Related papers: Quadratic character sums with multiplicative coeff…

There are two main interrelated goals of this paper. Firstly we investigate the sums \[ S_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{n\|n\alpha-\gamma\|}~~~\text{and}~~~ R_N(\alpha,\gamma):=\sum_{n=1}^N\frac{1}{\|n\alpha-\gamma\|}\,, \] where…

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums…

Let $q$ be an odd prime power and let $\F_q$ be the finite field with $q$ elements. Let $\mathcal{P}(n)$ be the set of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$. For $f=t^n+f_{n-1}t^{n-1}+\cdots+f_0\in\mathcal{P}(n)$,…

Let $\UT_n(q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of $\UT_n(q)$ are precisely the integers $q^e$ with $0\leq e\leq \lfloor…

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds…

Recent work has introduced the study of graphical properties of cyclic supercharacters, functions $\mathbb{Z}/n\mathbb{Z}\to \mathbb{C}$ whose values are exponential sums with close connections to Gauss sums and Gaussian periods. Plots of…

We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…

In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} \lambda_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\Big ( \frac{8d}{\cdot}\Big )$ is the…

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with…

In this note we report about a method to deal with finite energy sum rules. With a reasonable knowledge of the main resonances of the spectrum, the method guarantees that we can find a nice duality matching between the low energy hadronic…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

We show that certain sums studied in two recent papers are basically character coordinates (as they are called in the literature). These sums involve values of Dirichlet characters and powers of $\cot(\pi k/n)$, $1\le k\le n-1$. We also…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

In this paper, we establish joint extreme values of Dirichlet (L)-functions and their logarithmic derivatives using the resonance method. Our results extend previous work of Aistleitner et al. (2019) and Yang (2023).

Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g :…

In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and…

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…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss…