International Journal of Number Theory

Bolytropes are bounded subsets of an affine building that consist of all points that have distance at most $r$ from some polytrope. We prove that the points of a bolytrope describe the set of all invariant lattices of a bolytrope order, generalizing the correspondence between polytropes and graduated orders.

We study the asymptotic behavior of Sudler products $P_N(\alpha )={\prod }_{r=1}^{N}2|sin\pi r\alpha |$ for quadratic irrationals $\alpha \in ℝ$. In particular, we verify the convergence of certain perturbed Sudler products along subsequences, and show that ${lim\; inf}_N{P}_N(\alpha )=0$ and ${lim\; sup}_N{P}_N(\alpha )/N=\infty$ whenever the maximal digit in the period of the continued fraction expansion of $\alpha$ exceeds $23$. This generalizes known results for the period one case $\alpha =[0;\overline{a}]$.

Let $f$ be a Hecke–Maass cusp form for ${\mathrm{\text{SL}}}_2(\mathbb{Z})$ with Laplace eigenvalue ${\lambda }_f(\mathrm{\Delta })=1/4+{\mu }^2$ and let ${\lambda }_f(n)$ be its $n$th normalized Fourier coefficient. It is proved that, uniformly in $\alpha ,\beta \in ℝ$, ${\sum }_{n\le X}{\lambda }_f(n)e(\alpha n^2+\beta n)\ll X^{7/8+𝜀}{\lambda }_f{(\mathrm{\Delta })}^{1/2+𝜀},$ where the implied constant depends only on $𝜀$. We also consider the summation function of ${\lambda }_f(n)$ and under the Ramanujan conjecture we are able to prove ${\sum }_{n\le X}{\lambda }_f(n)\ll X^{1/3+𝜀}{\lambda }_f{(\mathrm{\Delta })}^{4/9+𝜀}$ with the implied constant depending only on $𝜀$.

We give an alternative and relatively simple proof of the functional equation (Funct. Approx. Comment. Math. 55(2) (2016) 227–241) of Mordell–Tornheim multiple zeta function.

Recently, the second author and Qin numerically verified some Mahler measure identities of genus 2 and 3 polynomial families. In this paper, we use the elliptic regulator to prove an identity involving shifted Mahler measure for Boyd’s family.

A complete solution of Mumford’s second problem about representation of theta derivatives with rational characteristics in terms of theta constants with rational characteristics is found. An explicit formula for computing such an expression for theta derivative with an arbitrary rational characteristic is derived, and illustrated with examples. This approach fails only on four characteristics: $\left[\right.\begin{array}{c}\hfill 1/2\hfill \\ \hfill 1/6\hfill \end{array}\left]\right.$, $\left[\right.\begin{array}{c}\hfill 1/6\hfill \\ \hfill 1/2\hfill \end{array}\left]\right.$, $\left[\right.\begin{array}{c}\hfill 1/6\hfill \\ \hfill 1/6\hfill \end{array}\left]\right.$ and $\left[\right.\begin{array}{c}\hfill 1/6\hfill \\ \hfill 5/6\hfill \end{array}\left]\right.$; however expressions for theta derivatives with these four characteristics are found by other authors. Expressions for theta derivatives appear to be homogeneous of degree 3 with respect to theta constants.

In this paper, we study the existence of the Miller basis for $S_k(N)$, the space of cusp forms of weight $k$ and level $N$. We seek an easy criterion to determine whether or not the space $S_k(N)$ admits the Miller basis. In particular, we prove that if the level $N$ is sufficiently composite, then $S_k(N)$ does not have the Miller basis. We also prove that there exists a constant $w(N)$ such that $S_k(N)$ admits the Miller basis if and only if $S_{k+w(N)}(N)$ admits the Miller basis when $k\ge 4$.

In this paper, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\mathrm{\text{GL}}}_4({𝔸}_{\mathbb{Q}})$ to cuspidal cohomology of the ${\mathrm{\text{GL}}}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\mathrm{\text{GL}}}_2({𝔸}_{\mathbb{Q}})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\mathrm{\text{GL}}}_2$ and the automorphic tensor product of two representations of ${\mathrm{\text{GL}}}_2$ cannot be equal (up to a twist by a character of ${\mathrm{\text{GL}}}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.

In this paper, we prove asymptotic 0–1 laws satisfied by diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane, and the upper boundary is called the shape. For various types, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve — that is, there is a single limit shape. Similar phenomena have been well studied for integer partitions, so this work is a natural extension. One notable corollary is a transferred limit shape for overpartitions.

The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation $x^2+d{y}^6=z^p$ for square-free values $1\le d\le 20$. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, $\mathbb{Q}$-curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations.

Let $p$ be an odd prime. We study the distribution of consecutive quadratic residues modulo $p$ in non-homogeneous Beatty sequences ${{ℬ}}_{\alpha ,\beta }=\{\lfloor \alpha n+\beta \rfloor :n=1,2,\dots \}$, where $\alpha ,\beta \in ℝ$ and $\alpha$ is irrational. Furthermore, we study the pseudorandom properties of sequences formed by the Legendre symbol and Beatty sequences. In addition, we give upper bounds for certain character sums over ${{ℬ}}_{\alpha ,\beta }$.

For a function $f:\mathbb{N}\to \mathbb{N}$, define $N_f^{\times }(x)=\#\{n\le x:n=kf(k)\text{ for some }k\}$. Let $\tau (n)={\sum }_{d|n}1$ be the divisor function, $\omega (n)={\sum }_{p|n}1$ be the prime divisor function, and $\phi (n)=\#\{1\le k\le n:(k,n)=1\}$ be Euler’s totient function. We prove that