Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T20:40:10.227Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distribution of αp modulo 1

Published online by Cambridge University Press:  26 February 2010

R. C. Vaughan
R. C. Vaughan
Affiliation:
Imperial College, London, S.W.7.
Get access

Extract

In [4] we have given a simple method of estimating trigonometrical sums over prime numbers. Here we show how the argument can be adapted in order to give estimates for the distribution of αp modulo 1 which are sharper than those obtained by I. M. Vinogradov [5], [6]. Vinogradov uses the sieve of Eratosthenes to relate the sum

to the bilinear form

the function μ being the Mobius function. When d1 … ds is small compared with N this can be treated in a fairly straightforward manner. However, in order to treat the terms with d1 …ds close to N, Vinogradov has to introduce an argument of a rather recondite combinatorial nature.

Type
Research Article
Information
Mathematika , Volume 24 , Issue 2 , December 1977 , pp. 135 - 141
Copyright
Copyright © University College London 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Huxley, M. N.. The distribution of prime numbers (Clarendon Press, Oxford, 1972).Google Scholar
2.Montgomery, H. L.. Topics in multiplicative number theory, Lecture notes in mathematics, volume 227 (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
2.Vaughan, R. C.. “Mean value theorems in prime number theory”, J. London Math. Soc. (2), 10 (1975), 153–162.CrossRefGoogle Scholar
3.Vaughan, R. C.. “Sommes trigonométriques sur les nombres premiers”, Comptes Rendus Acad. Sc. Paris, Série A, to appear.Google Scholar
4.Vinogradov, I. M.. The method of trigonometrical sums in the theory of numbers, translated from the Russian, revised and annotated by Roth, K. F. and Davenport, A. (Interscience Publishers, 1954).Google Scholar
6.Vinogradov, I. M.. “An elementary proof of a theorem from the theory of prime numbers”, Izvestia Akad. Nauk, SSSR Ser. Mat., 17 (1953), 3–12.Google Scholar