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Sumaia Saad Eddin

Publications and source records attributed to Sumaia Saad Eddin.

24 records · Page 2Linked to original sources
arXiv2017

Applications of the Laurent-Stieltjes constants for Dirichlet $L$-series

The Laurent Stieltjes constants $γ_n(χ)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $χ$ is non principal, $(-1)^nγ_n(χ)$ is simply the value of the $n$-th derivative of $L(s,χ)$ at $s=1$. In this paper, we give an approximation of the Dirichlet L-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $ζ(s)$ has no zeros in the region $|s-1|\leq 2.2093,$ with $0\leq \Re{(s)}\leq 1.$

math.NT↗
arXiv2016

On Weyl products and uniform distribution modulo one

In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(πx_k)$ for $N \to \infty$, where the numbers $ω=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $ω$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $ω$ are the initial segment of a Kronecker or van der Corput sequence. The paper concludes with some probabilistic analogues.

math.NT↗
arXiv2016

Products of two proportional primes

In RSA cryptography numbers of the form $pq$, with $p$ and $q$ two distinct proportional primes play an important role. For a fixed real number $r>1$ we formalize this by saying that an integer $pq$ is an RSA-integer if $p$ and $q$ are primes satisfying $p<q\le rp$. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p, q$ both prime, satisfy $p\equiv q\equiv 3\pmod{4}$. In this paper we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers $\le x$ which is much more precise than an earlier one derived by Decker and Moree in 2008.

math.NT↗
arXiv2015

An asymptotic distribution for $\left|L^\prime/L(1,χ)\right|$

Let $χ$ be a Dirichlet character modulo $q$, let $L(s, χ)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, χ)$ denotes its derivative with respect to the complex variable $s$. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $\left|L^\prime/L(1, χ)\right|$ when $χ$ ranges a primitive Dirichlet character modulo $q$ for $q$ prime. We derive some consequences, in particular a bound for the number of $χ$ such that $\left|L^\prime/L(1, χ)\right|$ is large.

math.NT↗