International Journal of Number Theory

We attach Hecke polynomials [Formula: see text]([Formula: see text]) to weak Hecke eigenforms [Formula: see text] of weight 2 – [Formula: see text] and show that, for large [Formula: see text], every zero is simple and lies in [0, 1728]. The construction pulls back a weakly holomorphic Hecke combination of [Formula: see text] along [Formula: see text]; the analysis follows Hecke orbits on the uni…

Let [Formula: see text] be a number field, [Formula: see text] a real number, and [Formula: see text] integers. Let [Formula: see text] and [Formula: see text] be sequences of integers such that [Formula: see text] and [Formula: see text] for all 1 [Formula: see text]. We give an upper bound for the number of integers [Formula: see text] for which [Formula: see text], for each [Formula: see text]…

Recently, Andrews and Hopkins provided a partial extension of the Crank-Mex theorem by refining the equinumerosity among partitions with even mex, with a fixed point, with negative crank and with positive crank in terms of the number of parts greater than one. In this work, we first give a further refinement of Andrews and Hopkins' result by including the 2-measure and the size of the Durfee squa…

We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.

Let [Formula: see text] denote the number of partitions of [Formula: see text] into distinct odd parts. In this paper, we prove that the quartic Jensen polynomial associated with [Formula: see text]: [Formula: see text] has four distinct real roots for all [Formula: see text]. To achieve this, we establish several inequalities for [Formula: see text] by utilizing its arbitrary-precision error est…

We prove Conjecture 10 of Cai in A Modern Introduction to Classical Number Theory, World Scientific Publishing Co. Pte., Singapore (2021).

Let [Formula: see text] be nonzero integers satisfying [Formula: see text]. We consider solutions of the unequal powers equation [Formula: see text], where [Formula: see text] and [Formula: see text] belong to two certain sets [Formula: see text] and [Formula: see text] of Piatetski-Shapiro primes corresponding to [Formula: see text], respectively. We prove a Roth-type result that for [Formula: s…

We present an infinite family of identities that represent Ramanujan’s tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level 1 cusp forms from modular forms of higher levels.

This paper provides a modular construction of p-adic L-functions for real quadratic fields. The key ingredient in this construction is the use of period integrals of automorphic forms. This work extends the previous results of Darmon and Dasgupta.

Let [Formula: see text] be the continued fraction expansion of [Formula: see text]. In this paper, for any [Formula: see text], we study the multifractal spectrum of the convergence exponent of the weighted product of consecutive partial quotients [Formula: see text] defined by [Formula: see text] The weighted products of more than two consecutive partial quotients are also discussed.

Let [Formula: see text] be a smooth projective curve of genus [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero. For integers [Formula: see text] sufficiently large and [Formula: see text], we provide a bound for the heights of solutions in the function field [Formula: see text] to the equation [Formula: see text], where [Formula: see text] are nonz…

Covering systems of the integers were introduced by Erdős in 1950. Since then, many beautiful questions and conjectures about these objects have been posed. Most famously, Erdős asked whether the minimum modulus of a covering system with distinct moduli can be arbitrarily large. This problem was resolved in 2015 by Hough, who proved that the minimum modulus is bounded. In 2022, Balister et al. de…

We prove explicit Erdős–Wintner bounds for Cantor numeration systems via a trailing-window decomposition. We temporarily discard the last block of digits (the “window”) and analyze the remaining prefix, viewed through a short-range Markov/window chain. The resulting Kolmogorov-distance estimate splits into three contributions: (i) a bridge loss coming from discarding the window; (ii) a variance-t…

The structure of the multiplicative group [Formula: see text] encodes a great deal of arithmetic information about the integer [Formula: see text] (examples include [Formula: see text], the Carmichael function [Formula: see text], and the number [Formula: see text] of distinct prime factors of [Formula: see text]). We examine the invariant factor structure of [Formula: see text] for typical integ…

Let [Formula: see text] be a multi-quadratic totally real number field. Let [Formula: see text] denote its distinct embeddings. Given [Formula: see text] we give an explicit formula for [Formula: see text] and [Formula: see text] where [Formula: see text] Let [Formula: see text] be a fractional ideal in [Formula: see text] and [Formula: see text] The set of shortest nonzero lattice points for [Fo…

Let [Formula: see text] be a fixed positive integer. Let [Formula: see text] be a linear recurrence sequence for every [Formula: see text], and we set [Formula: see text], where [Formula: see text]. In this paper, we obtain sufficient conditions on [Formula: see text] so that the intervals [Formula: see text] do not contain any prime numbers for infinitely many integers [Formula: see text], where…

Due to a study of multi-dimensional Kronecker sequences, we realise the importance of the growth rate and the distribution of multiple sums of partial quotients in continued fraction expansions. This leads us to consider an extension of Philipp’s Limit Theorem, or the strong law of large numbers on continued fractions, to multiple sums of partial quotients. In this paper, we extend the first half…

Let [Formula: see text] be a cuspidal Hermitian eigenform of degree two over [Formula: see text], with first Fourier–Jacobi coefficient not identically zero. Building on a paper by Das and Jha, we prove the meromorphic continuation to [Formula: see text] and the functional equation of a degree six [Formula: see text]-function attached to [Formula: see text] by Gritsenko, twisted by a Dirichlet ch…

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If [Formula: see text] is an irrational number having a continued fraction expansion with bounded terms (in particular, if [Formula: see text] is a quadratic irrational), then the number of primes [Formula: see text] in…

In 2021, Andrews studied the generating function for [Formula: see text], where [Formula: see text] denotes the total number of parts in the partitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text], and then proved two congruences for [Formula: see text] conjectured by Beck. Later, Chern introduced [Formula: see text], which denotes the total number …