International Journal of Number Theory

In this article, we investigate systems of fundamental units of quartic fields which are constructed by adjoining a root of a certain parametric quartic polynomial to ℚ.

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

In this paper, we will study the pseudo-nullity of the dual fine Selmer group and its related question. We investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over an admissible p-adic Lie extension F_∞ from the knowledge of the pseudo-nullity of the Galois group of the maximal abelian unramified pro-p extension of F_∞ at which every prime of F_∞ above p splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside p. We will illustrate our results with many examples.

This paper is concerned with the metric properties of the generalized continued fractions (GCF_ϵ) with the parameter function ϵ(k_n), where k_n is the nth partial quotient of the GCF_ϵ expansion. When -1 < ϵ(k_n) ≤ 1, Zhong [Metrical properties for a class of continued fractions with increasing digits, J. Number Theory128 (2008) 1506–1515] obtained the following metrical properties:

In this paper, we indicate a method to obtain asymptotic evaluations for double sums from the asymptotic evaluations of a single sum. As applications, we show that all results in the recent paper [New extensions of some classical theorems in number theory, J. Number Theory133(13) (2013) 3771–3795] can be used to obtain some asymptotic evaluations for double sums. Further, we give asymptotic evaluations for double sums which involve the divisor and the Euler totient function.

We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have been proved under the assumption of the Riemann hypothesis. The result mirrors central limit theorems in random matrix theory that have been proved by Szegő, Spohn, and Soshnikov among others, and therefore provides support for the view that the zeros of the zeta function are distributed like the eigenvalues of a random matrix. A key ingredient in our proof is a simple bootstrapping of classical zero density estimates of Selberg and Jutila for the zeta function, which may be of independent interest.

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a) - f(b) ≡ 0 (mod (a - b)) for all a > b. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance, all functions x ↦ ⌊e^1/a a^x x!⌋, with a ∈ ℤ\{0, 1}, and a function equal to ⌊e x!⌋ except on 0. Finally, to study the complement class, we look at functions ℕ → ℝ which are not uniformly close to any function having integral difference ratios.

Let G be an additive finite abelian group, and let disc(G) denote the smallest positive integer t such that every sequence S over G of length ∣S∣ ≥ t has two nonempty zero-sum subsequences of distinct lengths. We determine disc(G) for some groups including the groups , the groups of rank at most two and the groups C_mpⁿ ⊕ H, where m, n are positive integers, p is a prime and H is a p-group with pⁿ ≥ D*(H).

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct modulo powers of 3. In this paper, we focus on congruences for pod(n) modulo 5. For example, for α ≥ 2 and n ≥ 0, we have

We prove an analogue for Drinfeld modules of a theorem of Romanoff. Specifically, let ϕ be a Drinfeld A-module over a global function field L and denote by ϕ(L) the A-module structure on L coming from ϕ. Let Γ ⊂ ϕ (L) be a free A-submodule of finite rank. For each effective divisor of L, let f_Γ(𝔇) be the cardinality of the image of the reduction map if all elements of Γ are relatively prime to the divisor 𝔇; otherwise, just set f_Γ(𝔇) = ∞. We give explicit upper bounds for the series and .

In this paper, the Hausdorff dimensions of level sets described by two kinds of moving averages are determined. The dissimilar results complement the work of Pfaffelhuber [Moving shift averages for ergodic transformation, Metrika22 (1975) 97–101] and del Junco and Steele [Moving averages of ergodic processes, Metrika24 (1977) 35–43], and reveal simultaneously that the two moving averages are of different convergence processes in the ergodic theory.

This paper contains results on a strange smallest parts function related to the second Atkin–Garvan moment. Some new identities are discovered in relation to Andrews spt function as well as one of Borweins' two-dimensional theta functions.

Given a discriminant form D of level N, there is a natural lifting which maps elliptic modular forms of level N to vector-valued modular forms for the Weil representation associated to D. We show that in some cases the zero component of a lifting of a newform f is just a scalar multiple of f. In order to do so, we split the lifting map into certain partial liftings corresponding to the prime powers exactly dividing N and then proceed to compute the zero components of these partial maps explicitly. As an application, we show that the L-function L_𝒜(f, s) of a newform f and an ideal class 𝒜 as defined by Gross and Zagier can be written as a certain L-series of the lifting of f.

We find several new parity results for 7- and 23-regular partitions.

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He also proved that . Chen, Hirschhorn and Sellers then found infinite families of congruences modulo 3 and modulo powers of 2 for , and . In this paper, we find new congruences for modulo 4, 18 and 36, infinite families of congruences modulo 2 and 4 for , congruences modulo 2 and 3 for , , and congruences modulo 2 for and . We use simple p-dissections of Ramanujan's theta functions.