International Journal of Number Theory

We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such functions are Jacobi forms. In establishing these results, we construct other functions which are also Jacobi forms. These results are motivated by applications in the theory of vertex operator algebras.

For a cyclic group G and a ∈ G, we define 〈a〉 as the subgroup generated by a and the atom of a as the set of all elements generating 〈a〉. In this paper, given t (t ≥ 2) atoms, we obtain an exact formula for the number of representations of each element in the sumset of these t atoms. We also show explicitly which atoms are part of the union which constitutes the sumset of t given atoms. The case t = 2 was recently obtained by Sander and Sander.

Using modular forms we determine the number of representations of a positive integer by certain diagonal octonary quadratic forms with coefficients 1, 3 or 9.

The theory of quasimodular forms is used to evaluate the convolution sums

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ₀ = ℕ ∪{0}. Let a₁x² + b₁xy + c₁y² and a₂z² + b₂zt + c₂t² be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ₀ as a sum of these two binary quadratic forms is

In this paper, the convolution sum ∑_i+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams.

In this paper, we give formulas for the coordinates of the Greenberg transform of a polynomial over rings of Witt vectors. Besides being of independent theoretical interest, these formulas can be used to obtain improvement on computations with Witt vectors of arbitrary length in cases where these cannot be reduced to computation on well-known local rings.

Let be a commutative ring with identity and be a polynomial. In the present paper we consider digit representations in the residue class ring . In particular, we are interested in the question whether each can be represented modulo P in the form e₀ + e₁x + ⋯ + e_hx^h, where the are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

We prove that the dimension of the Hecke module generated by a certain eta-quotient is equal to the class number of an imaginary quadratic field. To do this, we relate the eta-quotient to the Hecke theta series attached to a ray class character of the imaginary quadratic field.

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point τ. These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on τ and a prime number p that, if satisfied, imply that p^m divides the algebraic part of all the Taylor coefficients of f of sufficiently high degree. We also give effective bounds on the largest n such that p^m does not divide the algebraic part of the nth Taylor coefficient of f at τ that are sharp under certain additional hypotheses.

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form φ(z; τ) are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of φ at each pole, as well as some well-known real analytic functions, that appear for instance in the completion of Appell–Lerch sums.

We disprove a conjecture of Guo and Krattenthaler on the distribution of binomial coefficients modulo primes by providing counterexamples for p = 5.

In this paper, we obtain two effective bounds for the j-invariant of integral points on certain modular curves which have positive genus and less than three cusps.

In this paper, we investigate the limiting behavior of short incomplete Gauss sums at random argument as the number of terms goes to infinity. We prove that the limit distribution is given by the distribution of theta sums and differs from the limit law for long Gauss sums studied by the author and Marklof. The key ingredient in the proof is an equidistribution theorem for rational points on horocycles in the metaplectic cover of SL(2, ℝ).

We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression of real numbers and a sufficiently large integer n (depending on ), there is a need of at least Cn geometric progressions to cover the first n terms of . A similar result is presented, with the role of arithmetic and geometric progressions reversed.

For a positive integer n, let ped(n) be the number of partitions of n where the even parts are distinct, and be the number of overpartitions of n into odd parts. Moreover, let Q(n) denote the number of the partition pairs of n into two colors (say, red and blue), where the parts colored red satisfy restrictions of partitions counted by ped(n), while the parts colored blue satisfy restrictions of partitions counted by . We establish several congruences for Q(n). We also obtain an asymptotic formula for Q(n).