International Journal of Number Theory

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums $s(m,n)$ with $m$ and $n$ running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums $\{s(m,n):m\in {({\mathbb{Z}}/n{\mathbb{Z}})}^{\ast }\}$ on average for $n\in [1,N]$.

We prove a formula for the coefficients of a weight $3/2$ Cohen–Eisenstein series of square-free level $N$. This formula generalizes a result of Gross, and in particular, it proves a conjecture of Quattrini. Let $l$ be an odd prime number. For any elliptic curve $E$ defined over $\mathbb{Q}$ of rank zero and square-free conductor $N$, if $l\left|\right|E(\mathbb{Q})|$, under certain conditions on the Shafarevich–Tate group ${\mathrm{\text{III}}}_D$, we show that $l$ divides $\left|{\mathrm{\text{III}}}_D\right|$ if and only if $l$ divides the class number $h(-D)$ of $\mathbb{Q}(\sqrt{-D}).$

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $\mathrm{\Gamma }$ of SL₂(ℤ). We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis ${\sum }_j{y}^k|f_j(z){|}^2$ and analyzing this sum by means of a Bergman kernel and the Fourier coefficients of Poincaré series. Under some weak assumptions, we further prove the right order of magnitude of ${sup}_{z\in ℍ}{\sum }_j{y}^k|f_j(z){|}^2$. Our results are valid without any assumption that the forms are Hecke eigenfunctions.

The aim of this note is to improve an approximation formula of Ramanujan-type discussed by Hirschhorn and Villarino.

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. In this paper, we will give a complete determination of $\overline{p}(n)$ modulo $16$ by relating it to some binary quadratic forms; this will generalize the result of Kim on $\overline{p}(n)$ modulo $8$. Moreover, we give infinitely many new Ramanujan-like congruences for overpartition function modulo $16$.

Noether’s problem asks whether, for a given field $K$ and finite group $G$, the fixed field $L:=K{(x_h:h\in G)}^G$ is a purely transcendental extension of $K$, where $G$ acts on the $x_h$ by $g\cdot x_h=x_{gh}$. The field $L$ is naturally the function field for a quotient variety $V:=V(K,G)$. We study the degree of irrationality $Irr(V)$ of $V$ for an abelian group $G$, which is defined to be the minimal degree of a dominant rational map from $V$ to projective space. In particular, we give bounds for $Irr(V)$ in terms of the arithmetic of cyclotomic extensions $K$.

Let

In this paper, we determine the asymptotic behavior of moments of class numbers for discriminants in a given arithmetic progression and for fundamental discriminants in the case that the class numbers are ordered by the size of the regulator. Furthermore, we give the limit distribution of class numbers in these cases.

Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer. Let $\{B_n\}$ and $\{B_n(x)\}$ denote the Bernoulli numbers and Bernoulli polynomials given by $B_0=1,{\sum }_{k=0}^{n-1}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_k=0(n\ge 2)$ and $B_n(x)={\sum }_{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_k{x}^{n-k}(n\ge 0)$. In this paper, we show that

We show that if a sequence of non-zero polynomials in $\mathbb{Z}[X_1,X_2]$ takes small values at translates of a fixed point $(\xi ,\eta )$ by multiples of a fixed rational point within the group $ℂ\times {ℂ}^{\ast }$, then $\xi$ and $\eta$ are both algebraic over $\mathbb{Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on their absolute values at these translates. It is essentially best possible in some range of the parameters.

We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for $p$-adic Arakawa–Kaneko zeta functions, including a $p$-adic analogue of Ohno’s sum formula. These $p$-adic identities serve to illustrate the relationships between real periods and their $p$-adic analogues.

In this paper, we consider a refinement, due to Nathanson, of the Calkin–Wilf tree. In particular, we study the properties of such trees associated with the matrices $L_u=\left[\begin{array}{cc}1\hfill & 0\hfill \\ u\hfill & 1\hfill \end{array}\right]$ and $R_v=\left[\begin{array}{cc}1\hfill & v\hfill \\ 0\hfill & 1\hfill \end{array}\right]$, where $u$ and $v$ are non-negative integers. We extend several known results of the original Calkin–Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin–Wilf tree.

A well-known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.

We express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.

Let $p$ be a prime. Back in 1878, Lucas proved that the congruence

Let ${𝔽}_q[t]$ denote the ring of polynomials over ${𝔽}_q$, the finite field of $q$ elements. We prove an estimate for fractional parts of polynomials over ${𝔽}_q[t]$ satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

In this paper, on the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

Euler showed that there can be no more than three integer squares in arithmetic progression. In quadratic number fields, Xarles has shown that there can be arithmetic progressions of five squares, but not of six. Here, we prove that there are no cubic number fields which contain five squares in arithmetic progression.

Bourgain, Konyagin and Shparlinski obtained a lower bound for the size of the product set $AB$ when $A$ and $B$ are sets of positive rational numbers with the numerator and denominator less than or equal to $Q$. We extend and slightly improve that lower bound using a different approach.