Vivian Kuperberg

activity

20142026

most citedOdd moments in the distribution of primes

1 citations · 2 across the 17 of their papers we have counts for

collaborators

20 papers

math.NT2026

Distributions of left prime truncations

Vivian Kuperberg, Matilde Lalín

The prime number 357686312646216567629137 is notable because of the unusual property that it remains prime successively on removing the left digit until there are no remaining digi…

math.NT2026

Murmurations in the depth aspect

Claire Burrin, Vivian Kuperberg, Min Lee +2

We compute the murmuration density function for the family of Hecke forms of weight $k$ and prime power level $N = ℓ^{a}$ , with $ℓ$ a fixed odd prime and $a \to \infty$ .

math.NT2024

Arithmetic constants for symplectic variances of the divisor function

Vivian Kuperberg, Matilde Lalín

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_{k} (n)$ over number fields, which were inspired by analogous res…

math.NT2024★ 1 cited

Positive density for consecutive runs of sums of two squares

Noam Kimmel, Vivian Kuperberg

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus $q$ , any two reduced congruence classes $a_{1}$ a…

math.NT2024

The gaps in the multiplication table

Emmanuel Kowalski, Vivian Kuperberg

We determine the complete list of the gaps between successive elements of the multiplication table of the first N integers.

math.NT2024

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Muhammad Afifurrahman, Vivian Kuperberg, Alina Ostafe +1

We consider the set of $n \times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matri…

math.NT2023

Odd moments and adding fractions

Thomas F. Bloom, Vivian Kuperberg

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an applicat…

math.NT2023

Lower bounds on weighted moments of primes in short intervals in number fields

Régis de la Bretèche, Vivian Kuperberg

We consider an analog of a conjecture of Montgomery and Soundararajan on the moments of primes in short intervals in number fields; this analog was discussed and heuristically deri…

math.NT2023

Consecutive runs of sums of two squares

Noam Kimmel, Vivian Kuperberg

We study the distribution of consecutive sums of two squares in arithmetic progressions. If ${E_{n}}_{n \in N}$ is the sequence of sums of two squares in increasing order,…

math.NT2023

Sums of singular series along arithmetic progressions and with smooth weights

Vivian Kuperberg

Sums of the singular series constants that appear in the Hardy--Littlewood $k$ -tuples conjectures have long been studied in connection to the distribution of primes. We study const…

math.NT2022

Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

Vivian Kuperberg, Matilde Lalín

In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_{k} (f)$ over the function field $F_{q} [T]$ in the limit as $q \to \infty$ …

math.NT2022

Sums of singular series with large sets and the tail of the distribution of primes

Vivian Kuperberg

In 1976, Gallagher showed that the Hardy--Littlewood conjectures on prime $k$ -tuples imply that the distribution of primes in log-size intervals is Poissonian. He did so by computi…

math.NT2021★ 1 cited

Odd moments in the distribution of primes

Vivian Kuperberg

Montgomery and Soundararajan showed that the distribution of $ψ (x + H) - ψ (x)$ , for $0 \leq x \leq N$ , is approximately normal with mean $\sim H$ and variance $\sim H lo g (N / H)$ , whe…

math.NT2021

Sums of divisor functions and von Mangoldt convolutions in $F_{q} [T]$ leading to symplectic distributions

Vivian Kuperberg, Matilde Lalín

In [arXiv:1504.07804], Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_{k} (f)$ over short intervals and…

math.NT2020

On pseudo-polynomials divisible only by a sparse set of primes and $\a$-primary pseudo-polynomials

Vivian Kuperberg

We explore two questions about pseudo-polynomials, which are functions $f : N \to Z$ such that $k$ divides $f (n + k) - f (n)$ for all $n, k$ . First, for certain arbitrari…

math.NT2020

Sums of singular series and primes in short intervals in algebraic number fields

Vivian Kuperberg, Brad Rodgers, Edva Roditty-Gershon

Gross and Smith have put forward generalizations of Hardy - Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series tha…

math.GR2017

Soficity and variations on Higman's group

Martin Kassabov, Vivian Kuperberg, Timothy Riley

A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalga…

math.CO2017

Chip-firing on trees of loops

Sameer Kailasa, Vivian Kuperberg, Nicholas Wawrykow

Cools, Draisma, Payne, and Robeva proved that generic metric graphs that are "paths of loops" are Brill-Noether general. We show that Brill-Noether generality does not hold for "tr…

math.MG2015

Packings of equal disks in a square torus

Robert Connelly, Matthew Funkhouser, Vivian Kuperberg +1

Packings of equal disks in the plane are known to have density at most $π / 12$ , although this density is never achieved in the square torus, which is what we call the plane m…

math.CO2014

Hadamard matrices modulo p and small modular Hadamard matrices

Vivian Kuperberg

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$ -modular and $11$ -modular versions of the Hadamard conjecture fo…

Distributions of left prime truncations

Murmurations in the depth aspect

Arithmetic constants for symplectic variances of the divisor function

Positive density for consecutive runs of sums of two squares

The gaps in the multiplication table

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Odd moments and adding fractions

Lower bounds on weighted moments of primes in short intervals in number fields

Consecutive runs of sums of two squares

Sums of singular series along arithmetic progressions and with smooth weights

Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime

Sums of singular series with large sets and the tail of the distribution of primes

Odd moments in the distribution of primes

Sums of divisor functions and von Mangoldt convolutions in Fq​[T] leading to symplectic distributions

On pseudo-polynomials divisible only by a sparse set of primes and $\a$-primary pseudo-polynomials

Sums of singular series and primes in short intervals in algebraic number fields

Soficity and variations on Higman's group

Chip-firing on trees of loops

Packings of equal disks in a square torus

Hadamard matrices modulo p and small modular Hadamard matrices

Sums of divisor functions and von Mangoldt convolutions in $F_{q} [T]$ leading to symplectic distributions