arXiv subjects
Publications and source records attributed to Lazhar Fekih-Ahmed.
Using the reflection formula of the Gamma function, we derive a new formula for the Taylor coefficients of the reciprocal Gamma function. The new formula provides effective asymptotic values for the coefficients even for very small values of the indices. Both the sign oscillations and the leading order of growth are given.
We formulate a parametrized uniformly absolutely globally convergent series of $ζ$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$ζ$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the polylogarithm function. As an immediate first application of the new parametrized series, a new expression of $ζ$(s) follows: (s -- 1)$ζ$(s) = -- 1 0 Li s z z -- 1 dz. As a second important application, using the functional equation and exploiting uniform convergence of the series defining Z(s, x), we have for any non-trivial zero s
The author has found today an error in the denominator of the residue equation (4.5). This unfortunate mistake makes the conclusions and the title of the paper incorrect. The function $Z(s,x)$ is regular at $x=1$ and the multivalued part disappears. The original motivation of the paper was to derive an old (and correct!) result due to the author using a different approach.
We derive a new integral formula for the Stieltjes constants. The new formula permits easy computations as well as an exact approximate asymptotic formula. Both the sign oscillations and the leading order of growth are provided. The formula can also be easily extended to generalized Euler constants.
As a first application of a very old theorem, known as Herschel's theorem, we provide direct elementary proofs of several explicit expressions for some numbers and polynomials that are known in combinatorics. The second application deals with the analytical continuation of the polylogarithmic function of complex argument beyond the circle of convergence.
We give new integral and series representations of the Hurwitz zeta function. We also provide a closed-form expression of the coefficients of the Laurent expansion of the Hurwitz-zeta function about any point in the complex plane.
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of complex order.
This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation are briefly outlined. The second part exposes a totally different approach. Using the new series representation of the zeta function of the first part, exact information on its zeros is provided.