Recently, Alanazi et al. [‘Refining overpartitions by properties of nonoverlined parts’, Contrib. Discrete Math. 17(2) (2022), 96–111] considered overpartitions wherein the nonoverlined parts must be $\ell $-regular, that is, the nonoverlined parts cannot be divisible by the integer $\ell $. In the process, they proved a general parity result for the corresponding enumerating functions. They also proved some specific congruences for the case $\ell =3$. In this paper we use elementary generating function manipulations to significantly extend this set of known congruences for these functions.

Given a positive integer m, let $\mathbb {Z}_m$ be the set of residue classes mod m. For $A\subseteq \mathbb {Z}_m$ and $n\in \mathbb {Z}_m$, let $\sigma _A(n)$ be the number of solutions to the equation $n=x+y$ with $x,y\in A$. Let $\mathcal {H}_m$ be the set of subsets $A\subseteq \mathbb {Z}_m$ such that $\sigma _A(n)\geq 1$ for all $n\in \mathbb {Z}_m$. Let

$$ \begin{align*} \ell_m=\min\limits_{A\in \mathcal{H}_m}\bigg\lbrace m^{-1}\sum_{n\in \mathbb{Z}_m}\sigma_A(n)\bigg\rbrace. \end{align*} $$

Ding and Zhao [‘A new upper bound on Ruzsa’s numbers on the Erdős–Turán conjecture’, Int. J. Number Theory 20 (2024), 1515–1523] showed that $\limsup _{m\rightarrow \infty }\ell _m\le 192$. We prove

$$ \begin{align*} \limsup\limits_{m\rightarrow\infty}\ell_m\leq 144 \end{align*} $$

and investigate parallel results on subtractive bases of $ \mathbb {Z}_m$.

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

Let G be a split connected reductive group defined over $\mathbb {Z}$. Let F and $F'$ be two non-Archimedean m-close local fields, where m is a positive integer. D. Kazhdan gave an isomorphism between the Hecke algebras $\mathrm {Kaz}_m^F :\mathcal {H}\big (G(F),K_F\big ) \rightarrow \mathcal {H}\big (G(F'),K_{F'}\big )$, where $K_F$ and $K_{F'}$ are the mth usual congruence subgroups of $G(F)$ and $G(F')$, respectively. On the other hand, if $\sigma $ is an automorphism of G of prime order l, then we have Brauer homomorphism $\mathrm {Br}:\mathcal {H}(G(F),U(F))\rightarrow \mathcal {H}(G^\sigma (F),U^\sigma (F))$, where $U(F)$ and $U^\sigma (F)$ are compact open subgroups of $G(F)$ and $G^\sigma (F),$ respectively. In this article, we study the compatibility between these two maps in the local base change setting. Further, an application of this compatibility is given in the context of linkage – which is the representation theoretic version of Brauer homomorphism.

We prove that any increasing sequence of real numbers with average gap $1$ and Poisson pair correlations has some gap that is at least $3/2+10^{-9}$. This improves upon a result of Aistleitner, Blomer, and Radziwiłł.

We prove the existence of a vector-valued cusp form for the full modular group for which the nth derivative of its L-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen’s plus space and prove an analogous result for Jacobi forms.

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

For $ \beta>1 $, let $ T_\beta $ be the $\beta $-transformation on $ [0,1) $. Let $ \beta _1,\ldots ,\beta _d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define

$$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}}$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}} W(\mathcal P)$

Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weight $k_{1}$ and $k_{2}$ for the full modular group $SL(2,\mathbb {Z})$, respectively. Suppose that $\lambda _{f\times f\times f}(n)$ and $\lambda _{g\times g\times g}(n)$ are the n-th Dirichlet coefficient of the triple product L-functions $L(s,f\times f\times f)$ and $L(s,g\times g\times g)$. In this paper, we consider the sign changes of the sequence $\{\lambda _{f\times f\times f}(n)\}_{n\geq 1}$ and $\{\lambda _{f\times f\times f}(n)\lambda _{g\times g\times g}(n)\}_{n\geq 1}$ in short intervals and establish quantitative results for the number of sign changes for $n \leq x$, which improve the previous results.

We give an explicit formula for the Frobenius number of triples associated with the Diophantine equation $x^2+y^2=z^3$, that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of $x^2+y^2=z^3$. For the equation $x^2+y^2=z^2$, the Frobenius number has already been given. Our approach can be extended to the general equation $x^2+y^2=z^r$ for $r>3$.

We prove a comparison theorem between Greenberg–Benois $\mathcal {L}$-invariants and Fontaine–Mazur $\mathcal {L}$-invariants. Such a comparison theorem supplies an affirmative answer to a speculation of Besser–de Shalit.

We study a certain class of arithmetic functions that appeared in Klurman’s classification of $\pm 1$ multiplicative functions with bounded partial sums; c.f., Comp. Math. 153(2017), 2017, no. 8, 1622–1657. These functions are periodic and $1$-pretentious. We prove that if $f_1$ and $f_2$ belong to this class, then $\sum _{n\leq x}(f_1\ast f_2)(n)=\Omega (x^{1/4})$. This confirms a conjecture by the first author. As a byproduct of our proof, we studied the correlation between $\Delta (x)$ and $\Delta (\theta x)$, where $\theta $ is a fixed real number. We prove that there is a nontrivial correlation when $\theta $ is rational, and a decorrelation when $\theta $ is irrational. Moreover, if $\theta $ has a finite irrationality measure, then we can make it quantitative this decorrelation in terms of this measure.

We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$-covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

Oda’s problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda’s prediction.

We establish explicit constructions of Mahler’s p-adic $U_{m}$-numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $. We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p, $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$-module by elements with norm in S. The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

Let A be an abelian surface over ${\mathbb {Q}}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur’s theorem for elliptic curves, we show that the torsion subgroup of $A({\mathbb {Q}})$ is $12$-torsion and has order at most $18$. Under the additional assumption that A is of $ {\mathrm{GL}}_2$-type, we give a complete classification of the possible torsion subgroups of $A({\mathbb {Q}})$.