Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.01363

Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums

László Tóth
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce new analogues of the Ramanujan sums, denoted by $\widetilde{c}_q(n)$, associated with unitary divisors, and obtain results concerning the expansions of arithmetic functions of several variables with respect to the sums $\widetilde{c}_q(n)$. We apply these results to certain functions associated with $σ^*(n)$ and $ϕ^*(n)$, representing the unitary sigma function and unitary phi function, respectively.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

László Tóth. 2018-06-11. Expansions of arithmetic functions of several variables with respect to certain modified unitary Ramanujan sums. https://arxiv.org/abs/1705.01363

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the Pair Correlation of Zeros of $L$-Functions for Non-CM Newforms in Shifted Ranges

We study the pair correlation between zeros of a shifted auxiliary $ L $-function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional asymptotic result for the pair correlation and introduce a simplicity hypothesis for the zeros of this function, which if true means that multiple zeros of the original $ L $-function cannot be separated by the same fixed distance. Our results provide macroscopic information in contrast to the pair correlation of the original $ L $-function which is of microscopic nature.

math.NT↗
arXiv2026

Prime Solutions to a Binary Additive Equation and Mixed Moments of Character Sums

We obtain an asymptotic formula with a power-saving error term for counting the integer points $(a,b,c,d)$ in an expanding box that satisfy the determinant equation $x_1x_2-x_3x_4 =r$ for $r \neq 0 $ with two of entries to be prime. Finally, these estimates are applied to evaluate mixed fourth moments of Dirichlet character sums over integers and primes, yielding non-trivial bounds. The method involves the Poisson summation formula and the estimation for the average of the sums of the Kloosterman fractions over primes.

math.NT↗
arXiv2026

On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

Given a maximal order $\mathcal{D}$ of a central division algebra $D$ over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $\mathcal{D}^\times$-shtukas to be proper over a finite field (modulo a suitable central action) in terms of the \emph{local invariants} of $D$ and \emph{bounds}. Our proof is a refinement of E.~Lau's result (Duke Math. J. \textbf{140} (2007)), which showed the properness of the \emph{leg morphism} (or \emph{characteristic morphism}) away from the ramification locus of $D$. %, by carefully measuring the contribution of ``ramified legs''. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $\mathcal{B}^\times$-shtukas, where $\mathcal{B}$ is a maximal order of a central simple algebra over $F$.

math.NT↗