International Journal of Number Theory

In this paper, we prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a $p$-adic variable admits an analytic continuation in a non-trivial way, as it was shown previously by Dwork for a class of hypergeometric series.

In this paper, we prove that, for any positive constants $\delta$ and $\epsilon$ and every large enough $x$, the interval $[x,x+\sqrt{x}{(logx)}^{7/3+\delta }]$ contains numbers whose all prime factors are smaller than $x^{\epsilon }$.

Let $K$ be a finite geometric separable extension of the rational function field ${𝔽}_q(T)$, and let $L$ be a finite cyclic extension of $K$ of prime degree $l$. Assume that the ideal class number of the integral closure ${{𝒪}}_K$ of ${𝔽}_q[T]$ in $K$ is not divisible by $l$. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the $l$-class group of $\mathrm{\text{Cl}}({{𝒪}}_L)$ (i.e. the Sylow $l$-subgroup of the ideal class group of ${{𝒪}}_L$) as Galois module, where ${{𝒪}}_L$ is the integral closure of ${{𝒪}}_K$ in $L$. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.

In this note, we improve earlier results towards the Bruinier–Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász’ theorem. Moreover, by applying a result of Serre, we remove all unproved assumptions.

In this paper, we prove the following theorem: Let $𝔽$ be an algebraic closure of a finite field of characteristic $p>3$. Let $\rho$ be a continuous homomorphism from the absolute Galois group of $\mathbb{Q}$ to $\mathrm{\text{GL}}(3,𝔽$) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of $\rho$ is contained in the intersection of the stabilizers of the line spanned by $e_2$ and the plane spanned by $e_1,e_3$, where $\{e_i\}$ denotes the standard basis. Such $\rho$ will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of $\rho$ is squarefree, that the predicted weight $(a,b,c)$ lies in the lowest alcove, and that $c\not\equiv b+1(modp-1)$, we prove that $\rho$ is attached to a Hecke eigenclass in $H_2(\mathrm{\Gamma },M)$, where $\mathrm{\Gamma }$ is a subgroup of finite index in $\mathrm{\text{SL}}(3,\mathbb{Z})$ and $M$ is an $𝔽\mathrm{\Gamma }$-module. The particular $\mathrm{\Gamma }$ and $M$ are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.

Let ${{𝒵}}_l^{(n)}$ be the vector space spanned by multiple zeta values (MZVs) of weight $l$ and depth less than $n+1$. Let ${\overline{{𝒵}}}_l^{(n-1)}$ be the subspace spanned by MZVs of weight $l$ and depth $n-1$, and let ${{𝒫}}_l^{(n)}$ be the subspace given by the intersection of ${{𝒵}}_l^{(n)}$ and the vector space spanned by products of MZVs. In the present paper, we prove congruence identities of regularized multiple zeta values involving a pair of index sets modulo ${\overline{{𝒵}}}_l^{(n-1)}+{{𝒫}}_l^{(n)}$. We also obtain a proof of the parity result, and a congruence sum formula for MZVs.

Let $f$ be a normalized Maass cusp form for ${\mathrm{\text{SL}}}_2(Z)$. For $\frac{1}{2}<\sigma <1$, we define $m(\sigma )$ $(\ge 2)$ as the supremum of all numbers $m$ such that

For a field of definition $k$ of an abelian variety ${𝒜}$ and prime ideal $𝔭$ of $k$ which is of a good reduction for ${𝒜}$, the structure of ${𝒜}({𝔽}_{𝔭})$ as abelian group is:

In this study, we investigate the half-factorial property in non-maximal real quadratic orders and investigate some necessary and sufficient conditions for these orders to be half-factorial domains in terms of their conductor.

Dedekind sums $s(m,n)$ occur in many fields of mathematics. Since $s(m_1,n)=s(m_2,n)$ if $m_1\equiv m_2\mathrm{\text{mod}}n$, it is natural to ask which of the Dedekind sums $s(m,n)$, $0\le m<n$, take equal values. So far no simple criterion is known by which the equality of $s(m_1,n)$ and $s(m_2,n)$ could be decided. In this note, we show how to obtain non-obvious examples of equal Dedekind sums. We consider two cases which mark the extreme possibilities for the argument $n$, namely, $n$ a prime power and $n$ square-free. Whereas we can give a partial overview of equal Dedekind sums in the prime power case, such an overview seems to be much more difficult to obtain in the square-free case.

Let $f(z)={\sum }_{n=1}^{\infty }{\lambda }_f(n)e^{2\pi inz}\in S_k^{\mathrm{\text{new}}}({\mathrm{\Gamma }}_0(N))$ be a newform of even weight $k\ge 2$ on ${\mathrm{\Gamma }}_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence ${\{{\lambda }_f(p)\}}_{p\in \mathbb{P}}$ does not satisfy Benford’s Law in any integer base $b\ge 2$. However, given a base $b\ge 2$ and a string of digits $S$ in base $b$, the set

We consider sums of the Gaussian $q$-binomial coefficients with a parametric rational weight function. We use the partial fraction decomposition technique to prove the claimed results. We also give some interesting applications of our results to certain generalized Fibonomial sums weighted with finite products of reciprocal Fibonacci or Lucas numbers.

In this paper, we prove a conjecture of Sun concerning the arithmetic progression of integers which contains no integers of the form $F_n+p$. In fact, we show that $d-F_n$ has at least two distinct prime factors whenever $d\equiv r(modM)$, where $F_n$ is a Fibonacci number and $r$ and $M$ are suitable positive integers.

In this article, we will give a new proof of the reciprocity law for Dedekind sums, as well as a proof of the transformation formula for the Dedekind $\eta$-function using the Chinese Remainder Theorem.

The harmonic numbers $H_n={\sum }_{0<k\le n}1/k(n=0,1,2,\dots )$ play important roles in mathematics. Let $p>3$ be a prime. With the help of some combinatorial identities, we establish the following two new congruences:

We provide several results on the Diophantine properties of continued fractions on the Heisenberg group, many of which are analogous to classical results for real continued fractions. In particular, we show an analog of Khinchin’s theorem and we also show that convergents are the best rational approximations up to a constant factor of the denominator.

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

In the sense of the theory of lattice points in large bodies, a fairly general body of rotation ${{ℬ}}_{\mathrm{\text{rot}}}$ in ${ℝ}^3$ is considered whose convexity is being hampered by a “dent” (see Fig. 1). It turns out that, for $t$ a large real parameter, the true order of magnitude of the lattice discrepancy of $t{{ℬ}}_{\mathrm{\text{rot}}}$ essentially can be evaluated, contrary to the convex case.