数论

In this paper, we give explicit expressions about $q$-harmonic sums on $1-\cdots-1,A,1-\cdots-1$ indices. When $A=1$, many previous authors have studied and showed the identities, expressions, and properties. There are many results for…

Generalising the two-dimensional case, we provide estimates for the mean-values of the lengths of the edges of an integral box with given volume and minimal surface.

Determining the explicit forms and modularity for string functions and branching coefficients for Kac--Moody algebras after Kac, Peterson, and Wakimoto is an important problem. For positive admissible-level string functions for the affine…

We present an algorithm for computing all the solutions in not necessarily distinct integers to the decomposition of the unit into a sum of unit fractions with denominators $p^a.q^b$ where $p$ and $q$ are two distinct primes, each appearing…

In this paper we study the product of two central values of $L$-functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. For the proof, we…

In this paper, we prove an explicit upper bound on the number of rational points on a smooth projective curve of genus at least two over a number field. This gives explicit constants in the uniform Mordell conjecture proposed by Mazur and…

We prove that the thin parts of arithmetically defined locally symmetric space take up a negligible part of their volume and deduce asymptotic results on their Betti numbers.

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $\lambda_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq…

We study weighted simultaneous rational approximation to points of the form $(1,\xi,\xi^2)$, for a class of extremal real numbers $\xi$, within the framework of multi-parametric geometry of numbers.

We establish a classification of the values of \( N \) for which an elliptic curve defined over \( \mathbb{Q} \) with square discriminant admits an \( N \)-isogeny. Furthermore, we determine the values of \( N \) for which two elliptic…

Let $D$ be a square-free integer. Under certain conditions on $D$, we characterize non-constant arithmetic progressions of squares over quadratic extensions of $\mathbb{Q}(\sqrt{D})$.

Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a…

In this paper, we obtain the meromorphic continuation of a q-analogue of multiple zeta function using an elementary formula called translation formula. We then obtain the matrix representation of the translation formula and using it, we…

In this article, we are interested in finding rational points on certain superelliptic curves.

Let $C$ be a smooth plane quartic curve over $\mathbb{Q}$. Costa, Harvey and Sutherland provide an algorithm with an implementation, improving Harvey's average polynomial-time algorithm, to compute the $\bmod \ p$ reduction of the numerator…

We study fast Monte-Carlo methods for testing irreducibility and detecting arithmetic imprimitivity of polynomials over $\mathbb{Q}$. Building on the subset-sum criterion of Pemantle-Peres-Rivin, we develop a probabilistic irreducibility…

In this paper, we obtain a general t-shuffle product formula, using which we derive a generalized Euler decomposition formula for interpolated multiple zeta values. We also provide the same formula in case of height one through two…

We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie…

We prove that, over a field of characteristic $0$, the degrees of factors of a binomial $t^n-\alpha$ are divisible by the least such degree. As a consequence, we deduce that for relatively prime natural numbers $m,n$, a polynomial has the…

Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $\rho$ with $\rho^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,\rho}$ on…