Reducing character sums to Kloosterman sums

We apply a bound for very short Kloosterman type sums, due to A. A. Karatsuba, to deduce a bound for a mean-value of short sums of Dirichlet characters.     

Large character sums

We make conjectures and give estimates for how large character sums can be as we vary over all characters mod , and as we vary over real, quadratic characters. In particular we show that the largest sums seem to depend on the value of the character at ``smooth numbers'.

     

On twisted character sums

In this paper we give a simple proof of a result by Burgess about short sums involving Dirichlet characters and exponentials. Indeed we establish a slightly stronger and more general bound that applies to sums of the form \({\sum_{n=M+1}^{M+N}f(\alpha n)\chi(n)}\), where χ is a non-principal character to the modulus p and f is a smooth 1-periodic function.     

Estimates for character sums in number fields

We obtain asymptotic formulae for the moments of coefficients of Artin-WeilL-functions (defined by equation (1)). As a by-product of our investigations, we strengthen and generalise the results of several authors cited in the references.     

Kloosterman sums and traces of singular moduli

We give a new proof of some identities of Zagier relating traces of singular moduli to the coefficients of certain weakly holomorphic half integral weight modular forms. These identities play a central role in Zagier's work on the infinite product isomorphism introduced by Borcherds. In addition, we derive a simple expression for writing twisted traces of singular moduli as infinite series.     

Degenerate and character Dedekind sums

In this paper, we study on two subjects. We first construct degenerate analogues of Dedekind sums in the sense of Apostol, Carlitz and Takács, and prove the corresponding reciprocity formulas. Secondly, we define generalized Dedekind character sums, which are explicit extensions of Berndt's definition, and prove the reciprocity laws. From the derived reciprocity laws, we obtain Berndt's reciprocity laws as special cases.     

Cubefree ideal modulus character sums

Archiv der Mathematik -     

On character sums with determinants

We estimate weighted character sums with determinants ad-bc of 2×2 matrices modulo a prime p with entries a,b,c and d vary ing over the interval [1,N].Our goal is to obtain non-trivial bounds for values of N as small as possible.In particular,we achieve this goal,with a power saving,for N≥p^1/8+ε with any fixed ε>0,which is very likely to be the best possible unless the celebrated Burgess bound is improved,By other techniques,we also treat more general sums but sometimes for larger ...     

A note on character sums

We investigate certain character sums and prove some discrepancy-type inequalities for incomplete sums.      

On the character degree sums

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we get the exact lower bound of |G|∕T(G) for a non-r-solvable group G.     

A note on character sums

Let $\text{S(k) = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{p?1}}\text{(}\text{n}\text{p}\text{)n}{}^{\mathrm{k}}$ where p is a prime ≡ 3 mod 4 and k is an integer ≥ 3. Then S(k) frequently takes large values of each sign.     

Proximity for sums of composite functions

We propose an algorithm for computing the proximity operator of a sum of composite convex functions in Hilbert spaces and investigate its asymptotic behavior. Applications to best approximation and image recovery are described.     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

Riemann sums for stochastic integrals and L p moduli of continuity

Summary The Brownian stochastic integral for a deterministic function f can be approximated by Riemann sums . Here is a partion of [0,1], and f is obtained by replacing f by its mean value in each interval of . In an earlier paper [3], the author determined which Lipschitz conditions on f imply f db f db a.s. as becomes fine. This is now done for L ^p Lipschitz conditions. The results are stated in terms of Besov spaces. Nonanticipating are considered when p = l. Further, we determine on which L ^p Lipschitz spaces f db defines a bounded linear functional.     

A hybird mean value involving gauss sums and character sums

Let χ be a Dirichlet character modulo q > 2, and L(s, χ) denotes the Dirichlet L-function corresponding to χ. The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value properties of $ \frac{{L'}} {L}(1,\chi ) $ \frac{{L'}} {L}(1,\chi ) with the weight of Gauss sums and character sums, and give an interesting mean value formula for it.     

Short Kloosterman sums with weights

We obtain a new estimate for Kloosterman sums with weights in which the number of summands is significantly less than any arbitrarily small fixed power of the modulus.     

Quadratic class numbers and character sums

We present an algorithm for computing the class number of the quadratic number field of discriminant . The algorithm terminates unconditionally with the correct answer and, under the GRH, executes in steps. The technique used combines algebraic methods with Burgess' theorem on character sums to estimate . We give an explicit version of Burgess' theorem valid for prime discriminants and, as an application, we compute the class number of a 32-digit discriminant.

     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form