Second Moment Of Dirichlet <i>L</i>-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers

Upper bound estimate of character sums over Lehmer's numbers

*Correspondence: [email protected] Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China Abstract Let p be an odd prime. For each integer awith 1≤ a≤ p – 1, it is clear that there exists one and only one awith 0≤ a≤ p – 1 such that a · a≡ 1 mod p. LetA denote the set of all integers 1≤ a≤ p – 1, in which a and a are of opposite parity. The main purpose of this paper is us...

Double Character Sums over Subgroups and Intervals

We estimate double sums Sχ(a, I,G) = ∑

On Certain Character Sums over Smooth Numbers

Quadratic class numbers and character sums

We present an algorithm for computing the class number of the quadratic number field of discriminant d. The algorithm terminates unconditionally with the correct answer and, under the GRH, executes in Oε(|d|) steps. The technique used combines algebraic methods with Burgess’ theorem on character sums to estimate L(1, χd). We give an explicit version of Burgess’ theorem valid for prime discrimin...

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulae and estimates for character sums of the type S(χ, f, pm) = ∑pm x=1 χ(f(x)), where p m is a prime power with m ≥ 2, χ is a multiplicative character (mod pm), and f = f1/f2 is a rational function over Z. In particular, if p is odd, d = deg(f1) + deg(f2) and d∗ = max(deg(f1), deg(f2)) then we obtain |S(χ, f, pm)| ≤ (d− 1)pm(1− 1 d∗ ) for any non constant f (mod p) and primitive ...