Adrian W. Dudek

activity

20142026

most citedPrimes in explicit short intervals on RH

7 citations · 21 across the 19 of their papers we have counts for

collaborators

21 papers

math.NT2026

An Explicit Result for the Sum of Two Almost Primes

Adrian Dudek, Lachlan Dunn

We show that every $N \geq 2$ can be written as the sum of positive integers $a$ and $b$ where $Ω (ab) \leq 40$ . The result is obtained through the direct application of an explicit…

math.NT2022

An explicit mean-value estimate for the PNT in intervals

Michaela Cully-Hugill, Adrian W. Dudek

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 p…

math.NT2019

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

Michaela Cully-Hugill, Adrian W. Dudek

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

math.NT2018★ 1 cited

Note on the number of divisors of reducible quadratic polynomials

Adrian W. Dudek, Łukasz Pańkowski, Victor Scharaschkin

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 $d (n)$ denot…

math.NT2016

Explicit Estimates in the Theory of Prime Numbers

Adrian Dudek

It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is consider…

math.NT2016★ 2 cited

Explicit Short Intervals for Primes in Arithmetic Progressions on GRH

Adrian W. Dudek, Loïc Grenié, Giuseppe Molteni

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

math.HO2016

On the Success of Mishandling Euclid's Lemma

Adrian Dudek

We examine Euclid's lemma that if $p$ is a prime number such that $p ∣ ab$ , then $p$ divides at least one of $a$ or $b$ . Specifically, we consider the common misapplication of this…

math.NT2015★ 6 cited

On the Number of Divisors of $n^{2} - 1$

Adrian Dudek

We prove an asymptotic formula for the sum $\sum_{n \leq N} d (n^{2} - 1)$ , where $d (n)$ denotes the number of divisors of $n$ . During the course of our proof, we also furnish an asym…

math.NT2015

On the Distribution of Products of Primes and Powers

Adrian Dudek

We prove several results regarding the distribution of numbers that are the product of a prime and a $k$ -th power. First, we prove an asymptotic formula for the counting function o…

math.NT2015★ 3 cited

On the Sum of the Square of a Prime and a Square-Free Number

Adrian Dudek, Dave Platt

We prove that every integer $n \geq 10$ such that $n \neq \equiv 1 mod 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a t…

math.NT2015★ 7 cited

Primes in explicit short intervals on RH

Adrian Dudek, Loïc Grenié, Giuseppe Molteni

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

math.CO2015

Gaps Between Almost-Primes and a Construction of Almost-Ramanujan Graphs

Adrian Dudek

For all $k \geq 3$ , we show how one can explicitly construct an infinite family of $k$ -regular graphs all of which have second largest eigenvalue satisfying the bound $O (k^{1/2})$ .…

math.CO2014

On the Spectrum of the Generalised Petersen Graphs

Adrian Dudek

We show that the gap between the two greatest eigenvalues of the generalised Petersen graphs $P (n, k)$ tends to zero as $n \to \infty$ . Moreover, we provide explicit upper b…

math.NT2014

On the Sum of a Prime and a Square-free Number

Adrian Dudek

We prove that every integer greater than two may be written as the sum of a prime and a square-free number.

math.NT2014

An Explicit Result for $∣ L (1 + i t, χ) ∣$

Adrian Dudek

We give an explicit upper bound for non-principal Dirichlet $L$ -functions on the line $s = 1 + i t$ . This result can be applied to improve the error in the zero-counting formulae for th…

math.NT2014

Some Notes on Digit Strings in the Primes

Adrian Dudek

Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$ . We also show, as a corollary of t…

math.NT2014★ 1 cited

On Solving a Curious Inequality of Ramanujan

Dave Platt, Adrian Dudek

Ramanujan proved that the inequality $π (x)^{2} < \frac{e x}{l o g x} π (\frac{x}{e})$ holds for all sufficiently large values of $x$ . Using an explicit estimate for the error i…

math.NT2014

Almost-Ramanujan Graphs and Prime Gaps

Adrian Dudek

The method of Murty and Cioabă shows how one can use results about gaps between primes to construct families of almost-Ramanujan graphs. In this paper we give a simpler constructio…

math.NT2014★ 1 cited

On the Riemann Hypothesis and the Difference Between Primes

Adrian Dudek

We prove some results concerning the distribution of primes on the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}π \…

math.NT2014

An Explicit Result for Primes Between Cubes

Adrian Dudek

We prove that there is a prime between $n^{3}$ and $(n + 1)^{3}$ for all $n \geq exp (exp (33.217))$ . Our new tool which we derive is a version of Landau's explicit formula for the Riema…

math.NT2014

An Elementary Proof of an Asymptotic Formula of Ramanujan

Adrian Dudek

We derive the mean square of the divisor function using only elementary techniques.

An Explicit Result for the Sum of Two Almost Primes

An explicit mean-value estimate for the PNT in intervals

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

Note on the number of divisors of reducible quadratic polynomials

Explicit Estimates in the Theory of Prime Numbers

Explicit Short Intervals for Primes in Arithmetic Progressions on GRH

On the Success of Mishandling Euclid's Lemma

On the Number of Divisors of n2−1

On the Distribution of Products of Primes and Powers

On the Sum of the Square of a Prime and a Square-Free Number

Primes in explicit short intervals on RH

Gaps Between Almost-Primes and a Construction of Almost-Ramanujan Graphs

On the Spectrum of the Generalised Petersen Graphs

On the Sum of a Prime and a Square-free Number

An Explicit Result for ∣L(1+it,χ)∣

Some Notes on Digit Strings in the Primes

On Solving a Curious Inequality of Ramanujan

Almost-Ramanujan Graphs and Prime Gaps

On the Riemann Hypothesis and the Difference Between Primes

An Explicit Result for Primes Between Cubes

An Elementary Proof of an Asymptotic Formula of Ramanujan

On the Number of Divisors of $n^{2} - 1$

An Explicit Result for $∣ L (1 + i t, χ) ∣$