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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

An upper bound for the sum $ \sum\sp {a+H}\sb {n=a+1}f(n)$ for a certain class of functions $ f$


Authors: Edward Dobrowolski and Kenneth S. Williams
Journal: Proc. Amer. Math. Soc. 114 (1992), 29-35
MSC: Primary 11L40
MathSciNet review: 1068118
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Abstract: For a certain class of functions $ f:Z \to C$ an upper bound is obtained for the sum $ \sum\nolimits_{n = a + 1}^{a + H} {f\left( n \right)} $. This bound is used to give a proof of a classical inequality due to Pólya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols $ \sum\nolimits_{x = 1}^H {( ( {R{g^x} + S} )/p} ) $, where $ g$ is a primitive root of the odd prime $ p,1 \leq H \leq p - 1$ and $ RS$ is not divisible by $ p$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1068118-6
PII: S 0002-9939(1992)1068118-6
Keywords: Inequality, character sum, Pólya-Vinogradov inequality
Article copyright: © Copyright 1992 American Mathematical Society