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Abdelmejid Bayad

Publications and source records attributed to Abdelmejid Bayad.

7 recordsLinked to original sources
arXiv2026

Stieltjes constants for Hurwitz-type L-series attached to periodic arithmetic functions

We investigate generalized Stieltjes constants attached to Hurwitz-type Dirichlet series of the form $$ L(s, f, x)=\sum_{n \geq 0} \frac{f(n)}{(n+x)^s} $$ where $f$ is a periodic arithmetic function and $x\in\mathbb C\backslash\{0,-1,-2,\dots\}.$ We prove that these series admit meromorphic continuation to the whole complex plane, with at most a simple pole at $s=1$. We further derive functional equations satisfied by $L(s,f,x).$ Our main result gives an explicit formulas of the Stieltjes constants $γ_n(f, x)$ appearing in the Laurent expansion at $s=1$. In particular, we obtain explicit evaluations in the case of rational $x$. As an application, we derive explicit closed-form expressions for the first three generalized Stieltjes constants associated with nonprincipal real primitive odd Dirichlet characters arising from quadratic fields. These expressions are given in terms of the class number of the corresponding quadratic number field. These results extend and unify recent work of Blagouchine and others on generalized Stieltjes constants and Hurwitz-type $L$-functions.

math.NT↗
arXiv2026

Appell Sequences in Analytic Number Theory: Shifted Expansions and Dirichlet L-Functions

We place the Appell shift theorem for asymptotic expansions inside a Borel--Laplace framework and apply it to two classical functions of analytic number theory: the Hurwitz zeta function \(ζ(m,q)\) and the Lerch transcendent \(Φ(z,m,a)\). The resulting shifted expansions have coefficients given by Appell polynomials evaluated at the shift, with remainders controlled uniformly on compact sets. The generating sequences are identified explicitly in terms of the Taylor coefficients of the Borel kernels. In a second direction, we extend the Appell--Mellin framework of Navas, Ruiz and Varona to Dirichlet \(L\)-functions. For a primitive non-principal character \(χ\) modulo \(q\), we construct an explicit Appell--Mellin generating function \(A_χ(t)\) whose associated Appell sequence satisfies \(R_n^χ(x) = L(-n,χ,x)\), and we prove the functional equation \(A_χ(t) = χ(-1)A_χ(-t)\). We further establish five structural properties of the polynomials \(R_n^χ\): identification with generalized Bernoulli polynomials, a difference equation, an explicit formula, a Fourier expansion, and the trivial zeros of \(L(s,χ)\). To the best of our knowledge, these results do not appear in the literature.

math.NT↗
arXiv2017

Reciprocity Theorems for Bettin--Conrey Sums

Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{πmh}{k}\right)ζ\left(-a,\frac{m}{k}\right), \] where $a\in\mathbb{C}$, $h$ and $k$ are positive coprime integers, and $ζ(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these Bettin--Conrey sums, which in the case of an odd negative integer $a$ can be explicitly given in terms of Bernoulli numbers. This, in turn, implies explicit formulas for the period functions appearing in Bettin--Conrey's work. We study generalizations of Bettin--Conrey sums involving zeta derivatives and multiple cotangent factors and relate these to special values of the Estermann zeta function.

math.NT↗
arXiv2016

New Characterization of Appell polynomials

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli-Euler's one.

math.NT↗
arXiv2013

Relations for Bernoulli--Barnes Numbers and Barnes Zeta Functions

The \emph{Barnes $ζ$-function} is \[ ζ_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to $\C$. Specialized at negative integers $-k$, the Barnes $ζ$-function gives \[ ζ_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) \] where $B_k(x; \a)$ is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing $B_k(0; \a)$ gives the \emph{Bernoulli--Barnes numbers}. We exhibit relations among Barnes $ζ$-functions, Bernoulli--Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

math.NT↗