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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.
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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.

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

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References

1.Huxley, M. N.. The distribution of prime numbers (Clarendon Press, Oxford, 1972).Google Scholar
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