Brad Rodgers

activity

20122026

most citedArithmetic functions in short intervals and the symmetric group

23 citations · 141 across the 21 of their papers we have counts for

collaborators

21 papers

math.CA2026

On the optimal constant in the Montgomery-Vaughan weighted Hilbert inequality

Brad Rodgers

We give a proof that the optimal constant in the weighted Hilbert inequality is strictly greater than $π$ , answering in the negative a question asked by Montgomery and Vaughan. The…

math-ph2023

Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble

Brad Rodgers, Harshith Sai Vallabhaneni

We find closed formulas for arbitrarily high mixed moments of characteristic polynomials of the Alternative Circular Unitary Ensemble (ACUE), as well as closed formulas for the ave…

math.NT2022★ 1 cited

Averages of quadratic twists of long Dirichlet polynomials

Brian Conrey, Brad Rodgers

We investigate averages of long Dirichlet polynomials twisted by Kronecker symbols and we compare our result with the recipe of [CFKRS]. We are able to compute these averages in th…

math.NT2021

Squarefrees are Gaussian in short intervals

Ofir Gorodetsky, Alexander P. Mangerel, Brad Rodgers

We show that counts of squarefree integers up to $X$ in short intervals of size $H$ tend to a Gaussian distribution as long as $H \to \infty$ and $H = X^{o (1)}$ . This answers…

math.NT2020★ 14 cited

Moments of polynomials with random multiplicative coefficients

Jacques Benatar, Alon Nishry, Brad Rodgers

For $X (n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $P_{N} (θ) = \frac{1}{N} n \leq N \sum X (n) e (n θ),$ and show that the…

math.NT2020★ 11 cited

On the variance of squarefree integers in short intervals and arithmetic progressions

Ofir Gorodetsky, Kaisa Matomäki, Maksym Radziwiłł +1

We evaluate asymptotically the variance of the number of squarefree integers up to $x$ in short intervals of length $H < x^{6/11 - ε}$ and the variance of the number of s…

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.GR2019★ 6 cited

Traces of powers of matrices over finite fields

Ofir Gorodetsky, Brad Rodgers

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $U (n, C)$ . Diaconis and Shahshahani proved that the traces of $M, M^{2}, \dots, M^{k}$ …

math.PR2019★ 3 cited

Band-limited mimicry of point processes by point processes supported on a lattice

Jeffrey C. Lagarias, Brad Rodgers

We say that one point process on the line $R$ mimics another at a bandwidth $B$ if for each $n \geq 1$ the two point processes have $n$ -level correlation functions that agr…

math.NT2019★ 4 cited

Higher Correlations and the Alternative Hypothesis

Jeffrey C. Lagarias, Brad Rodgers

The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced which one would like to rule out. In the Alternative Hy…

math.NT2018★ 4 cited

The variance of the number of sums of two squares in $F_{q} [T]$ in short intervals

Ofir Gorodetsky, Brad Rodgers

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short interval…

math.NT2018★ 7 cited

The De Bruijn-Newman constant is non-negative

Brad Rodgers, Terence Tao

For each $t \in R$ , define the entire function $H_{t} (x) := \int_{0}^{\infty} e^{t u^{2}} Φ (u) cos (xu) d u$ where $Φ$ is the super-exponentially decaying function $$ Φ(u) := \sum_…

math.PR2017

The limiting characteristic polynomial of classical random matrix ensembles

Reda Chhaibi, Emma Hovhannisyan, Joseph Najnudel +2

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ens…

math.NT2016★ 14 cited

The variance of divisor sums in arithmetic progressions

Brad Rodgers, Kannan Soundararajan

We study the variance of sums of the $k$ -fold divisor function $d_{k} (n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted ra…

math.NT2016★ 23 cited

Arithmetic functions in short intervals and the symmetric group

Brad Rodgers

We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements o…

math.CA2016★ 10 cited

On the distribution of Rudin-Shapiro polynomials and lacunary walks on $S U (2)$

Brad Rodgers

We characterize the limiting distribution of Rudin-Shapiro polynomials, showing that, normalized, their values become uniformly distributed in the disc. This resolves conjectures o…

math.NT2015★ 2 cited

Tail bounds for counts of zeros and eigenvalues, and an application to ratios

Brad Rodgers

Let $t$ be random and uniformly distributed in the interval $[T, 2 T]$ , and consider the quantity $N (t + 1/ lo g T) - N (t)$ , a count of zeros of the Riemann zeta function in a box of he…

math.NT2014★ 6 cited

Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function

Kenneth Maples, Brad Rodgers

We unconditionally prove a central limit theorem for linear statistics of the zeros of the Riemann zeta function with diverging variance. Previously, theorems of this sort have bee…

math.NT2013★ 5 cited

The covariance of almost-primes in $F_{q} [T]$

Brad Rodgers

We estimate the covariance in counts of almost-primes in $F_{q} [T]$ , weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This genera…

math.PR2012★ 19 cited

A central limit theorem for the zeroes of the zeta function

Brad Rodgers

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic inter…

math.NT2012★ 12 cited

Macroscopic pair correlation of the Riemann zeroes for smooth test functions

Brad Rodgers

On the assumption of the Riemann hypothesis, we show that over a class of sufficiently smooth test functions, a measure conjectured by Bogomolny and Keating coincides to a very sma…

On the optimal constant in the Montgomery-Vaughan weighted Hilbert inequality

Autocorrelations of characteristic polynomials for the Alternative Circular Unitary Ensemble

Averages of quadratic twists of long Dirichlet polynomials

Squarefrees are Gaussian in short intervals

Moments of polynomials with random multiplicative coefficients

On the variance of squarefree integers in short intervals and arithmetic progressions

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

Traces of powers of matrices over finite fields

Band-limited mimicry of point processes by point processes supported on a lattice

Higher Correlations and the Alternative Hypothesis

The variance of the number of sums of two squares in Fq​[T] in short intervals

The De Bruijn-Newman constant is non-negative

The limiting characteristic polynomial of classical random matrix ensembles

The variance of divisor sums in arithmetic progressions

Arithmetic functions in short intervals and the symmetric group

On the distribution of Rudin-Shapiro polynomials and lacunary walks on SU(2)

Tail bounds for counts of zeros and eigenvalues, and an application to ratios

Bootstrapped zero density estimates and a central limit theorem for the zeros of the zeta function

The covariance of almost-primes in Fq​[T]

A central limit theorem for the zeroes of the zeta function

Macroscopic pair correlation of the Riemann zeroes for smooth test functions

The variance of the number of sums of two squares in $F_{q} [T]$ in short intervals

On the distribution of Rudin-Shapiro polynomials and lacunary walks on $S U (2)$

The covariance of almost-primes in $F_{q} [T]$