Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Euler-Ramanujan formula, Dirichlet series and minimal surfaces
130
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we rewrite two forms of an Euler-Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass-Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.
rate research
Read More
There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-Burmann formula for power series. The proof is probabilistic in nature and is based on Kendalls identity, which arises in the fluctuation theory of Levy processes.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
In this paper, we study the alternating Euler $T$-sums and $S$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $S$-sums, from which, the parity theorems of Hoffmans double and triple $t$-values and Kaneko-Tsumuras double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $S$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.
We develop an approach to establish $1/pi$-series from bimodular forms. Utilizing this approach, we obtain new families of $2$-variable $1/pi$-series associated to Zagiers sporadic Apery-like sequences.
We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
