International Journal of Number Theory

In this paper, we provide an example of a nonmaximal order in a cubic number field $K$ whose Witt ring is embedded into the Witt ring of $K$.

We provide comparison results for anticyclotomic $p$-adic $L$-functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable $p$-adic $L$-function introduced by Skinner and Urban, and the anticyclotomic $L$-function constructed by means of $p$-adic families of Gross points in the setting of definite quaternion algebras. Several auxiliary results involving anticyclotomic $p$-adic $L$-functions introduced by Chida-Hsieh and Büyükboduk–Lei are also proved.

Conditionally on the $abc$ conjecture, we generalize the previous work of Clark and the author to show that a superelliptic curve $C:y^n=f(x)$ of sufficiently high genus has infinitely many twists violating the Hasse Principle if and only if $f(x)$ has no $\mathbb{Q}$-rational roots. We also show unconditionally that a curve defined by $C:y^{pN}=f(x)$ (for $p$ prime and $N>1$) has infinitely many twists violating the Hasse Principle over any number field $k$ such that $k$ contains the $p$th roots of unity and $f(x)$ has no $k$-rational roots.

It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals.

For a given quadratic irrational $\alpha$, let us denote by $D(\alpha )$ the length of the periodic part of the continued fraction expansion of $\alpha$. We prove that for every positive integer $d$, which is not a perfect square, there are infinitely many even integers $k\ge 1$, for which the equality $D(n\sqrt{d})=k$ holds for infinitely many integers $n\ge 1$.

We study the modularity of the function $u(\tau )=C(\tau )C(2\tau )$, where $C(\tau )$ is Ramanujan’s cubic continued fraction. It is an analogue of Ramanujan’s function $k(\tau )=r(\tau )r{(2\tau )}^2$, where $r(\tau )$ is the Rogers–Ramanujan continued fraction. We first prove the modularity of $u(\tau )$ and express $C(\tau )$ and $C(2\tau )$ in terms of $u(\tau )$. Subsequently, we find modular equations of $u(\tau )$ of level $n$ for every positive integer $n$ by using affine models of modular curves. Finally, we demonstrate that the value of $u(\tau )$ generates the ray class field over an imaginary quadratic field modulo 2 for some $\tau$ in an imaginary quadratic field.

In the eleventh paper in the series on MacMahon’s partition analysis, Andrews and Paule introduced the $k$-elongated partition diamonds. Recently, they revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the $k$-elongated plane partition diamonds of length $n$. Andrews and Paule obtained several generating functions and congruences for $d_1(n)$, $d_2(n)$, and $d_3(n)$. Da Silva et al. further found many congruences modulo 4, 5, 7, 8, 9, and 11 for $d_k(n)$. In this paper, we extend some individual congruences found by Andrews and Paule and da Silva et al. to their respective families as well as find new families of congruences for $d_k(n)$. We also present a refinement in an existence result for congruences of $d_k(n)$ found by da Silva et al. and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64, and 128.

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of $k$-measure of an integer partition, and proved a surprising identity that the number of partitions of $n$ which have $2$-measure $m$ is equal to the number of partitions of $n$ with a Durfee square of side $m$. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of $n$ which have $k$-measure $m$ for $k\ge 3$. In this paper, we perform these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we obtain a natural generalization for $k$-measures.

We obtain an explicit zero-free region for a weighted sum of $L$-functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose $L$ value does not vanish at a given point inside this zero-free region.

In this paper, we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright’s generalized Bessel functions are also treated.

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said large if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.

Let $\mathrm{\Gamma }$ be a congruence subgroup of ${\mathrm{\text{Sp}}}_{2n}(\mathbb{Z})$. Using Poincaré series of $K$-finite matrix coefficients of integrable discrete series representations of ${\mathrm{\text{Sp}}}_{2n}(ℝ)$, we construct a spanning set for the space $S_m(\mathrm{\Gamma })$ of Siegel cusp forms of weight $m\in {\mathbb{Z}}_{>2n}$. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of ${\mathrm{\text{Sp}}}_{2n}(ℝ)$, we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of $S_m(\mathrm{\Gamma })$. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

We continue the study of the $(a,b,m)$-copartition function ${\text{cp}}_{a,b,m}(n)$, which arose as a combinatorial generalization of Andrews’ partitions with even parts below odd parts. The generating function of ${\text{cp}}_{a,b,m}(n)$ has a nice representation as an infinite product. In this paper, we focus on the parity of ${\text{cp}}_{a,b,m}(n)$. We find specific cases of $a,b,m$ such that ${\text{cp}}_{a,b,m}(n)$ is even with density 1. Additionally, we show that the sequence ${\{{\text{cp}}_{a,m-a,m}(n)\}}_{n=0}^{\infty }$ takes both even and odd values infinitely often.

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient $F$, over any arithmetic progression. Namely, if $g_{a,b}(x)$ denotes the number of even coefficients of $F$ in degrees $n\equiv b$ (mod $a$) such that $n\le x$, then we show that $g_{a,b}(x)/\sqrt{x}$ is unbounded for $x$ large.

Note that our result is very close to the best bound currently known even in the special case of the partition function $p(n)$ (namely, $\sqrt{x}loglogx$, proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre’s classical theorem on the number of even values of $p(n)$, combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients.

Interestingly, even in the case of $p(n)$ first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre’s theorem.

In this paper, we pursue positive characteristic analogs of the results of Furusho, Komori, Matsumoto, and Tsumura on $p$-adic multiple $L$-functions. We consider $\infty$-adic and $v$-adic multiple zeta functions concerned by Anglès, Ngo Dac, and Tavares Ribeiro. Our main results in this paper consist of: (1) integral expressions of special values of $\infty$-adic multiple zeta functions, (2) integral expressions of $v$-adic multiple zeta functions themselves that are similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type congruences for the special values of $v$-adic multiple zeta functions at integers, (4) relationships between special values of $\infty$-adic and those of $v$-adic multiple zeta functions at negative integers, and (5) orthogonal properties of multiple zeta functions and multiple zeta star functions.