arXiv subjects

Adrian W. Dudek

Publications and source records attributed to Adrian W. Dudek.

6 recordsLinked to original sources

arXiv2025

Almost primes between all squares

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach relies on a recent computation by Sorenson and Webster, along with an explicit version of the linear sieve. As part of our proof, we also prove an explicit version of Kuhn's weighted sieve. This is done for generic sifting sets to enhance the future applicability of our methods.

math.NT↗

arXiv2024

On Counting Constructions and Isomorphism Classes of $I$-Graphs

We prove a collection of asymptotic density results for several interesting classes of the $I$-graphs. Specifically, we quantify precisely the proportion of $I$-graphs that are generalised Petersen graphs as well as those that are connected. Our results rely on the estimation of sums over tuples satisfying various coprimality conditions along with other techniques from analytic number theory.

math.CO↗

arXiv2023

An explicit mean-value estimate for the PNT in intervals

This paper gives an explicit version of Selberg's 1943 mean-value estimate for the prime number theorem in intervals under the Riemann hypothesis. Two applications are given: for primes in short intervals, and Goldbach numbers (sums of two primes) in short intervals. Under the Riemann hypothesis, we show there exists a prime in $(y,y+32277\log^2 y]$ for at least half the $y\in[x,2x]$ for all $x\geq 2$, and at least one Goldbach number in $(x,x+9696 \log^2 x]$ for all $x\geq 2$.

math.NT↗

arXiv2022

A Conditional Explicit Result for the Prime Number Theorem in Short Intervals

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

math.NT↗

arXiv2019

Explicit Short Intervals for Primes in Arithmetic Progressions on GRH

We prove explicit versions of Cramér's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

math.NT↗

arXiv2018

Note on the number of divisors of reducible quadratic polynomials

In a recent paper, Lapkova uses a Tauberian theorem to derive the asymptotic formula for the divisor sum $\sum_{n \leq x} d( n (n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove her result by following a suggestion of Hooley, namely investigating the relationship between this sum and the well-known sum $\sum_{n \leq x} d( n ) d (n+v)$. As such, we are able to furnish additional terms in the asymptotic formula.

math.NT↗