数论

We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with…

In our earlier work with Christopher Skinner (J. Eur. Math. Soc 24 (2022), no. 2; DOI 10.4171/JEMS/1124; Arxiv 1706.00201), we constructed Euler systems for the 4-dimensional spin Galois representations corresponding to automorphic forms…

Let $k = \mathbb{Q}(\sqrt {-m})$ and $p \geq 3$ split in $k$. We prove new properties of the $\mathbb{Z}_p$-extensions $K/k$, distinct from the cyclotomic one; we do not assume $K/k$ totally ramified, nor the triviality of the $p$-class…

Let us consider the symmetric square transfer of the automorphic representation $\pi$ associated to a modular form $f \in S_k(N,\epsilon)$. In this article, we study the variation of the epsilon factor of ${\mathrm{sym}}^2(\pi)$ under…

In this article, we study two important properties of ${\rm{sym}}^3$ transfers of the automorphic representation $\pi$ associated to a modular form. First we compute the conductor of ${\rm{sym}}^3(\pi)$. Then we detect the types of local…

$A$ be an abelian variety over a number field $K$ of dimension $r$, $a_1, \dots, a_g \in A(K)$ and $F/K$ a finite Galois extension. We consider the density of primes $\frak p$ of $K$ such that the quotient $\bar{A}(k({\frak p}))/\langle…

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric…

This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for…

Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated…

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and…

Khintchine's theorem on the measure dichotomy for the set of $\psi$-approximable numbers has been generalized to inhomogeneous and higher-dimensional settings. Allen and Ram\'irez conjectured that the monotonicity condition can be removed…

We study a class of square matrices with non-negative elements which have cyclically monotone rows in the sense that each row of a matrix from the class consists of a cyclically non-increasing sequence of numbers starting from a maximal…

Let $f$ be the Hecke eigenform for the modular group $SL_2(\mathbb{Z})$, and $L(s, \text{sym}^2 f)$ be the symmetric square $L$-function associated with $f$. For $\frac{1}{2}<\sigma<1$, define $m(\sigma)$ as the supremum of all numbers $m$…

In this research article, we obtain few theta function identities of level ten employing Ramanujan's $_1 \psi_1$ summation formula. Using these identities, we derive a new modular equation of degree five. Further, we establish Eisenstein…

We prove that for any pair of irreducible principal series representations $(\pi_1,\pi_2)$ of $\operatorname{GL}_n(\mathbb{R})$ in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification,…

Assume that $\lambda_1, \lambda_2, \lambda_3,\lambda_4,\lambda_5,\lambda_6,\lambda_7$ are non-zero real numbers , $\lambda_1/\lambda_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $\delta >0$. For any given…

We establish a relationship between error terms appearing in estimates for the counting functions of smooth and rough numbers. We then apply this link to obtain an explicit upper bound for the error term in de Bruijn's approximation…

We give an independent eta-product derivation of the level-8 Apery limit lim B_n^{(8)}/s_n = (7/32) zeta(3), where s_n = sum_{k=0}^n C(n,k)^2 C(2k,n)^2 and B_n^{(8)} is the rational companion sequence satisfying the same cubic recurrence…

We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally…

We study generalized Skewes' numbers, which are the locations of the first sign change between two comparable prime counting functions. In the context of the race between quadratic residues and quadratic nonresidues, we construct sequences…