Bhattacharya, Soumya
Published in
Research in the Mathematical Sciences

The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann’s finiteness theorem. On the other hand, for checking whether f is irreducible, it is essential to know at least an explicit upper bound for the ...

Hamieh, Alia Raji, Wissam
Published in
Research in Number Theory

In this paper, we show that, on average, the derivatives of L-functions of cuspidal Hilbert modular forms with sufficiently large parallel weight k do not vanish on the line segments I(s)=t0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \us...

Zemel, Shaul
Published in
The Ramanujan Journal

We extend the relation between quasi-modular forms and modular forms to a wider class of functions. We then relate both forms to vector-valued modular forms with symmetric power representations, and prove a general structure theorem for these vector-valued forms.

Bringmann, Kathrin Kaszian, Jonas Zhou, Jie
Published in
Research in Number Theory

We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and sing...

Booker, Andrew R. Lee, Min Strömbergsson, Andreas

We derive a fully explicit version of the Selberg trace formula for twist‐minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2‐dimensional Artin representations of small conductor; in particular, we show that the even ic...

Abo Touk, Sarah Al Houchan, Zina El Bachraoui, Mohamed
Published in
Analysis

In this paper we will give q-analogues for the Pythagorean trigonometric identity sin 2 z + cos 2 z = 1 {\sin^{2}z+\cos^{2}z=1} in terms of Gosper’s q-trigonometry. We shall also give new q-analogues for the duplicate trigonometric identity sin ( x - y ) sin ( x + y ) = sin 2 x - sin 2 y {\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y} . More...

Bringmann, Kathrin Nazaroglu, Caner
Published in
Research in the Mathematical Sciences

False theta functions closely resemble ordinary theta functions; however, they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications ...

Purkait, Soma

Lau, Yuk-Kam Ng, Ming Ho Wang, Yingnan
Published in
Forum Mathematicum

A two-dimensional central limit theorem for the eigenvalues of GL ( n ) {\mathrm{GL}(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in...

Jana, Subhajit
Published in
The Ramanujan Journal

Let ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaterni...