Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Turán inequalities and zeros of Dirichlet series associated with certain cusp forms

Authors: J. B. Conrey and A. Ghosh
Journal: Trans. Amer. Math. Soc. 342 (1994), 407-419
MSC: Primary 11F66; Secondary 11N75
MathSciNet review: 1207582
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11F66, 11N75

Retrieve articles in all journals with MSC: 11F66, 11N75

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society