Abstract
The main purpose of this paper is applying the analysis method, the properties of Lucas polynomials and Gauss sums to study the estimation problems of some kind hybrid character sums. In the end, we obtain several sharp upper bound estimates for them. As some applications, we prove some new and interesting combinatorial identities.
Highlights
As usual, let q ≥ 3 be an integer, χ denotes any Dirichlet character mod q
The classical Gauss sum τ (χ) is defined by q a τ (χ) = χ(a)e, q a=1 where e(y) = e2πiy. We know that this sum plays a very important role in analytic number theory; plenty of number theory problems are closely related to it
Concerning the various elementary properties of τ (χ), some authors studied it and obtained a series of interesting results, some conclusions can be found in Refs. [1] and [2]
Summary
Let q ≥ 3 be an integer, χ denotes any Dirichlet character mod q. The classical Gauss sum τ (χ) is defined by q a τ (χ) = χ(a)e , q a=1 where e(y) = e2πiy We know that this sum plays a very important role in analytic number theory; plenty of number theory problems (such as Dirichlet L-functions and distribution of primes) are closely related to it. We shall use the analytic method, the properties of Lucas polynomials and Gauss sums to do research on these problems, and obtain some sharp upper bound estimates for them. Theorem 1 If q is an integer with q > 2, for any positive integer n and primitive character χ mod q, we have the estimate (a) q–1 χ (a) cos2n. Theorem 2 If q is an integer with q > 2, for any positive integer n and primitive character χ mod q, we have the estimate (c) q–1 χ (a) sin2n (d) q–1 χ (a) sin2n–1. The estimates in our Theorem 1 and Theorem 2 are the best
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