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JP Journal of Algebra, Number Theory and Applications

and

it is clear that there exists one and only one

with

such that

Let k be any fixed integer with

denote the number of all k‑tuples with positive integer coordinates

such that

and

is an odd prime and

is an even number, then we let

denote the number of all k-tuples with primitive roots coordinates

such that

and

The main purpose of this paper is to use the estimates of general Kloosterman’s sums and the properties of trigonometric sums to study the asymptotic behavior of the mean values

and

and to give two interesting asymptotic formulae.

JP Journal of Algebra, Number Theory and Applications
Volume 14, Issue 1, Pages 37 - 50 (June 2009)

ON A GENERALIZATION OF D. H. LEHMER PROBLEM
Yi Yuan (P. R. China) and Hou Yiwei (P. R. China)

Abstract:
Let q be an odd number. For each integer x with

and

it is clear that there exists one and only one

with

such that

Let k be any fixed integer with

denote the number of all k‑tuples with positive integer coordinates

such that

and

is an odd prime and

is an even number, then we let

denote the number of all k-tuples with primitive roots coordinates

such that

and

The main purpose of this paper is to use the estimates of general Kloosterman’s sums and the properties of trigonometric sums to study the asymptotic behavior of the mean values

and

and to give two interesting asymptotic formulae.

Keywords and phrases: Lehmer problem, trigonometric sums, asymptotic formula.

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