Estimates For Certain Types Of Character Sums

It is well known that the estimate for character sum (?)χ(n) plays an importantrole in the proof of zero-free region for Dirichlet L-functions. The most frequently applied result is the Polya-Vinogradov bound(?) forχmod q non-principaland it leads immediately to the Dirichlet Theorem that there exist infinitely many primes in arithmetic progressions. In this paper, certain types of character sums will be under discussion and some new estimates will be proved.We first introduce some results related to our subject. Ifχis a primitive character modulo q, Sokolovski(?) [11] proved the existence of x satisfying the following inequalitywhere [y] denotes the greatest integer less than or equal to y. It implies that in general case, the Polya-Vinogradov bound cannot be substantially improved except the log q factor. For non-principal character, Burgess obtained the mean value estimate of character sumswhere h is any positive integer. This was conjectured by Norton [8], who obtained the weaker upper bound 9/8k·h. Zhefeng Xu and other authors in [13] and [14] studied the 2k-th power mean of the even and odd primitive character sums over the quarter interval [1,q/4) by using mean value theorems of Dirichlet L-functions, and obtained two asymptotic formulas. Following a result of Garaev and Karatsuba [5] and combining other materials, we will deal with character sums (?), especially when k is not large or N is relatively small compared with q. We have reached the following results:Theorem 1. Assuming that (?)＜N＜q for some (?)＞0, we haveif N＜(?) for any (?)＞0,it holds that (?).Theorem 2. For a prime p and integer k≥2, we haveAnd for character sums twisted by exponential functions, using analytic methods, we get the following three theorems:Theorem 3. Let q≥1 and x≥1. Letα,β∈(0,1]. Thenwhere T₀ = 1+|α|x^β.Theorem 4. Let q≥2 and x≥1. Letα,β∈(0,1]. Then for Q≥1,where T₀=1+|α|x^β.Theorem 5. Let integers k≥1,q≥2. Let x≥1. Suppose thatα=a/r+λwith (a,r) = 1,λ∈R. Thenwhere T₁=1+|λ|x^k.