. Math., 149, 130-144 (2000) · Zbl 1008.11032
[5] Balazard, M.; de Roton, A., Sur un critère de Báez-Duarte pour l’hypothèse de Riemann, Int. J. Number Theory, 6, 883-903 (2010) · Zbl 1201.11088
[6] Balazard, M.; Saias, E., Notes sur la fonction \(ζ\) de Riemann, 1, Adv. Math., 139, 310-321 (1998) · Zbl 0920.11062
[7] Bateman, P. T.; Diamond, H. G., Analytic Number Theory. An Introductory Course (2004), World Scientific: World Scientific London · Zbl 1074.11001
[8] Bercovici, H.; Foias, C., A real variable restatement of Riemann’s hypothesis, Israel J. Math., 48, 57-68 (1984) · Zbl 0569.46011
[9] Bettin, S.; Conrey, J. B.; Farmer, D. W., An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion, Proc. Steklov Inst. Math., 280, 38-44 (2013) · Zbl 1295.11086
[10] Beurling, A., A closure problem related to the Riemann zeta-function, Proc. Natl. Acad. Sci. USA, 41, 312-314 (1955) · Zbl 0065.30303
[11] Burnol, J. F., A lower bound in an approximation problem involving the zeroes of the Riemann zeta function, Adv. Math., 170, 56-70 (2002) · Zbl 1029.11045
[12] de Roton, A., Généralisation du critère de Beurling et Nyman pour l’hypothèse de Riemann, C. R. Math. Acad. Sci. Paris, 340, 191-194 (2005) · Zbl 1063.11029
[13] de Roton, A., Une approche hilbertienne de l’hypothèse de Riemann généralisée, Bull. Soc. Math. France, 134, 417-445 (2006) · Zbl 1204.11145
[14] de Roton, A., Généralisation du critère de Beurling-Nyman pour l’hypothèse de Riemann généralisée, Trans. Amer. Math. Soc., 12, 6111-6126 (2007) · Zbl 1136.11054
[15] de Roton, A., Une approche séquentielle de l’hypothèse de Riemann généralisée, J. Number Theory, 129, 2647-2658 (2009) · Zbl 1232.11087
[16] Delaunay, C.; Fricain, E.; Mosaki, E.; Robert, O., Zero-free regions for Dirichlet series, Trans. Amer. Math. Soc., 365, 3227-3253 (2013) · Zbl 1322.11091
[17] Dimitrov, D. K.; Oliveira, W. D., An extremal problem related to generalizations of the Nyman-Beurling and Báez-Duarte criteria
[18] Iwaniec, H.; Kowalski, E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1059.11001
[19] Lubinsky, D. S., Orthogonal Dirichlet polynomials with arctangent density, J. Approx. Theory, 177, 43-56 (2014) · Zbl 1480.42038
[20] Nikolski, N., Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann \(ζ\)-function, Ann. Inst. Fourier, 45, 143-159 (1995) · Zbl 0816.30026
[21] Nyman, B., On Some Groups and Semigroups of Translation (1950), Thesis, Uppsala · Zbl 0037.35401
[22] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0880.11001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.