International Journal of Number Theory

The author remembers Heini Halberstam and views their early joint work through the lens of additive and multiplicative arithmetic functions.

This memoir is concerned with a number of additive questions in which squares occur. For ternary additive problems with at least one square, the exceptional set is considered and it is shown that current methods cannot be sharpened substantially. In a different direction, but using cognate methods, an asymptotic formula is obtained for the number of partitions of a number into squares.

We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].

Davenport's theorem was established nearly a lifetime ago, but there have been some very interesting recent developments. The various proofs over the years bring in different ideas from number theory, probability theory, analysis and group theory. In this short survey, we shall not present complete proofs, but will describe instead some of these underlying ideas.

Here we try to have a glimpse of the polynomial method used in the study of some zero-sum problems in additive combinatorics. While the first few sections of this paper are of an expository nature, in the last section we present some new results on bounds of some zero-sum constants obtained by generalizing a polynomial method. We also mention some open questions.

In a very recent work, G. E. Andrews defined the combinatorial objects which he called singular overpartitions with the goal of presenting a general theorem for overpartitions which is analogous to theorems of Rogers–Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his work, Andrews noted two congruences modulo 3 which followed from elementary generating function manipulations. In this work, we show that Andrews' results modulo 3 are two examples of an infinite family of congruences modulo 3 which hold for that particular function. We also expand the consideration of such arithmetic properties to other functions which are part of Andrews' framework for singular overpartitions.

For each of the functions f ∈ {φ, σ, ω, τ} and every natural number K, we show that there are infinitely many solutions to the inequalities f(p_n - 1) < f(p_n+1 - 1) < ⋯ < f(p_n+K - 1), and similarly for f(p_n -1) > f(p_n+1 - 1) > ⋯ > f(p_n+K -1). We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

Let x, h and Q be three parameters. We show that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, every reduced arithmetic progression a(mod q) has approximately the expected number of primes p from the interval (y, y + h], provided that h > x^1/6+ϵ and Q satisfies appropriate bounds in terms of h and x. Moreover, we prove that, for most moduli q ≤ Q and for most positive real numbers y ≤ x, there is at least one prime p ∈ (y, y + h] lying in every reduced arithmetic progression a(mod q), provided that 1 ≤ Q² ≤ h/x^1/15+ϵ.

The objective in this paper is to present a general theorem for overpartitions analogous to Rogers–Ramanujan type theorems for ordinary partitions with restricted successive ranks.

In his lost notebook, Ramanujan recorded two identities involving double series of Bessel functions that are closely connected with the classical, unsolved circle and divisor problems. In a series of papers with Zaharescu, the authors proved these identities under various interpretations, as well as Riesz mean analogues. In this paper, logarithmic mean analogues, also involving double series of Bessel functions, are established. Weighted divisor sums involving characters play a central role.

The "strange" function of Kontsevich and Zagier is defined by

This series is defined only when q is a root of unity, and provides an example of what Zagier has called a "quantum modular form". In their recent work on congruences for the Fishburn numbers ξ(n) (whose generating function is F(1 - q)), Andrews and Sellers recorded a speculation about the polynomials which appear in the dissections of the partial sums of F(q). We prove that a more general form of their speculation is true. The congruences of Andrews–Sellers were generalized by Garvan in the case of prime modulus, and by Straub in the case of prime power modulus. As a corollary of our theorem, we reprove the known congruences for ξ(n) modulo prime powers.

Recently, in three separate papers, Andrews and Merca, Yee, and Kolitsch and Burnette proved some results concerning the Truncated Pentagonal Number Theorem. In this paper, a family of generating functions that keeps track of the number of different parts in a partition will be studied. This family of generating functions can be used to get the earlier results on the Truncated Pentagonal Number Theorem as well as several other results.

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

Kahane provided a proof of a conjecture of Bateman and Diamond on Beurling generalized primes. The conjecture says that the L²-condition

These results cover the preceding forms and extend further the Beurling PNT. As part of our argument, we extend first the theorems on Chebyshev bounds with correspondingly weaker conditions. These bounds play an important role in later proofs.

We construct a number system for representing numbers in [0, 1] that is based on iterations of an asymmetric tent map that incorporates the golden ratio. It is already known that in this system a number has a periodic representation if and only if it lies in . We investigate other aspects of this system such as non-uniquely representable numbers, an inherent semigroup structure, connections with Wythoff's game, related sequences of rational functions, and its connection with an iterative scheme reminiscent of paper folding analysis. Many details of the above-mentioned connections are conjectural.

Let σ be the usual sum-of-divisors function. We say that a and b form a harmonious pair if ; equivalently, the harmonic mean of and is 2. For example, 4 and 12 form a harmonious pair, since and . Every amicable pair is harmonious, but there are many others. We show that the count of numbers that belong to a harmonious pair having both members in [1, x] is at most , as x → ∞.

The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than 10¹⁴⁰. Both of their explicit estimates are on restricted ranges. In this paper, we prove an explicit estimate that works for any range. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime p be to ensure that there is a kth power non-residue less than p^1/6.

The Fishburn numbers ξ(n) are defined by the formal power series

Recently, Andrews and Sellers discovered congruences of the form ξ(pm + j) ≡ 0 modulo p, valid for all m ≥ 0. These congruences have then been complemented and generalized to the case of r-Fishburn numbers by Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.

Here, we show that the title equation has only finitely many positive integer solutions (X, m, n), where φ is the Euler function.

In this paper, we consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.

Let Ω be a non-zero holomorphic 2-form on a K3 surface S. Suppose that S is projective algebraic and is defined over . Let be the -vector space generated by the numbers given by all the periods ∫_γ Ω, γ ∈ H₂(S, ℤ). We show that, if , then S has complex multiplication, meaning that the Mumford–Tate group of the rational Hodge structure on H²(S, ℚ) is abelian. This result was announced in [P. Tretkoff, Transcendence and CM on Borcea–Voisin towers of Calabi–Yau manifolds, J. Number Theory 152 (2015) 118–155], without a detailed proof. The converse is already well known.