arXiv subjects

Jeremy Schlitt

Publications and source records attributed to Jeremy Schlitt.

5 recordsLinked to original sources

arXiv2026

Integers divisible by a shifted prime in a given interval

In this paper we study the behaviour of $H^*(x,y,z)$, the number of integers less than $x$ possessing a divisor in the interval $(y,z]$ of the form $p-1$, where $p$ is a prime, for all values of $y = y(x)$ and $z= z(y)$. We observe multiple phase transitions at critical values of $z$ in terms of $x$ and $y$ guided largely by the anatomy of $n$. Our results generalize a result of Ford from 2017, which corresponds to the case $z = x$.

math.NT↗

arXiv2026

Random Multiplicative Functions with Periodic Weights

Given a Steinhaus random multiplicative function $f$, a $1$-periodic function of bounded variation $g$, and an irrational number $\alpha$, we study the distribution of $\sum_{n=1}^N f(n) g(\alpha n)$. We determine a necessary and sufficient condition for these sums, normalized by their standard deviation, to converge to the standard complex Gaussian distribution. On the other hand, we show that if we restrict the summands to have all their prime factors $>z$, with $z$ tending to infinity arbitrarily slowly, then a central limit theorem always holds for such sums.

math.NT↗

arXiv2026

Multiplication Tables for Integers with Restricted Prime Factors

Let $Q$ be a set of primes with relative density $\delta$. We count integers in $[1,x]$ with prime factors all in $Q$ that also have a divisor in $(y,2y]$. We establish the order of magnitude for all $\delta \in (0,1]$. This generalizes the case $\delta = 1$ from the 2008 work of Ford. We also show that there is a phase transition at the critical point $\delta = 1/\log 4$, for which we explicitly determine the behaviour.

math.NT↗

arXiv2023

Biases Towards the Zero Residue Class for Quadratic Forms in Arithmetic Progressions

We examine a bias towards the zero residue class for the integers represented by binary quadratic forms. In many cases, we are able to prove that the bias comes from a secondary term in the associated asymptotic expansion (unlike Chebyshev's bias, which lives somewhere at the level of $O(x^{1/2+ε})$.) In some other cases, we are unable to prove that a bias exists, even though it is present numerically. We then make a conjecture on the general situation which includes the cases we could not prove. Many interesting results on the distribution of the integers represented by a quadratic form -- some of which are of independent interest -- are proven along the way. The paper concludes with some numerical data that is illustrative of the aforementioned bias.

math.NT↗

arXiv2021

Lemke Oliver and Soundararajan bias for consecutive sums of two squares

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares, which also explain the biases in the experimental data. In the process, we prove several results related to averages of the Hardy--Littlewood constant in the context of sums of two squares.

math.NT↗