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Evaluation of causal relationships between genetic liability to inflammatory bowel disease and autism spectrum disorder by Mendelian randomization analysis. [PDF]

Dialogues Clin Neurosci
Zeng R, Jiang R, Huang W, Wu H, Zhuo Z, Yang Q, Li J, Leung FW, Sha W, Chen H.   +9 more
europepmc   +1 more source

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Monomials of Eisenstein series

Journal of Number Theory, 2021
Abstract Let E k ( z ) be the normalized Eisenstein series of weight k for the modular group SL ( 2 , Z ) . We study the zeros of E k to prove that the equation ∏ i = 1 n E k i = ∏ j = 1 m E l j has no solutions, except for those given by known ...
Trevor Griffin, Nathan Kenshur, Abigail Price, Bradshaw Vandenberg-Daves, Hui Xue, Daozhou Zhu   +5 more
semanticscholar   +3 more sources

Partition Eisenstein series and semi-modular forms

Research in Number Theory, 2021
We identify a class of “semi-modular” forms invariant on special subgroups of GL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage ...
Matthew Just, Robert Schneider
semanticscholar   +1 more source

Schubert Eisenstein Series and Poisson Summation for Schubert Varieties

American Journal of Mathematics, 2021
:Schubert Eisenstein series are defined by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $\GL_3$ over $\QQ$ Bump and the first author proved that these Schubert Eisenstein series have ...
Y. Choie, Jayce R. Getz
semanticscholar   +1 more source

Polynomials and reciprocals of Eisenstein series

, 2021
Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane.
B. Heim, M. Neuhauser
semanticscholar   +1 more source

Eichler integrals of Eisenstein series as q-brackets of weighted t-hook functions on partitions

The Ramanujan journal, 2020
We consider the t -hook functions on partitions $$f_{a,t}: \mathcal {P}\rightarrow \mathbb {C}$$ f a , t : P → C defined by $$\begin{aligned} f_{a,t}(\lambda ):=t^{a-1} \sum _{h\in \mathcal {H}_t(\lambda )}\frac{1}{h^a}, \end{aligned}$$ f a , t ( λ ) : =
K. Bringmann, K. Ono, Ian Wagner
semanticscholar   +1 more source

ANALYTIC PROPERTIES OF EISENSTEIN SERIES AND STANDARD $L$-FUNCTIONS

Nagoya mathematical journal, 2020
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 $\mathbb{C}[(L^{\prime }/L)^{n}]$.
O. Stein
semanticscholar   +1 more source

The Eisenstein Series [PDF]

, 1973
Let ω1,ω2 be two complex numbers, different from zero, such that the quotient t =ω2/ω1 is not real. The totality Ω of all complex numbers m1ω1 + m2ω2 with m1m2 integers, forms a point-lattice in the complex plane.
openaire   +1 more source

Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers

Research in Number Theory, 2018
New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight k. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of ...
C. O’Sullivan
semanticscholar   +1 more source

Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect

Communications in Mathematical Physics, 2019
We prove a variety of quantum unique ergodicity (QUE) results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet L-function on the 1-line.
Jiakun Pan, M. Young
semanticscholar   +1 more source