Alexandru Zaharescu

Simultaneous Unimodularity of Positive Integer Matrices and Their Entrywise Cubes

Xinyi Li, Shuochen Wang, Zimeng Ye +1

We solve an open problem posed by Richard Guy in his book Unsolved Problems in Number Theory. We prove that, for every integer $n \geq 10$ , there exist infinitely many $n \times n$ in…

On the exponential sum over squarefree integers

Nicolas Robles, Alexandru Zaharescu, Dirk Zeindler

Let $μ$ be the Möbius function and $e (t) = e^{2 π i t}$ . We prove that if $N \geq 2$ , $α \in R$ , $(a, q) = 1$ , and $∣ α - a / q ∣ \leq q^{- 2}$ , then \[\bigg|\sum_{n\le N}μ^2(n)e(αn)\bigg|\ll\l…

Self-intersection Points of Billiard Trajectories in a Square with Small Pockets

Anji Dong, Colin Edsey, Shuta Iwai +1

We study the self-intersections of billiard trajectories in a square with small pockets of size $ε$ removed from its four corners. In particular, we establish an asymptot…

math.NT2026★ 1 cited

Representations as Sums of Icosahedral and Dodecahedral Numbers: Proof of Pollock's Conjectures

Debmalya Basak, Anji Dong, Katerina Saettone +1

On the occasion of George Andrews' and Bruce Berndt's combined 170th birthday, we prove two conjectures of Sir Frederick Pollock that are more than 170 years old. In 1843, Pollock…

Erdős-Moser Equation in Arithmetic Progressions

Anji Dong, Vi Anh Nguyen, Alexandru Zaharescu

We consider the Erdős-Moser equation $1^{k} + 2^{k} + \dots + (m - 1)^{k} = m^{k}$ in arithmetic progressions. We prove among other things that when $k = 2$ , for any solution to exist, the above sum…

Admissible Pairs: A Variation to Pollock Conjectures

Anji Dong, Vi Anh Nguyen, Alexandru Zaharescu

We introduce a notion of an admissible pair, and prove a variation to Pollock's conjectures on icosahedral and dodecahedral numbers.

Binomial coefficients with divisors avoiding an interval

Hung M. Bui, Slava Naprienko, Kyle Pratt +1

We solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient $(k n)$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\le…

Effective Estimates for a Class of Farey Fraction Sums and Bounds for Mundici-Type Constants

Anji Dong, Huy Xuan Nguyen, Vi Anh Nguyen +1

Let $D_{2} (Q)$ denote the sum of squared distances between consecutive Farey fractions in the full interval $(0, 1]$ . Daniele Mundici conjectured that $C(Q):=D_{2}(Q)\cdot Q^2/\log…

The $θ = \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$ -functions

Anji Dong, Nawapan Wattanawanichkul, Alexandru Zaharescu

The $θ = \infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an ana…

A statistical model for points expanding in higher dimensions while being tied to bijective involutions

Cristian Cobeli, The Nguyen, Alexandru Zaharescu

Let $M$ be a set with $M$ elements, let $ψ : M \to M$ be a bijective involution, and let~ $X_{ψ}$ be the set of sequences $(x_1,\dots,…

Zeros of Polynomials in Derivatives of Automorphic $L$ -functions

Anji Dong, Nawapan Wattanawanichkul, Alexandru Zaharescu

Let $F_{m}$ be the set of all cuspidal automorphic representations of $GL_{m} (A_{Q})$ , and let $F (s, π)$ be a polynomial in the derivati…

Cannonball Polygons with Multiplicities

Anji Dong, Katerina Saettone, Kendra Song +1

We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$ . We associate these functions $w$ with certain polygons, which we cal…

An Equidistribution Result for Differences Associated to Square Pyramidal Numbers II

Anji Dong, Katerina Saettone, Kendra Song +1

This paper presents some new results concerned with uniform distribution properties associated with the sequence $(a_{n})_{n \in N}$ , which is defined as the distance from the…

An Arithmetic Sum Associated with the Classical Theta Function

Bruce C. Berndt, Raghavendra N. Bhat, Jeffrey L. Meyer +2

The sum $S (h, k) := \sum_{j = 1}^{k - 1} (- 1)^{j + 1 + [hj / k]}$ appears in the modular transformation formulae of the classical theta function $ϑ_{3} (z)$ . The double sum $S(k) := \sum_{h…

Exponential sums twisted by general arithmetic functions

Anji Dong, Nicolas Robles, Alexandru Zaharescu +1

We examine exponential sums of the form $\sum_{n \leq X} w (n) e^{2 π i α n^{k}}$ , for $k = 1, 2$ , where $α$ satisfies a generalized Diophantine approximation and where $w$ are different arit…

On the order of 4-dimensional regular polytope numbers

Anji Dong, The Nguyen, Alexandru Zaharescu

In light of Kim's conjecture on regular polytopes of dimension four, which is a generalization of Waring's problem, we establish asymptotic formulas for representing any sufficient…

Strange and pseudo-differentiable functions with applications to prime partitions

Anji Dong, Nicolas Robles, Alexandru Zaharescu +1

Let $p_{P_{r}} (n)$ denote the number of partitions of $n$ into $r$ -full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_…

An Equidistribution Result for Differences Associated with Square Pyramidal Numbers

Anji Dong, Katerina Saettone, Kendra Song +1

We provide an asymptotic formula for the average value of the sequence A351830: $a_{n} = ∣ P_{n} - y_{n}^{2} ∣$ for $1 \leq n \leq x$ , where $P_{n}$ is the $n$ -th square pyramidal nu…

On the Pair Correlation of Zeros of $L$ -Functions for Non-CM Newforms in Shifted Ranges

Di Liu, Clayton Williams, Alexandru Zaharescu

We study the pair correlation between zeros of a shifted auxiliary $L$ -function attached to a non-CM newform, the scale of which is a fixed constant. We prove an unconditional as…