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Vivian Kuperberg

Publications and source records attributed to Vivian Kuperberg.

At least 19 recordsLinked to original sources
arXiv2026

Murmurations in the depth aspect

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

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arXiv2026

Odd moments and adding fractions

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 application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in \mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest.

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arXiv2026

Distributions of left prime truncations

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 digits. We explore here the distributions of the number of left prime truncations of integers and of the number of irreducible truncations of polynomials with coefficients over a finite field, focusing on the proportion among all $\ell$-digit numbers or polynomials, their variance, and the maximal proportion.

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arXiv2025

Positive density for consecutive runs of sums of two squares

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 $a_1$ and $a_2$ mod $q$, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod $q$, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod $q$. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class $a$ mod $q$ and for any $r \ge 1$, a positive density of primes begin a sequence of $r$ consecutive primes, all of which are $a$ mod $q$.

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arXiv2024

Consecutive runs of sums of two squares

We study the distribution of consecutive sums of two squares in arithmetic progressions. If $\{E_n\}_{n \in \mathbb{N}}$ is the sequence of sums of two squares in increasing order, we show that for any modulus $q$ and any congruence classes $a_1,a_2,a_3 \mod q$ which are admissible in the sense that there are solutions to $x^2 + y^2 \equiv a_i \mod q$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_i \mod q$, for $i = 1,2,3$. We also show that for any $r_1, r_2 \ge 1$, there exist infinitely many $n$ with $E_{n+i-1} \equiv a_1 \mod q$ for $1 \le i \le r_1$ and $E_{n+ i - 1} \equiv a_2 \mod q$ for $r_1 + 1 \le i \le r_1 + r_2$.

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arXiv2024

Arithmetic constants for symplectic variances of the divisor function

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 results over function fields proven in [arXiv:2107.01437]. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.

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arXiv2024

Odd moments in the distribution of primes

Montgomery and Soundararajan showed that the distribution of $ψ(x+H) - ψ(x)$, for $0 \le x \le N$, is approximately normal with mean $ \sim H$ and variance $\sim H \log (N/H)$, when $N^δ \le H \le N^{1-δ}$. Their work depends on showing that sums $R_k(h)$ of $k$-term singular series are $μ_k(-h \log h + Ah)^{k/2} + O_k(h^{k/2-1/(7k) + \varepsilon})$, where $A$ is a constant and $μ_k$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_k(h) \asymp h^{(k-1)/2}(\log h)^{(k+1)/2}$. We prove an upper bound with the correct power of $h$ when $k = 3$, and prove analogous upper bounds in the function field setting when $k =3$ and $k = 5$. We provide further evidence for this conjecture in the form of numerical computations.

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arXiv2024

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

In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in the limit as $q \to \infty$ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [arXiv:1504.07804] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.

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arXiv2023

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

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 computing average values of the singular series constants over different sets of a fixed size $k$ contained in an interval $[1,h]$ as $h \to \infty$, and then using this average to compute moments of the distribution of primes. In this paper, we study averages where $k$ is relatively large with respect to $h$. We then apply these averages to the tail of the distribution. For example, we show, assuming appropriate Hardy--Littlewood conjectures and in certain ranges of the parameters, the number of intervals $[n,n +λ\log x]$ with $n\le x$ containing at least $k$ primes is $\ll x\exp(-k/(λe)).$

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arXiv2023

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

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 derived in a paper of the second author, Rodgers, and Roditty-Gershon. Adapting work of the first author and Fiorilli in the integer case, we establish lower bounds on a weighted version of these moments which agree with the conjectured values.

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arXiv2023

Sums of singular series along arithmetic progressions and with smooth weights

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 constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.

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arXiv2021

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

We explore two questions about pseudo-polynomials, which are functions $f:\mathbb N \to \mathbb Z$ such that $k$ divides $f(n+k) - f(n)$ for all $n,k$. First, for certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider $α$-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for $k$ lying in a set of primes of density $α$. We show that if an $α$-primary pseudo-polynomial is $O(e^{(2/3-ε) n})$, then it is a polynomial.

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arXiv2021

Sums of divisor functions and von Mangoldt convolutions in $\mathbb F_q[T]$ leading to symplectic distributions

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 over arithmetic progressions for the function field $\mathbb F_q[T]$ to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.

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arXiv2020

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

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 that arises in these conjectures, up to lower order terms. More exactly, we find asymptotic formulas for smoothed sums of the singular series minus one. Based upon Gross and Smith's conjectures, we use our result to suggest that for large enough 'short intervals' in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramer model prediction by a universal factor, independent of K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers. Numerical data is provided supporting the conjecture.

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arXiv2017

Soficity and variations on Higman's group

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 amalgamation of four copies of the Baumslag--Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag--Solitar group by other groups $G$. We give an elementary condition on $G$, enjoyed for example by $\mathbb{Z} \wr \mathbb{Z}$ and the integral Heisenberg group, under which the resulting group is sofic. We then use soficity to deduce that there exist permutations of $\mathbb{Z} / n\mathbb{Z}$ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.

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arXiv2017

Chip-firing on trees of loops

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 "trees of loops": the only trees of loops that are Brill-Noether general are paths of loops. We study various notions of generality and examine which of these graphs satisfy them.

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