International Journal of Number Theory

We compare two approaches to the study of Galois module structures: on the one hand, factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand, regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We then use this comparison to derive a factorizability result on higher K-groups of rings of integers, which is a direct analogue of a theorem of de Smit on S-units.

In 1908, Schur raised the question of the irreducibility over ℚ of polynomials of the form f(x) = (x - a₁)(x - a₂)⋯(x - a_n) + 1, where the a_i are distinct integers. Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the analogous question when replacing the linear polynomials with cyclotomic polynomials and allowing the constant perturbation of the product to be any integer d ∉ {-1, 0}. One interesting consequence of our investigations is that we are able to construct, for any positive integer N, an infinite set S of cyclotomic polynomials such that 1 plus the product of any k (not necessarily distinct) polynomials from S, where k ≢ 0(mod 2^N+1), is reducible over ℚ.

In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T_l, both in families and for a single weight. We show that the U-operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U-operator.

We consider the Fermat elliptic curve E₂ : x³ + y³ = 2 and prove (using descent methods) that its quadratic twists have rank zero for a positive proportion of squarefree integers with fixed number of prime divisors. We also prove similar result for rank zero cubic twists of this curve. Then we present detailed description of rank zero quadratic and cubic twists of E₂ by primes and by products of two primes. We also consider twists of Jacobians of Fermat curves x⁵ + y⁵ = m and distribution of their root numbers.

A natural number n is called abundant if the sum of proper divisors of n exceeds n. In 1933 Behrend determined bounds for the natural density d𝒜 of abundant numbers, with subsequent improvements leading to the 1998 bounds of Deléglise, 0.2474 < d𝒜 < 0.2480. We present a new infinite series expression for the density of abundant numbers which can be used to calculate an improved lower bound.

Relations between doubled distinct and self-conjugate t-cores for some small values of t are derived. By employing Ramanujan's simple theta function identities, several infinite families of arithmetic identities for doubled distinct t-cores for t = 3, 4, …, 10 are found.

In this paper, the convolution sums ∑_i+3j=nσ(i)σ₃(j) and ∑_3i+j=nσ(i)σ₃(j) are evaluated for all n ∈ ℕ, which answers an open problem given by Huard, Ou, Spearman and Williams. As an application, the number of representations of a positive integer n by the form

Schinzel showed that the set of primes that divide some value of the classical partition function is infinite. For a wide class of sets 𝒜, we prove an analogous result for the function p_𝒜(n) that counts partitions of n into terms belonging to 𝒜.

We determine formulae for the numbers of representations of a positive integer by certain octonary quadratic forms whose coefficients are 1, 2, 3 and 6.

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL₂(𝔸_E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M_ét(π^∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M_ét(π^∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M_ét(π^∞) is spanned by algebraic cycles.

In this paper, we discuss the log-behavior of the Catalan–Larcombe–French sequence {P_n}_n≥0. We prove that {P_n}_n≥0 is log-balanced and {P_n/(n!)²}_n≥0 is unimodal. In addition, we show that {P_k/k!}_0≤k≤n is reverse ultra log-concave.

We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We obtain an asymptotic formula of the fourth moment for the central value of these Dirichlet L-functions. In addition, we find a lower bound for the 2kth moment of these L-functions. These results agree up to constants with the polynomial ring analog of the Keating and Snaith Conjecture for the asymptotic of leading terms.

We produce generalizations of Iwasawa's "Riemann–Hurwitz" formula for number fields. These generalizations apply to cyclic extensions of number fields of degree pⁿ for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa λ-invariants which generalizes a result of Fukuda et al. for totally real number fields.

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length T^1/11-∊. Our result generalizes a formula of Hughes and Young concerning the fourth moment of the Riemann zeta function.