Turán inequalities and zeros of Dirichlet series associated with certain cusp forms
HTML articles powered by AMS MathViewer
- by J. B. Conrey and A. Ghosh PDF
- Trans. Amer. Math. Soc. 342 (1994), 407-419 Request permission
Abstract:
The "Turan inequalities" are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros. We give a one-parameter family of generalized Dirichlet series, each with functional equation, for which the Turan inequalities hold for the associated $\xi$-function (normalized so that the critical line is the real axis). For a discrete set of values of the parameter the Dirichlet series has an Euler product and is the L-series associated to a modular form. For these we expect the analogue of the Riemann Hypothesis to hold. For the rest of the values of the parameter we do not expect an analogue of the Riemann Hypothesis. We show for one particular value of the parameter that the Dirichlet series in fact has zeros within the region of absolute convergence.References
- Ronald Alter, On a necessary condition for the validity of the Riemann hypothesis for functions that generalize the Riemann zeta function, Trans. Amer. Math. Soc. 130 (1968), 55–74. MR 218312, DOI 10.1090/S0002-9947-1968-0218312-X
- J. W. S. Cassels, Footnote to a note of Davenport and Heilbronn, J. London Math. Soc. 36 (1961), 177–184. MR 146359, DOI 10.1112/jlms/s1-36.1.177
- George Csordas, Timothy S. Norfolk, and Richard S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), no. 2, 521–541. MR 846596, DOI 10.1090/S0002-9947-1986-0846596-4
- H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, J. London Math. Soc. 21 (1946), 185–193. MR 20578, DOI 10.1112/jlms/s1-21.3.185
- N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226. MR 37351, DOI 10.1215/S0012-7094-50-01720-0 J. Grommer, Ganze transcendente Funktionen mit lauter reellen Nullstellen, J. Reine Angew. Math. 144 (1914), 114-165.
- E. Grosswald, Generalization of a formula of Hayman and its application to the study of Riemann’s zeta function, Illinois J. Math. 10 (1966), 9–23. MR 186797, DOI 10.1215/ijm/1256055197
- James Lee Hafner, Explicit estimates in the arithmetic theory of cusp forms and Poincaré series, Math. Ann. 264 (1983), no. 1, 9–20. MR 709858, DOI 10.1007/BF01458047 G. H. Hardy, Ramanujan, 3rd ed., Chelsea, New York, 1978.
- Yu. V. Matiyasevich, Yet another machine experiment in support of Riemann’s conjecture, Kibernetika (Kiev) 6 (1982), 10, 22 (Russian, with English summary); English transl., Cybernetics 18 (1982), no. 6, 705–707 (1983). MR 716432, DOI 10.1007/BF01069156 G. Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6-18.
- Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
- Robert Spira, The integral representation for the Riemann $\Xi$-function, J. Number Theory 3 (1971), 498–501. MR 282930, DOI 10.1016/0022-314X(71)90016-3
- 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 A. Wintner, A note on the Riemann $\xi$-function, J. London Math. Soc. 10 (1935), 82-83.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 342 (1994), 407-419
- MSC: Primary 11F66; Secondary 11N75
- DOI: https://doi.org/10.1090/S0002-9947-1994-1207582-2
- MathSciNet review: 1207582