wovepaper — the arXiv observatory

papers

Publications (48)

math.NT2013

There are no socialist primes less than 10^9

Tim Trudgian

There are no primes $p$ with $5 < p < 1 0^{9}$ for which $2!, 3!, \dots, (p - 1)!$ are all distinct modulo $p$ ; it is conjectured that there are no such primes.

math.NT2018

Existence results for primitive elements in cubic and quartic extensions of a finite field

Geoff Bailey, Stephen D. Cohen, Nicole Sutherland +1

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $I^^{3}$ generates $\Fqn$ over $\Fq$ and $I^^{2}$ is a non-zero element of $\Fqn$, is there always an…

math.NT2022

Primitive elements with prescribed traces

Andrew R. Booker, Stephen D. Cohen, Nicol Leong +1

Given a prime power $q$ and a positive integer $n$ , let $F_{q^{n}}$ denote the finite field with $q^{n}$ elements. Also let $a, b$ be arbitrary members of the ground field $\…

math.NT2018

An elementary bound on Siegel zeroes

Thomas Morrill, Tim Trudgian

We consider Dirichlet $L$ -functions $L (s, I ¨ )$ where $I ¨ $ is a real, non-principal character modulo $q$ . Using Pintz's refinement of Page's theorem, we prove that for $q \geq 3$ the…

math.NT2015

Bounds on the number of Diophantine quintuples

Tim Trudgian

We consider Diophantine quintuples ${a, b, c, d, e}$ . These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It…

math.NT2015

Resolving Grosswald's conjecture on GRH

Kevin McGown, Enrique TreviÃ±o, Tim Trudgian

In this paper we examine Grosswald's conjecture on $g (p)$ , the least primitive root modulo $p$ . Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by…

math.NT2014

Improvements to Turing's Method II

Tim Trudgian

Turing's method uses explicit bounds on $∣ \int_{t_{1}}^{t_{2}} S (t) d t ∣$ , where $I ¨ S (t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int_{…

math.NT2025

On the error term of the fourth moment of the Riemann zeta-function

Neea PalojÃ¤rvi, Tim Trudgian

We examine the size of $E_{2} (T)$ , the error term in the asymptotic formula for $\int_{0}^{T} ∣ I^¶ (1/2 + i t) ∣^{4} d t$ where $I^¶ (s)$ is the Riemann zeta-function. We make improveme…

math.NT2017

Square-full primitive roots

Marc Munsch, Tim Trudgian

We use character sum estimates to give a bound on the least square-full primitive root modulo a prime. Specifically, we show that there is a square-full primitive root mod $p$ less…

math.NT2025

New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach

Terence Tao, Tim Trudgian, Andrew Yang

We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and ne…

math.NT2019

Explicit upper bounds on the least primitive root

Kevin J. McGown, Tim Trudgian

We give a method for producing explicit bounds on $g (p)$ , the least primitive root modulo $p$ . Using our method we show that $g (p) < 2 r 2^{r I ¨ (p - 1)} p^{\frac{1}{4} + \frac{1}{4 r}}$ f…

math.CO2026

A conjecture of Glasby, Praeger, and Unger on permutations of $S_{n}$

Chiara Bellotti, Tim Trudgian

We prove a conjecture of Glasby, Praeger, and Unger concerning the symmetric group $S_{n}$ . Let $I ¨ _{n}$ denote the proportion of elements of $S_{n}$ that are pre- $p$ -cycles for so…

math.NT2015

Diophantine quintuples containing triples of the first kind

Dave Platt, Tim Trudgian

We consider Diophantine quintuples ${a, b, c, d, e}$ , sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of th…

math.NT2025

The determination of norm-Euclidean cyclic cubic fields

Gustav KjÃ¦rbye Bagger, Andrew R. Booker, Bryce Kerr +3

It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estima…

math.NT2014

On the first sign change of $I^¸ (x) - x$

Dave Platt, Tim Trudgian

Let $I^¸ (x) = \sum_{p \leq x} lo g p$ . We show that $I^¸ (x) < x$ for $2 < x < 1.39 \cdot 1 0^{17}$ . We also show that there is an $x < exp (727.951332668)$ for which $I^¸ (x) > x .$

math.NT2018

Primitive values of quadratic polynomials in a finite field

Andrew R. Booker, Stephen D. Cohen, Nicole Sutherland +1

We prove that for all $q > 211$ , there always exists a primitive root $g$ in the finite field $F_{q}$ such that $Q (g)$ is also a primitive root, where $Q (x) = a x^{2} + b x + c$ …

math.NT2017

Lehmer numbers and primitive roots modulo a prime

Stephen D. Cohen, Tim Trudgian

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p - 1$ whose inverse $a ˉ$ within the same range has opposite parity. Lehmer numbers that are also primit…

math.NT2015

Linnik's approximation to Goldbach's conjecture, and other problems

Dave Platt, Tim Trudgian

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most $K$ powers of 2. We outline an approach that only just falls short of im…

math.NT2020

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$

Dave Platt, Tim Trudgian

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3 \cdot 1 0^{12}$ . That is, all zeroes $I^^{2} + i I^^{3}$ of the Riemann z…

math.NT2020

The error term in the prime number theorem

Dave Platt, Tim Trudgian

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square…

math.NT2019

On integers $n$ for which $I ¨ (2 n + 1) \geq I ¨ (2 n)$

Mits Kobayashi, Tim Trudgian

We show that the natural density of positive integers $n$ for which $I ¨ (2 n + 1) \geq I ¨ (2 n)$ is between $0.053$ and $0.055$ .

math.NT2018

Improved bounds on Brun's constant

Dave Platt, Tim Trudgian

Brun's constant is $B = \sum_{p \in P_{2}} p^{- 1} + (p + 2)^{- 1}$ , where the summation is over all twin primes. We improve the unconditional bounds on Brun's constant to $1.840503 < B…

math.NT2014

The sum of the unitary divisor function

Tim Trudgian

This article establishes a new upper bound on the function $I ¨ ^{*} (n)$ , the sum of all coprime divisors of $n$ . The article concludes with two questions concerning this function.

math.NT2014

An improved explicit bound on $∣ I^¶ (1/2 + i t) ∣$

Dave Platt, Tim Trudgian

This article proves the bound $∣ I^¶ (\frac{1}{2} + i t) ∣ \leq 0.732 t^{\frac{1}{6}} lo g t$ for $t \geq 2$ , which improves on a result by Cheng and Graham. We also show that $|Î¶(\frac…

math.NT2026

Zero-free regions inspired by work of Heath-Brown

Chiara Bellotti, Tim Trudgian, Andrew Yang

We prove a new explicit zero-free region for the Riemann zeta-function, drawing substantially on Heath-Brown's seminal work on Linnik's constant. Using these ideas we are able to p…

math.NT2014

A short extension of two of Spira's results

Tim Trudgian

Two inequalities concerning the symmetry of the zeta-function and the Ramanujan $I ¨ $ -function are improved through the use of some elementary considerations.

math.NT2013

An incomplete variant of Wilson's congruence

Joel Beeren, David Harvey, Tim Trudgian

This article examines the nontrivial solutions of the congruence \[ (p-1)\cdots(p-r) \equiv -1 \pmod p. \] We discuss heuristics for the proportion of primes $p$ that have exactly…

math.NT2015

Searching for Diophantine quintuples

Mihai Cipu, Tim Trudgian

We consider Diophantine quintuples ${a, b, c, d, e}$ . These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It…

math.NT2014

A proof of the conjecture of Cohen and Mullen on sums of primitive roots

Stephen D. Cohen, TomÃ¡s Oliveira e Silva, Tim Trudgian

We prove that for all $q > 61$ , every non-zero element in the finite field $F_{q}$ can be written as a linear combination of two primitive roots of $F_{q}$ . This re…

math.NT2019

Sign changes in the prime number theorem

Thomas Morrill, Dave Platt, Tim Trudgian

Let $V (T)$ denote the number of sign changes in $I ¨ (x) - x$ for $x \in [1, T]$ . We show that $lim inf_{T \to \infty} V (T) / lo g T \geq I^_{1}^{3} / I ¨ + 1.867 \cdot 1 0^{- 30}$ , where $…

math.NT2014

Linear relations of zeroes of the zeta-function

Darcy Best, Tim Trudgian

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show…

math.NT2020

The distribution of $k$ -free numbers

Michael J. Mossinghoff, TomÃ¡s Oliveira e Silva, Tim Trudgian

Let $R_{k} (x)$ denote the error incurred by approximating the number of $k$ -free integers less than $x$ by $x / I^¶ (k)$ . It is well known that $R_{k} (x) = I^c ◯ (x^{\frac{1}{2 k}})$ , and widely…

math.NT2016

On the least square-free primitive root modulo $p$

Stephen D. Cohen, Tim Trudgian

Let $g^{□} (p)$ denote the least square-free primitive root modulo $p$ . We show that $g^{□} (p) < p^{0.96}$ for all $p$ .

math.NT2014

Updating the error term in the prime number theorem

Tim Trudgian

An improved estimate is given for $∣ I^¸ (x) - x ∣$ , where $I^¸ (x) = \sum_{p \leq x} lo g p$ . Three applications are given: the first to arithmetic progressions that have points in common…

math.NT2014

On consecutive primitive elements in a finite field

Stephen D. Cohen, TomÃ¡s Oliveira e Silva, Tim Trudgian

For $q$ an odd prime power with $q > 169$ we prove that there are always three consecutive primitive elements in the finite field $F_{q}$ . Indeed, there are precisely eleven…

math.NT2022

Primitive element pairs with a prescribed trace in the cubic extension of a finite field

Andrew R. Booker, Stephen D. Cohen, Nicol Leong +1

We prove that for any prime power $q \in / {3, 4, 5}$ , the cubic extension $F_{q^{3}}$ of the finite field $F_{q}$ contains a primitive element $I^¾$ such that $Î¾+Î¾^{…

math.NT2017

Uchiyama's conjecture on sums of squares

Tim Trudgian

Uchiyama showed that every interval $(n, n + c n^{1/4})$ contains an integer that is the sum of two squares, where $c = 2^{3/2}$ . He also conjectured a minimal value of $c$ such tha…

math.NT2021

Four consecutive primitive elements in a finite field

Tamiru Jarso, Tim Trudgian

For $q$ an odd prime power, we prove that there are always four consecutive primitive elements in the finite field $F_{q}$ when $q > 2401$ .

math.NT2017

Quadratic residues that are not primitive roots

Tamiru Jarso, Tim Trudgian

We prove that any prime $p$ satisfying $I ¨ (p - 1) \leq (p - 1) /4$ contains two consecutive quadratic non-residues modulo $p$ neither of which is a primitive root modulo $p$ .

math.NT2014

The $T_{4}$ and $G_{4}$ constructions of Costas arrays

Tim Trudgian, Qiang Wang

We examine two particular constructions of Costas arrays known as the Taylor variant of the Lempel construction, or the $T_{4}$ construction, and the variant of the Golomb construc…

math.NT2026

An update on the Linnik--Goldbach problem

Daniel R. Johnston, Tim Trudgian

We consider the Linnik--Goldbach problem of writing all large even integers as the sum of two primes and a fixed number of powers of 2. We show that, under the generalised Riemann…

math.NT2018

Linear combinations of primitive elements of a finite field

Stephen Cohen, TomÃ¡s Oliveira e Silva, Nicole Sutherland +1

We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p > 13$ has a pair of primitive…

math.NT2015

On Grosswald's conjecture on primitive roots

Stephen D. Cohen, TomÃ¡s Oliveira e Silva, Tim Trudgian

Grosswald's conjecture is that $g (p)$ , the least primitive root modulo $p$ , satisfies $g (p) \leq p - 2$ for all $p > 409$ . We make progress towards this conjecture by proving…

math.NT2016

On the sum of two squares and at most two powers of 2

Dave Platt, Tim Trudgian

We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two non-negative integer powers of 2.

math.NT2019

The least primitive root modulo $p^{2}$

Bryce Kerr, Kevin McGown, Tim Trudgian

We provide an explicit estimate on the least primitive root mod $p^{2}$ . We show, in particular, that every prime $p$ has a primitive root mod $p^{2}$ that is less than $p^{0.99}$ .

math.NT2018

Fujii's development on Chebyshev's conjecture

Dave Platt, Tim Trudgian

Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to $3 (mod 4)$ . His conjecture is equivalent to a version of the Generalised Riemann Hypo…

math.NT2026

A round of Pintz to celebrate oscillations in sums

Daniel R. Johnston, Tim Trudgian

We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$ -functions. In particular, we make expli…

math.NT2025

Applying hypersurface bounds to a conjecture by Carlet

ZoÃ« Gemmell, Tim Trudgian

A function from $F_{2^{n}}$ to $F_{2^{n}}$ is $k$ th order sum-free if the sum of its values over each $k$ -dimensional $F_{2}$ -affine subspace is nonzero. It is…

Publications (48)

There are no socialist primes less than 10^9

Existence results for primitive elements in cubic and quartic extensions of a finite field

Primitive elements with prescribed traces

An elementary bound on Siegel zeroes

Bounds on the number of Diophantine quintuples

Resolving Grosswald's conjecture on GRH

Improvements to Turing's Method II

On the error term of the fourth moment of the Riemann zeta-function

Square-full primitive roots

New exponent pairs, zero density estimates, and zero additive energy estimates: a systematic approach

Explicit upper bounds on the least primitive root

A conjecture of Glasby, Praeger, and Unger on permutations of Sn​

Diophantine quintuples containing triples of the first kind

The determination of norm-Euclidean cyclic cubic fields

On the first sign change of I^¸(x)−x

Primitive values of quadratic polynomials in a finite field

Lehmer numbers and primitive roots modulo a prime

Linnik's approximation to Goldbach's conjecture, and other problems

The Riemann hypothesis is true up to 3⋅1012

The error term in the prime number theorem

On integers n for which I¨(2n+1)≥I¨(2n)

Improved bounds on Brun's constant

The sum of the unitary divisor function

An improved explicit bound on ∣I^¶(1/2+it)∣

Zero-free regions inspired by work of Heath-Brown

A short extension of two of Spira's results

An incomplete variant of Wilson's congruence

Searching for Diophantine quintuples

A proof of the conjecture of Cohen and Mullen on sums of primitive roots

Sign changes in the prime number theorem

Linear relations of zeroes of the zeta-function

The distribution of k-free numbers

On the least square-free primitive root modulo p

Updating the error term in the prime number theorem

On consecutive primitive elements in a finite field

Primitive element pairs with a prescribed trace in the cubic extension of a finite field

Uchiyama's conjecture on sums of squares

Four consecutive primitive elements in a finite field

Quadratic residues that are not primitive roots

The T4​ and G4​ constructions of Costas arrays

An update on the Linnik--Goldbach problem

Linear combinations of primitive elements of a finite field

On Grosswald's conjecture on primitive roots

On the sum of two squares and at most two powers of 2

The least primitive root modulo p2

Fujii's development on Chebyshev's conjecture

A round of Pintz to celebrate oscillations in sums

Applying hypersurface bounds to a conjecture by Carlet

A conjecture of Glasby, Praeger, and Unger on permutations of $S_{n}$

On the first sign change of $I^¸ (x) - x$

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$

On integers $n$ for which $I ¨ (2 n + 1) \geq I ¨ (2 n)$

An improved explicit bound on $∣ I^¶ (1/2 + i t) ∣$

The distribution of $k$ -free numbers

On the least square-free primitive root modulo $p$

The $T_{4}$ and $G_{4}$ constructions of Costas arrays

The least primitive root modulo $p^{2}$