Estimates for Character Sums with Various Convolutions

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $\chi$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)\chi(n+a)\ll {N\over q^{1\over 4}}\log\log(6N)+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}. $$ We shall also prove that \begin{align*} &\sum_{n\leq N}f(n)\chi(n+a_1)\cdots\chi(n+a_t)\ll {N\over q^{1\over ...

The Burgess inequality is the best upper bound we have for the character sum $S_{\chi}(M,N) = \sum_{M<n\le M+N} \chi(n).$ Until recently, no explicit estimates had been given for the inequality. In 2006, Booker gave an explicit estimate for quadratic characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than $10^{140}$. Both of t...

We estimate mixed character sums of polynomial values over elements of a finite field $\mathbb F_{q^r}$ with sparse representations in a fixed ordered basis over the subfield $\mathbb F_q$. First we use a combination of the inclusion-exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large $q$. Then we focus on the particular case $q=2$, which is more intricate. The bounds depend on certain natural restrictions. We also provid...

In this paper we consider a variety of mixed character sums. In particular we extend a bound of Heath-Brown and Pierce to the case of squarefree modulus, improve on a result of Chang for mixed sums in finite fields, we show in certain circumstances we may improve on some results of Pierce for multidimensional mixed sums and we extend a bound for character sums with products of linear forms to the setting of mixed sums.

Let $\chi$ be a Dirichlet character modulo $p$, a prime. In applications, one often needs estimates for short sums involving $\chi$. One such estimate is the family of bounds known as \emph{Burgess' bound}. In this paper, we explore several minor adjustments one can make to the work of Enrique Trevi\~no on explicit versions of Burgess' bound. For an application, we investigate the problem of the existence of a $k$th power non-residue modulo $p$ which is less than $p^\alpha$ f...

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski. In particular, for any nontrivial multiplicative character $\chi$ modulo a prime $q$ and any integer $r\ge 2$, we show that $$ \sum_{M<n\le M+N}\chi(n) = O\left( N^{1-1/r}q^{(r+1)/4r^2}(\log q)^{1/4r}\right), $$ which sharpens previous resu...

We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X] $ and some complex weights $\{\alpha_u\}$, $\{\beta_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bo...

In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of $\mathbb A^n_k$, which generalize the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

In this paper, we prove a lower bound for $\underset{\chi \neq \chi_0}{\max}\bigg|\sum_{n\leq x} \chi(n)\bigg|$, when $x= \frac{q}{(\log q)^B}$. This improves on a result of Granville and Soundararajan for large character sums when the range of summation is wide. When $B$ goes to zero, our lower bound recovers the expected maximal value of character sums for most characters.