数论

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the…

Let \(Q \subseteq \mathbb{N}\) be a subset, and let \(\psi\colon \mathbb{N} \to [0, \tfrac{1}{2})\), \(\theta\colon \mathbb{N} \to \mathbb{R}\) be functions. Let \(\{A_q\}\) and \(\{B_q\}\) be sequences of integers such that \(\gcd(A_q,…

In this paper, we investigate the average behavior of ternary correlations for general $k$-divisor-bounded multiplicative functions, assuming certain second moment integral bounds for the associated $L$-functions. Our approach differs from…

We establish Burgess-type bounds for short multiplicative character sums over finite fields $\mathbb{F}_{p^n}$ under a purely volumetric condition. We show that for a box $B \subset \mathbb{F}_{p^n}$, nontrivial cancellation occurs whenever…

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials $\Xi_n,\Lambda_n\in\mathbb{Q}[x]$ arising from integral representations of $\beta(2n)/\pi^{2n-1}$ and $\zeta(2n+1)/\pi^{2n}$. In this paper we give explicit closed…

We study the strong approximation for classifying stacks $BG$, where $G$ is a linear algebraic group over a number field $k$. More specifically, we prove that the \'etale Brauer-Manin obstruction is the only obstruction to strong…

For a finite extension $F$ of $\mathbb{Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of…

We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when $a$ is a positive integer, $b=p^{m}$, $2p^{m}$ or $4p^{m}$, where $m$ is a non-negative integer, $p$ is prime, $\gcd…

We study the distribution modulo one of linear recurrent sequences of real numbers. We prove criteria for the finiteness of the set of limit values of the fractional parts of such a sequence and give lower bounds for the maximal distance…

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary…

We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms…

For $F \in \mathbb{Z}[s,t]$ a binary quadratic form which is irreducible over $\mathbb{Q}$, and $L$ an abelian number field with class number $1$, we obtain the order of magnitude for the number of values $F(s,t)$ which are a norm from $L$.…

Let $\spt(n)$ be the number of smallest parts in the partitions of $n$. In this paper, we give some generalized Euler-like recursive formulas for the $\spt$ function in terms of Hecke trace of values of special twisted quadratic Dirichlet…

In this paper, we study three integers arising naturally from B\'{e}zout's identity, the resultant and the reduced resultant of two coprime integer polynomials. We establish several new divisibility relations among them. We also pose two…

The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine…

We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$ and $\pi_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet…

We pursue the question how integers can be ordered or partitioned according to their divisibility properties. Based on pseudometrics on $\mathbb{Z}$, we investigate induced preorders, associated equivalence relations, and quotient sets. The…

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim,…

Let $d(\cdot)$ denote the natural density on the positive integers. We characterize all sets $A,B$ with positive density satisfying $d(A+B)=d(A)+d(B)$, under the assumption that the two sets are not both contained in a proper finite union…

In this paper, we apply the Taylor--Wiles--Kisin patching method to the coherent cohomology of modular curves at minimal level. We establish a multiplicity-one result for the patched module by the $q$-expansion principle and show that a…