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

DOI:
https://doi.org/10.1090/S0002-9939-1973-0327682-0

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.

**[1]**K. Chandrasekharan,*Arithmetical functions*, Die Grundlehren der mathematischen Wissenschaften, Band 167, Springer-Verlag, New York-Berlin, 1970. MR**0277490****[2]**P. X. Gallagher,*A large sieve density estimate near 𝜎=1*, Invent. Math.**11**(1970), 329–339. MR**0279049**, https://doi.org/10.1007/BF01403187**[3]**G. Hoheisel,*Primzahl probleme in der Analysis*, S.-B. Preuss. Akad. Wiss.**1930**, 580-588.**[4]**A. E. Ingham,*On the difference between consecutive primes*, Quart. J. Math. Oxford Ser.**8**(1937), 255-266.**[5]**Hugh L. Montgomery,*Topics in multiplicative number theory*, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR**0337847****[6]**Carlos Julio Moreno,*Prime number theorems for the coefficients of modular forms*, Bull. Amer. Math. Soc.**78**(1972), 796–798. MR**0299571**, https://doi.org/10.1090/S0002-9904-1972-13040-4**[7]**-,*Prime number theorems for the coefficients of modular forms and a problem of G. H. Hardy*(to appear).**[8]**-,*A density estimate for the Ramanujan zeta function*(to appear).

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0327682-0

Keywords:
Hoheisel phenomenon,
Dirichlet series,
Ramanujan functions ,
Littlewood theorem

Article copyright:
© Copyright 1973
American Mathematical Society