数论

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem in 1973 concerning the pair correlation of zeros of the Riemann zeta-function and applied this to prove that at least $2/3$ of the zeros are simple. In this paper, we…

We prove the Mixing Conjecture of Michel--Venkatesh for the class group action on Heegner points of large discriminant on compact arithmetic surfaces attached to maximal orders in rational quaternion algebras. The proof is conditional on…

Let $x\geqslant 2$ and assume that $a$ and $q$ are coprime positive integers. As usual, $\psi(x;q,a):=\sum_{n\leqslant x,n\equiv a(\!\!\!\mod{\!\!q})}\Lambda(n)$, where $\Lambda$ is the von Mangoldt function. In 2003, Friedlander and…

We prove formulas for generalized rank deviations for overpartitions. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and extend work of Lovejoy and the second author. As an application, we…

The Lehmer conjecture states that the non-constant Fourier coefficients of the 24th power of the Dedekind eta function are non-zero. In a recent preprint, Neuhauser and the first author exploited an easily accessible tool from algebraic…

In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights $k \in \{12, 16, 18, 20, 22, 26\}$ on $\Gamma_{0}(1) = \operatorname{SL}_{2}(\mathbb{Z})$ using the theory of modular Galois…

Zhixiang Wu has shown the existence of "change of weights" operation on $(\varphi,\Gamma)$-modules in families (arxiv:2405.16637). We interpret it in the trianguline case as pullbacks with a discussion on related stacks. Finally, we prove…

The Landau-Selberg-Delange method gives precise asymptotic formulas for the partial sums $\sum_{n \le x} \, a_n$ of a Dirichlet series $\sum_n \, a_n/n^s$ that behaves like a complex power of the Riemann zeta function. However, situations…

We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in…

Recently, Choie and Kumar extensively studied the Herglotz-Zagier-Novikov function $\mathfrak{F}(z;u,v)$, defined as \begin{align*} \mathfrak{F}(z;u,v) = \int_{0}^{1} \frac{\log(1-ut^z)}{v^{-1}-t} dt, \quad \textrm{for} \quad…

We find some equidistribution results connected to restriction quantum unique ergodicity problem in this paper. We shows that \begin{align*} \frac{1}{|\mathcal{B}_k|}\sum_{f\in \mathcal{B}_k} \int_{R}y^{k}|f(z)|^{2}\psi(z) d\mu_{R}(z)\to…

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…

In this paper, we develop a new method to reconstruct an abelian number field $K$ from the maximal $6$-step solvable quotient of $G_K$ group- theoretically. The new aspect of this paper is that the results in this paper are independent from…

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…

In a previous paper of the author, we establish a duality for the direct limit and the inverse limit of higher even $K$-groups over a $\mathbb{Z}_p^d$-extension. In this paper, we shall establish such a duality over certain non-commutative…

Ramanujan's famous formula for $\zeta(2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for…

We provide a multiple integral representation for each multiple zeta-star value, and utilize these representations to establish a natural order structure on the set of such values. This order structure allows for a one-to-one correspondence…

In recent work where Matsusaka generalizes the relationship between Habiro-type series and false theta functions after Hikami, five families of Hecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0…

In this paper, we establish the local converse theorem and the stability of local gamma factors for $\Mp_{2n}$ via the precise local theta correspondence between $\Mp_{2n}$ and $\SO_{2n+1}$ over local fields of characteristic not equal to…

This paper is a contribution to the algebraic study of contextuality in quantum theory. As an algebraic analogue of Kochen and Specker's no-hidden-variables result, we investigate rational subrings over which the partial ring of $d \times…