International Journal of Number Theory

For a positive integer N, define hypergeometric Cauchy numbers c_N,n by

A classical result of Chowla [Improvement of a theorem of Linnik and Walfisz, Proc. London Math. Soc. (2) 50 (1949) 423–429 and The Collected Papers of Sarvadaman Chowla, Vol. 2 (Centre de Recherches Mathematiques, 1999), pp. 696–702] states that for infinitely many fundamental discriminants D we have

In this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ₊ be fixed. Does there exist a positive integer k such that

The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that

The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems.

Let s ≥ 1, A_i ⊆ {1,2,…,N} (1 ≤ i ≤ s) and k ≥ 3 be an odd integer. In this paper, one of the main results is: if , then (a) each of A_i - A_j (1 ≤ i, j ≤ s) contains an arithmetic progression of length k with the same common difference; (b) all sets A_i - A_i (1 ≤ i ≤ s) contain a common arithmetic progression of length k. Finally, we pose several problems for further research.

We investigate the size of L^p-integrals for exponential sums over k-free numbers and prove essentially tight bounds.

In this paper we prove asymptotic formulas for general moments of partial Euler products and the first and the second moments of partial Hadamard products related to central values of the family of L-functions associated to the symmetric square lifts of holomorphic modular forms for SL₂(ℤ). Then using a hybrid Euler–Hadamard product formula for the central value, we relate these results with conjectures for general power moments of L-functions in this family and with Random Matrix Theory interpretations. This continues the work done previously by Gonek–Hughes–Keating and Bui–Keating for other families of L-functions.

We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant of elliptic curves. Our proof does not use the circle method. We use Laplace's method of steepest descent and the Hardy–Ramanujan asymptotic formula for the partition function. (The latter asymptotic formula can be derived without the circle method.)

In this paper we study totally definite quaternion fields over the rational field and over quadratic number fields. We establish a complete list of all such fields which are Euclidean. Moreover, we prove that every field in this list is in fact norm-Euclidean. The proofs are both theoretical and algorithmic.

Bringmann, Mahlburg, and Rhoades have found asymptotic expressions for all moments of the partition statistics rank and crank. In this work we extend their methods to higher ranks. The T-rank, introduced by Garvan, for odd integers T = 3 is a natural generalization of the rank (T = 3) and crank (T = 1).

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function and two smallest parts functions: for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for , , and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ²).

We prove that crystalline points are dense in the spectrum of the completed Hecke algebra on unitary Shimura varieties.

The ring of p-typical Witt vectors is an indispensable tool in number theory and mixed characteristic commutative algebra. Witt vectors were significantly generalized by Dress and Siebeneicher by producing for any profinite group G, a ring-valued functor W_G. The p-typical Witt vectors are recovered as the example G = Z_p. This paper explores the structure of the ring W_D2^∞(k) where k is a field of characteristic 2 and .

We call the polynomial a Barker polynomial of degree n-1 if each a_j ∈{-1, 1} and

Theorem.Ifn ≔ 2m + 1 > 13and

A clever usage of Newton's identities plays a central role in our elegant proof.

In this paper the metric theory of Diophantine approximation associated with the small linear forms is investigated. Khintchine–Groshev theorems are established along with Hausdorff measure generalization without the monotonic assumption on the approximation function.

Let 𝔄 be a commutative ℚ-algebra, A be an alphabet of non-commutative letters and 𝔥¹ be the non-commutative polynomial algebra generated by the set A over the algebra 𝔄. The harmonic algebra 𝔥¹ is closely related to multiple zeta values and multiple zeta-star values in some special case. Some identities in the algebra 𝔥¹ are given.

We evaluate the convolution sums ∑_{l,m∈ℕ,l+15m=n}σ(l)σ(m) and ∑_{l,m∈ℕ,3l+5m=n}σ(l)σ(m) for all n ∈ ℕ using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form