arXiv subjects

Tim Trudgian

Publications and source records attributed to Tim Trudgian.

48 records · Page 3Linked to original sources

arXiv2014

On consecutive primitive elements in a finite field

For $q$ an odd prime power with $q>169$ we prove that there are always three consecutive primitive elements in the finite field $\mathbb{F}_{q}$. Indeed, there are precisely eleven values of $q \leq 169$ for which this is false. For $4\leq n \leq 8$ we present conjectures on the size of $q_{0}(n)$ such that $q>q_{0}(n)$ guarantees the existence of $n$ consecutive primitive elements in $\mathbb{F}_{q}$, provided that $\mathbb{F}_{q}$ has characteristic at least~$n$. Finally, we improve the upper bound on $q_{0}(n)$ for all $n\geq 3$.

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arXiv2014

A short extension of two of Spira's results

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

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arXiv2014

Updating the error term in the prime number theorem

An improved estimate is given for $|θ(x) -x|$, where $θ(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the third to a conjecture by C. Pomerance.

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arXiv2014

An improved explicit bound on $|ζ(1/2 + it)|$

This article proves the bound $|ζ(\frac{1}{2} + it)|\leq 0.732 t^{\frac{1}{6}} \log t$ for $t \geq 2$, which improves on a result by Cheng and Graham. We also show that $|ζ(\frac{1}{2}+it)|\leq 0.732 |3.3081+it|^{\frac{1}{6}} \log |3.3081+it|$ for all $t$.

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arXiv2014

Improvements to Turing's Method II

Turing's method uses explicit bounds on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$, where $πS(t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$ given in an earlier paper by the author.

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arXiv2014

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

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 construction, or the $G_{4}$ construction. We connect these constructions with the concept of Fibonacci primitive roots, and show that under the Extended Riemann Hypothesis the $T_{4}$ and $G_{4}$ constructions are valid infinitely often.

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arXiv2014

Linear relations of zeroes of the zeta-function

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 that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf M(x)x^{-1/2}\leq -1.6383$.

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arXiv2014

The sum of the unitary divisor function

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

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arXiv2014

On the first sign change of $θ(x) - x$

Let $θ(x) = \sum_{p\leq x} \log p$. We show that $θ(x) x.$

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arXiv2014

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

We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.

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arXiv2013

There are no socialist primes less than 10^9

There are no primes $p$ with $5<p<10^{9}$ for which $2!, 3!, \ldots, (p-1)!$ are all distinct modulo $p$; it is conjectured that there are no such primes.

math.NT↗

arXiv2013

An incomplete variant of Wilson's congruence

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 $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calculations.

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