Related papers: Improving the Burgess bound via Polya-Vinogradov

Our objective in this paper is to introduce and investigate a newly-constructed subclass of normalized analytic and bi-univalent functions by means of the Chebyshev polynomials of the second kind. Upper bounds for the second and third…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

In this work, we undertake a comprehensive reformulation, modification, and extension of Smith and Spiegelhalter's (1980) and (1982) Bayes Factor work within the evolving subject of Objective Bayes Factors. Our primary focus centers on…

We refine and extend quantitative bounds, on the fraction of nonnegative polynomials that are sums of squares, to the multihomogenous case.

We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of \psi(y, \chi), the twisted summatory function associated to the von Mangoldt function \Lambda and a Dirichlet character \chi.…

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\;…

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered.…

We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_\nu} = A^{x_{\nu+1}} + \ldots + A^{x_{2\nu}}, \quad 1 \le x_1, \ldots,x_{2\nu} \le \tau, $$ with a fixed $n\times n$ matrix $A$ over a finite…

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set…

In this paper, we prove capacitary versions of the fractional Sobolev--Poincar\'e inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincar\'e inequalities through uniform fatness condition of the domain in…

We consider the series expansion of the $L^p$-Hardy inequality of \cite{BFT2}, in the particular case where the distance is taken from an interior point of a bounded domain in $\mathbb{R}^n$ and $1<p\neq n$. For $p<n$ we improve it by…

In this paper, we investigate the size of moments of quadratic character sums averaged over the family of fundamental discriminants. We obtain an asymptotic formula for all integer moments in a restricted range of parameters using a…

In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive…

We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.

Let $g$ be a fixed Hecke cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of conductor $M$. The best known subconvex bound for $L(1/2,g\otimes \chi)$ is of Burgess strength. The bound was proved by a…

We obtain new effective results in best approximation theory, specifically moduli of uniqueness and constants of strong unicity, for the problem of best uniform approximation with bounded coefficients, as first considered by Roulier and…

We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear…

We show that a short truncation of the Fourier expansion for a character sum gives a good approximation for the average value of that character sum over an interval. We give a few applications of this result. One is that for any $b$ there…

The polynomials of degree $\frac{p-1}{2}$ of range sum $p$ was determined in {\tt arXiv:2311.06136 [math.NT]} for large enough primes. We extend this result by reducing the lower bound for the primes to $23$ by introducing a new and…