Related papers: Dissections of a "strange" function
Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of…
Yui and Zagier made some fascinating conjectures on the factorization on the norm of the difference of Weber class invariants $ f(\mathfrak a_1) - f(\mathfrak a_2)$ based on their calculation in \cite{YZ}. Here $\mathfrak a_i$ belong two…
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…
In 1986, Andrews studied the function $\sigma(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved…
Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later,…
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a…
In order to understand the deformations of determinants and Pfaffians resulting from deformations of matrices, we study the deformation theory of composites $f\circ F$, with isolated singularities, where $f:Y\to\C$ has Cohen-Macaulay…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the $q$-bracket, is a quasimodular form. More generally, if a graded algebra $A$ of functions on…
An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue…
In this article we prove a version of Kolyvagin's conjecture for modular forms at non-ordinary primes. In particular, we generalize the work of Wang on a converse to a higher weight Gross-Zagier-Kolyvagin theorem in order to prove the…
Denote by $\mathbb F_q$ the finite field of order $q$ and by $\mathbb F_{q^n}$ its extension of degree $n$. Some $a\in\mathbb F_{q^n}$ is called primitive if it generates the multiplicative group $\mathbb F_{q^n}^*$ and it is called…
A nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$…
It is argued that it is valid to use QCD sum rules to determine the scalar and pseudoscalar two-point functions at zero momentum, which in turn determine the ratio of the strange to non-strange quark condensates $R_{su} = \frac{<\bar{s}…
The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly…
We study a new class of functions that arise naturally in quaternionic analysis, we call them "quasi regular functions". Like the well-known quaternionic regular functions, these functions provide representations of the quaternionic…
In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended…
In the genus 1 case, mirror symmetry reduces to the statement that a certain family of generating functions, relating to an elliptic curve, are quasimodular. In their proof of this fact, Kaneko and Zagier used a related family of generating…
After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group…