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Proceedings of the American Mathematical Society

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The Hoheisel phenomenon for generalized Dirichlet series

Author: Carlos Julio Moreno
Journal: Proc. Amer. Math. Soc. 40 (1973), 47-51
MSC: Primary 10H10; Secondary 10H25
MathSciNet review: 0327682
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Abstract: Hoheisel's proof that the difference between two consecutive primes is of smaller order of magnitude than either prime depends on Littlewood's estimate for the zero-free region of the Riemann zeta function and a density estimate for the number of zeros in certain rectangles in the critical strip. In this note we derive Hoheisel's result without appealing to Littlewood's theorem, thus enlarging the range of applicability of Hoheisel's argument to a more general class of Dirichlet series. Applications of the results to number theory are given.

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Keywords: Hoheisel phenomenon, Dirichlet series, Ramanujan functions $ \tau (n)$, Littlewood theorem
Article copyright: © Copyright 1973 American Mathematical Society

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