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Truncated convolution of Möbius function and multiplicative energy of an integer $n$Mar 13 2019We establish an interesting upper bound for the moments of truncated Dirichlet convolutions of M\"obius functions, a function noted $M(n,z)$. Our result implies that $M(n,j)$ is usually quite small for $j \in \{1,\dots,n\}$. Also, we establish an estimate ... More

Improving an inequality for the divisor functionNov 16 2017Jan 12 2018We improve using elementary means an explicit bound on the divisor function due to Friedlander and Iwaniec. Consequently we modestly improve a result regarding a sieving inequality for Gaussian sequences.

The Prime Grid. Introducing a geometric representation of natural numbersNov 08 2017In this report we present an off-the-number-line representation of the positive integers by expressing each integer by its unique prime signature as a grid point of an infinite dimensional space indexed by the prime numbers, which we term the prime grid. ... More

Moyennes effectives de fonctions multiplicatives complexesSep 05 2017Jan 22 2019We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several applications are ... More

Generalised divisor sums of binary forms over number fieldsSep 13 2016Sep 16 2016Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, ... More

Moments of averages of generalized Ramanujan sumsAug 07 2015Let $\beta$ be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by \begin{align} c_{q,\beta }(n) := \sum\limits_{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \nonumber \end{align} where $h$ ranges over the ... More

On the number of prime factors of values of the sum-of-proper-divisors functionMay 14 2014Sep 14 2015Let $\omega(n)$ (resp. $\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\omega(n)$ is $\log\log n$, and the same is true of $\Omega(n)$; roughly ... More