scholarly journals Around the Lipschitz Summation Formula

2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Wenbin Li ◽  
Hongyu Li ◽  
Jay Mehta
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Boundary Behavior ◽  
Summation Formula ◽  
Boundary Function ◽  
Essential Information ◽  
Long Run ◽  
Underlying Principle ◽  
Ramanujan’S Formula ◽  
Lerch Transcendent

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

scholarly journals Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 48
Author(s):  
Kottakkaran Sooppy Nisar
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Summation Formula ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

Analysis on arithmetic schemes. II

2010 ◽  
Vol 5 (3) ◽  
pp. 437-557 ◽  
Author(s):  
Ivan Fesenko
Keyword(s):  
Zeta Function ◽  
Logarithmic Derivative ◽  
Boundary Function ◽  
Boundary Integral ◽  
Integration Theory ◽  
Meromorphic Continuation ◽  
Two Dimensional ◽  
Fundamental Properties ◽  
Arithmetic Surfaces ◽  
Dimensional Version

AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

On Jackson’s proof of Ramanujan’s 1ψ1 summation formula

2018 ◽  
Vol 14 (02) ◽  
pp. 313-328
Author(s):  
Jorge Luis Cimadevilla Villacorta
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Summation Formula ◽  
Ramanujan’S Formula

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

scholarly journals ON A CERTAIN CONVOLUTION OF POLYLOGARITHMS

2011 ◽  
Vol 86 (3) ◽  
pp. 461-472 ◽  
Author(s):  
HIROFUMI TSUMURA
Keyword(s):  
Zeta Function ◽  
Eisenstein Series ◽  
Riemann Zeta Function ◽  
Double Series ◽  
The Riemann Zeta Function ◽  
Sum Of Products ◽  
Riemann Zeta ◽  
Real Analytic ◽  
Double Integrals

AbstractIn this paper, we consider certain double series analogous to Tornheim’s double series and real analytic Eisenstein series. By computing double integrals in two ways, we express the double series as a sum of products of polylogarithms. The technique generalises one given by Kanemitsu, Tanigawa and Yoshimoto. Evaluating the double series at particular points gives new evaluations for certain double series in terms of values of the Riemann zeta function and the dilogarithm which are analogues of formulas of Mordell and Goncharov.

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