Related papers: A note on exponential-M\"{o}bius sums over $\mathb…

In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an…

We extend bounds, proved by R.C. Thompson in 1966, on the sum of the $j$-th largest eigenvalues of the $(n-1) \times (n-1)$ principal matrices of an $n \times n$ Hermitian matrix. Our bounds are stronger than just summing up Thompson's…

We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…

We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points on hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of…

In this paper, we develop some basic techniques towards the Riemann hypothesis for higher rank non-abelian zeta functions of an integral regular projective curve of genus $g$ over a finite field $\mathbb F_q$. As an application of the…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds under…

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli.

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this…

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index…

In this paper we prove new upper bounds for the sum $\sum_{n=a+1}^{a+N}f(n)$, for a certain class of arithmetic functions $f$. Our results improve the previous results of G. Bachman and L. Rachakonda.

Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) =…

In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}\tau(n) e(\alpha n)$ where $\tau(n)$ is the divisor function.

We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

Let $\mu(t) = \sum_{\tau\in S} \alpha_\tau \delta(t-\tau)$ denote an $|S|$-atomic measure defined on $[0,1]$, satisfying $\min_{\tau\neq \tau'}|\tau - \tau'|\geq |S|\cdot n^{-1}$. Let $\eta(\theta) = \sum_{\tau\in S} a_\tau D_n(\theta -…

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$.…

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic…

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as…

A well known result of Newman says that upto a limit, multiples of $3$ with even number of 1's in binary representation always exceed multiples of $3$ with odd number of 1's. The phenomenon of preponderance of even number of 1's is now…