Upper bound estimate of incomplete Cochrane sum

Abstract By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.


Introduction
Let q be a positive integer, then for an arbitrary integer h, the famous Dedekind sum S.h; q/ is defined as where ..x// D ( x OEx 1 2 ; if x is not an integer, 0; if x is an integer.
In October 2000, during his visit to Xi'an, Todd Cochrane introduced a sum analogous to it as follows: where q X 0 aD1 denotes the summation over all 1 6 a 6 q such that .a; q/ D 1, and a is defined by aa Á 1.mod q/.
Since then, various properties of C.h; q/ are studied by many scholars. For example, Zhang and Yi [1] obtained the upper bound estimate jC.h; q/j p qd.q/ ln 2 q; where d.q/ is the classical divisor function. For the case q D p; an odd prime, they also gave a sharp asymptotic formula where .q/ denotes the Euler function and Y p˛kq the product over all prime divisors of q with p˛j q and p˛C 1 − q.
Later Lu and Yi [3] gave the mean square value of C.h; q/ over incomplete intervals. In fact, under the conditions that q > 3 is a square-free integer and ı a real number with ı 2 .0; 1, they got where !.q/ D X pjq 1, " is a sufficiently small positive constant and the O constant depends only on ".
For arbitrary integers m and n, Estermann [9] gave an upper bound estimate of the classical Kloosterman sum S.m; nI q/ as jS.m; nI q/j 6 .m; n; q/ (2) where S.m; nI q/ is defined by where e.x/ D e 2 ix ; and .m; n; q/ denotes the greatest common divisor of m, n, q. By completing method, one can derive immediately from (2) where I is an interval with length not exceeding q. Now we define an incomplete Cochrane sum as follows: where 2 .0; 1: By using the properties of Kloosterman sum and Dirichlet character, we shall prove the following: Theorem. Let q; h be integers with q 2 and .h; q/ D 1, be a real number with 2 .0; 1: Then we have the upper bound estimate jC.h; qI /j q 1 2 C" : Taking D 1 in Theorem, we may immediately obtain Corollary. Let q; h be integers with q 2 and .h; q/ D 1. Then we have which is almost the same estimate as (1).

Some lemmas
To prove Theorem, we need the following several lemmas.
Lemma 1.1. Let q; h be integers with q 2 and .h; q/ D 1, be a real number with 2 .0; 1: Then we have the identity where denotes a Dirichlet character modulo q, G. ; mI / D Ã denotes the partial Gauss sum corresponding to , and G. ; m/ WD G. ; mI 1/.
Proof. From (4) and the orthogonality relation for characters modulo q, we have From these identities, we have So we have .a/e .a/e nab C mb q ! : By the orthogonality relation for characters modulo q as the following:  we have where we have used the upper bound of (3). Then from the estimates for trigonometric sum and Gauss sum we can also get jA.y; /j DˇX Then we shall estimate M 3 . Since  Similarly, we can also get X mod q . This completes the proof of Theorem.