KOECHER–MAAß SERIES FOR REAL ANALYTIC SIEGEL EISENSTEIN SERIES

2006 ◽  
Author(s):  
TOMOYOSHI IBUKIYAMA ◽  
HIDENORI KATSURADA
Keyword(s):  
Eisenstein Series ◽  
Real Analytic

scholarly journals Eisenstein series and an asymptotic for the K-Bessel function

2021 ◽  
Author(s):  
Jimmy Tseng
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Eisenstein Series ◽  
Uniform Estimate ◽  
Positive Real ◽  
Dominant Term ◽  
Complex Order ◽  
Fixed Parameter ◽  
Real Argument ◽  
Real Analytic

AbstractWe produce an estimate for the K-Bessel function $$K_{r + i t}(y)$$ K r + i t ( y ) with positive, real argument y and of large complex order $$r+it$$ r + i t where r is bounded and $$t = y \sin \theta $$ t = y sin θ for a fixed parameter $$0\le \theta \le \pi /2$$ 0 ≤ θ ≤ π / 2 or $$t= y \cosh \mu $$ t = y cosh μ for a fixed parameter $$\mu >0$$ μ > 0 . In particular, we compute the dominant term of the asymptotic expansion of $$K_{r + i t}(y)$$ K r + i t ( y ) as $$y \rightarrow \infty $$ y → ∞ . When t and y are close (or equal), we also give a uniform estimate. As an application of these estimates, we give bounds on the weight-zero (real-analytic) Eisenstein series $$E_0^{(j)}(z, r+it)$$ E 0 ( j ) ( z , r + i t ) for each inequivalent cusp $$\kappa _j$$ κ j when $$1/2 \le r \le 3/2$$ 1 / 2 ≤ r ≤ 3 / 2 .

scholarly journals The Saito–Kurokawa lift and Siegel’s theta series

2017 ◽  
Vol 13 (07) ◽  
pp. 1679-1693
Author(s):  
Roland Matthes
Keyword(s):  
Spectral Analysis ◽  
Eisenstein Series ◽  
Short Proof ◽  
Theta Series ◽  
Converse Theorem ◽  
Selberg Integral ◽  
Real Analytic

The aim of this paper is to give another short proof of the Saito–Kurokawa lift based on a converse theorem of Imai as was already done by Duke and Imamoglu. In contrast to their proof we avoid spectral analysis but use a real analytic Eisenstein series in a suitable Rankin–Selberg integral involving Siegel’s theta series.

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.

ANALYTIC PROPERTIES OF EISENSTEIN SERIES AND STANDARD -FUNCTIONS

2020 ◽  
pp. 1-36
Author(s):  
OLIVER STEIN
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Analytic Continuation ◽  
Eisenstein Series ◽  
Weil Representation ◽  
Jacobi Forms ◽  
Doubling Method ◽  
Real Analytic ◽  
Forms Of Higher Degree ◽  
Vector Valued

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$ . By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$ , a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$ -function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

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