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A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM*

Published online by Cambridge University Press:  09 December 2011

ZHANG WENPENG
ZHANG WENPENG*
Affiliation:
Department of Mathematics, Northwest University, Xi'an, Shaanxi, P.R. China e-mail: [email protected]
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Abstract

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Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.

Keywords

Primary 11L4011F20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 155 - 162
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

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