On Some Weighted Average Values of L-functions

Let $q\ge 2$ and $N\ge 1$ be integers. W. Zhang (2008) has shown that for any fixed $\epsilon> 0$, and $q^{\epsilon} \le N \le q^{1/2 -\epsilon}$, $$\sum_{\chi \ne \chi_0} |\sum_{n=1}^N \chi(n)|^2 |L(1, \chi)|^2 = (1 + o(1)) \alpha_q q N$$ where the sum is take over all nonprincipal characters $\chi$ modulo $q$, $L(s, \chi)$ is the $L$-functions $L(1, \chi)$ corresponding to $\chi$ and $\alpha_q = q^{o(1)}$ is some explicit function of $q$. Here we show that the same formula holds in the range $q^{\epsilon} \le N \le q^{1 -\epsilon}$.Comment: Bull. Aust. Math. Soc. (to appear

Multiple Exponential and Character Sums with Monomials

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables

Cayley graphs generated by small degree polynomials over finite fields

We improve upper bounds of F. R. K. Chung and of M. Lu, D. Wan, L.-P. Wang, X.-D. Zhang on the diameter of some Cayley graphs constructed from polynomials over finite fields

Additive Decompositions of Subgroups of Finite Fields

We say that a set $S$ is additively decomposed into two sets $A$ and $B$, if $S = \{a+b : a\in A, \ b \in B\}$. Here we study additively decompositions of multiplicative subgroups of finite fields. In particular, we give some improvements and generalisations of results of C. Dartyge and A. Sarkozy on additive decompositions of quadratic residues and primitive roots modulo $p$. We use some new tools such the Karatsuba bound of double character sums and some results from additive combinatorics

Evasive Properties of Sparse Graphs and Some Linear Equations in Primes

We give an unconditional version of a conditional, on the Extended Riemann Hypothesis, result of L. Babai, A. Banerjee, R. Kulkarni and V. Naik (2010) on the evasiveness of sparse graphs.Comment: This version corrects a mistake made in the previous version, which was pointed out to the author by Laszlo Baba

On Small Solutions to Quadratic Congruences

We estimate the deviation of the number of solutions of the congruence $$m^2-n^2 \equiv c \pmod q, \qquad 1 \le m \le M, \ 1\le n \le N,$$ from its expected value on average over $c=1, ..., q$. This estimate is motivated by the recently established by D. R. Heath-Brown connection between the distibution of solution to this congruence and the pair correlation problem for the fractional parts of the quadratic function $\alpha k^2$, $k=1,2,...$ with a real $\alpha$

On Solutions to Some Polynomial Congruences in Small Boxes

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solution in boxes with the side length below $p^{1/2}$, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties

On Point Sets in Vector Spaces over Finite Fields That Determine Only Acute Angle Triangles

For three points $\vec{u}$,$\vec{v}$ and $\vec{w}$ in the $n$-dimensional space \F_q^n over the finite field \F_q of $q$ elements we give a natural interpretation of an acute angle triangle defined by this points. We obtain an upper bound on the size of a set \cZ such that all triples of distinct points \vec{u}, \vec{v}, \vec{w} \in \cZ define acute angle triangles. A similar question in the real space \cR^n dates back to P. Erd{\H o}s and has been studied by several authors