arXiv subjects

Sumaia Saad Eddin

Publications and source records attributed to Sumaia Saad Eddin.

At least 19 recordsLinked to original sources

arXiv2026

Weighted averages of Bernoulli polynomials and the Gamma function over regular integers modulo $n$

We investigate weighted averages involving Bernoulli polynomials and the Gamma function over regular integers modulo n. Building upon recent identities of Kiuchi and Matsuoka together with our previous transformation formulas for weighted averages over regular integers, we derive explicit asymptotic formulas for a broad class of arithmetic functions. As applications, asymptotic formulas are obtained for the identity, Möbius, divisor and Jordan totient functions.

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arXiv2026

Weighted averages of arithmetic functions over regular integers modulo $n$

We investigate weighted averages of arithmetic functions over regular integers modulo $n$, extending previous identities of Kiuchi and Matsuoka. We establish general transformation formulas for the associated partial sums corresponding to arbitrary arithmetic functions. These formulas provide a unified framework for deriving asymptotic results. As applications, we obtain explicit asymptotic formulas for weighted averages involving several classical multiplicative functions, including the identity, Möbius, divisor, and Jordan totient functions.

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arXiv2024

Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields

The Euler--Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_K(s)$ at $s=1$. We study the distribution of the Euler--Kronecker constant $γ_q^+$ of the maximal real subfield of $\mathbb Q(ζ_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $γ_q^+-γ_q$, with $γ_q$ the Euler--Kronecker constant of $\mathbb Q(ζ_q)$ and show how it is connected with Kummer's conjecture, which predicts the asymptotic growth of the relative class number of $\mathbb Q(ζ_q)$. We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for $γ_q^+-γ_q$. The methods employed are partly inspired by those used by Granville (1990) and Croot and Granville (2002) to investigate Kummer's conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.

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arXiv2024

The Kummer ratio of the relative class number for prime cyclotomic fields

Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and assuming the Riemann Hypothesis for the Dirichlet $L$-series attached to odd characters only. The numerical work in this paper extends and improves on our earlier preprint (arXiv:1908.01152) and demonstrates our theoretical results.

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arXiv2024

The summatory function of the Möbius function involving the greatest common divisor

Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers $m$ and $n$, and let $μ$ represent the M\" obius function. For any real number $x>5$, we define the summatory function of the M\" obius function involving the greatest common divisor as $ S_μ(x) := \sum_{mn\leq x} μ(\gcd(m,n)). $ In this paper, we present an asymptotic formula for $S_μ(x)$. Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of $S_μ(x)$ and derive a mean square estimate for its error term. Our proof employs the Perron formula, Parseval's theorem, complex integration techniques, and the properties of the Riemann zeta-function.

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arXiv2021

On sums of arithmetic functions involving the greatest common divisor

Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $τ$ and $μ$ denote the divisor function and the Möbius function, respectively. For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we give asymptotic formulas for $S_{τ,k}(x)$ and $S_{μ,k}(x)$ for $k\geq 3$.

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arXiv2021

On the multivariable generalization of Anderson-Apostol sums

In this paper, we give various identities for the weighted average of the product of generalized Anderson-Apostol sums with weights concerning completely multiplicative function, completely additive function, logarithms, the Gamma function, Bernoulli polynomials and binomial coefficients respectively. Our results generalize many known results.

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arXiv2020

On the weighted average number of subgroups of ${\mathbb {Z}}_{m}\times {\mathbb {Z}}_{n}$ with $mn\leq x$

Let $\mathbb{Z}_{m}$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{n}$, respectively. Define $$ \widetilde{D}_{s}(x) = \sum_{mn\leq x}s(m,n)\log\frac{x}{mn} \quad \quad \widetilde{D}_{c}(x) = \sum_{mn\leq x}c(m,n)\log\frac{x}{mn}. $$ In this paper, we study the asymptotic behaviour of functions $\widetilde{D}_{s}(x)$ and $\widetilde{D}_{c}(x)$.

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arXiv2020

Sums of averages of gcd-sum functions II

Let $\gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) $$ for any large real number $x\geq 5$, where $f$ is any arithmetical function. Let $ϕ$, and $ψ$ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $M_r(x; {\rm id})$, $M_r(x;ϕ)$ and $M_r(x;ψ)$. Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $M_r(x;{\rm id})$ for any large positive number $x>5$ satisfying $x=[x]+\frac{1}{2}$.

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arXiv2020

An asymptotic formula for the $2k$-th power mean value of $\left| (L'/L)(1+it_0, χ)\right|$

Let $q$ be a positive integer ($\geq 2$), $χ$ be a Dirichlet character modulo $q$, $L(s, χ)$ be the attached Dirichlet $L$-function, and let $L^\prime(s, χ)$ denote its derivative with respect to the complex variable $s$. Let $t_0$ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $\left|(L^\prime/L)(1+it_0, χ)\right|$ when $χ$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_0=0$).

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arXiv2019

Sums of weighted averages of gcd-sum functions II

In this paper, we establish the following two identities involving the Gamma function and Bernoulli polynomials, namely $$ \sum_{k\leq x}\frac{1}{k^s} \sum_{j=1}^{k^s}\logΓ\left(\frac{j}{k^s}\right) \sum_{\substack{d|k \\ d^{s}|j}}f*μ(d) \quad {\rm and } \quad \sum_{k\leq x}\frac{1}{k^s}\sum_{j=0}^{k^{s}-1} B_{m}\sum_{\substack{d|k \\ d^{s}|j}} f*μ(d) $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We give asymptotic formulas for them with various multiplicative functions $f$. We also consider several formulas of the Dirichlet series associated with the above identities. This paper is a continuation of an earlier work of the authors.

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arXiv2019

Computation of the Kummer ratio of the class number for prime cyclotomic fields

Let $ζ_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(ζ_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{http://www.math.unipd.it/~languasc/rq-comput.html}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.

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arXiv2019

Jordan totient quotients

The Jordan totient $J_k(n)$ can be defined by $J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})$. In this paper, we study the average behavior of fractions $P/Q$ of two products $P$ and $Q$ of Jordan totients, which we call Jordan totient quotients. To this end, we describe two general and ready-to-use methods that allow one to deal with a larger class of totient functions. The first one is elementary and the second one uses an advanced method due to Balakrishnan and Pétermann. As an application, we determine the average behavior of the Jordan totient quotient, the $k^{th}$ normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=1$, the second normalized derivative of the $n^{th}$ cyclotomic polynomial $Φ_n(z)$ at $z=-1$, and the average order of the Schwarzian derivative of $Φ_n(z)$ at $z=1$.

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arXiv2019

On the distribution of products of two primes

For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some similar bias in the RSA integers, but they proved that at least for fixed $r$, there is no such bias. In this paper, we provide an asymptotic formula for the number of the RSA integers available in wider ranges of $r$, and give some observations of the bias of the RSA integers, by interpolating the results of Dummit, Granville, and Kisilevsky and of Moree and the first author.

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arXiv2018

Constrained ternary integers

An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.

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arXiv2018

On sums of logarithmic averages of gcd-sum functions

Let $\gcd(k,j)$ be the greatest common divisor of the integers $k$ and $j$. For any arithmetical function $f$, we establish several asymptotic formulas for weighted averages of gcd-sum functions with weight concerning logarithms, that is $$\sum_{k\leq x}\frac{1}{k} \sum_{j=1}^{k}f(\gcd(k,j)) \log j.$$ More precisely, we give asymptotic formulas for various multiplicative functions such as $f=id$, $ϕ$, $id_{1+a}$ and $ϕ_{1+a}$ with $-1<a<0$. We also establish some formulas of Dirichlet series having coefficients of the sum function $\sum_{j=1}^{k}s_{k}(j)\log j$ where $s_{k}(j)$ is Anderson--Apostol sums.

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arXiv2018

On sums of weighted averages of $\gcd$-sum functions

Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$. In this paper, we give several interesting asymptotic formulas for weighted averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function $\sum_{d|k, d^{s}|j}(f*μ)(d) $ for any positive integers $j$ and $k$, namely $$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad \text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r} \sum_{\substack{d|k d^{s}|j}}(f*μ)(d), $$ with any fixed integer $s> 1$ and any arithmetical function $f$. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of $\gcd$-sum functions $f(\gcd(j,k)). $

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arXiv2017

The signs of the Stieltjes constants associated with the Dedekind zeta function

The Stieltjes constants $γ_n(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $ζ_K(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $γ_n(K)$ as Stieltjes obtained in 1885 for $γ_n(\mathbb Q)$. We also study the signs of $γ_n(K)$.

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