arXiv subjects

Mario Ziller

Publications and source records attributed to Mario Ziller.

7 recordsLinked to original sources

arXiv2026

Pseudometrics and preorders on sets of integer sequences induced by arithmetic functions functions

Starting from pseudometrics and preorders on sets of integers, we extend the focus to sets of finite sequences of integers, in particular sequences of consecutive integers. We outline existing concepts for deriving centred pseudometrics and preorders in a given pseudometric space and their application to $\mathbb{Z}$ and develop approaches to generalize the ideas to $\mathbb{Z}^m$. Sequences of consecutive integers represent a special case here and are examined in more detail. Another main topic is the use of arithmetic functions in this context. The types of pseudometrics and preorders examined in this paper can be induced by suitable arithmetic functions. We derive fundamental conclusions about relationships between functions and preorders, as well as about equivalent and potentially distinct types of preorders.

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arXiv2023

Orders and partitions of integers induced by arithmetic functions

We pursue the question how integers can be ordered or partitioned according to their divisibility properties. Based on pseudometrics on $\mathbb{Z}$, we investigate induced preorders, associated equivalence relations, and quotient sets. The focus is on metrics or pseudometrics on $\mathbb{D}_n$, the set of divisors of a given modulus $n\in\mathbb{N}$, that can be extended to pseudometrics on $\mathbb{Z}$. Arithmetic functions can be used to generate such pseudometrics. We discuss several subsets of additive and multiplicative arithmetic functions and various combinations of their function values leading to binary metric functions that represent different divisibility properties of integers. We conclude this paper with numerous examples and review the most important results. As an additional result, we derive a necessary condition for the truth of the odd k-perfect number conjecture.

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arXiv2020

On differences between consecutive numbers coprime to primorials

We consider the ordered sequence of coprimes to a given primorial number and investigate differences between consecutive elements. The Jacobsthal function applied to the concerning primorial turns out to represent the greatest of these differences. We will explore the smallest even number which does not occur as such a difference. Little is known about even natural numbers below the respective Jacobsthal function which cannot be represented as a difference between consecutive numbers coprime to a primorial. Existence and frequency of these numbers have not yet been clarified. Using the relation between restricted coverings of sequences of consecutive integers and the occuring differences, we derive a bound below which all even natural numbers are differences between consecutive numbers coprime to a given primorial $p_k\#$. Furthermore, we provide exhaustive computational results on non-existent differences for primes $p_k$ up to $k=44$. The data suggest the assumption that all even natural numbers up to $h(k-1)$ occur as differences of coprimes to $p_k\#$ where $h(n)$ is the Jacobsthal function applied to $p_n\#$.

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arXiv2019

New computational results on a conjecture of Jacobsthal

Jacobsthal's conjecture has been disproved by counterexample a few years ago. We continue to verify this conjecture on a larger scale. For this purpose, we implemented an extension of the Greedy Permutation Algorithm and computed the maximum Jacobsthal function for the product of $k$ primes up to $k=43$. We have found various new counterexamples. Their pattern seems to imply that the conjecture of Jacobsthal only applies to several small $k$. Our results raise further questions for discussion. In addition to this paper, we provide exhaustive information about all covered sequences of the appropriate maximum lengths in ancillary files.

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arXiv2017

A short note on the computation of the generalised Jacobsthal function for paired progressions

Jacobsthal's function was recently generalised for the case of paired progressions. It was proven that a specific bound of this function is sufficient for the truth of Goldbach's conjecture and of the prime pairs conjecture as well. We extended and adapted algorithms described for the computation of the common Jacobsthal function, and computed respective function values of the paired Jacobsthal function for primorial numbers for primes up to 73. All these values fulfil the conjectured specific bound. In addition to this note, we provide a detailed review of the algorithmic approaches and the complete computational results in ancillary files.

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arXiv2017

Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs

We investigate progressions in the set of pairs of integers $\mathbb{Z}^2$ and define a generalisation of the Jacobsthal function. For this function, we conjecture a specific upper bound and prove that this bound would be a sufficient condition for the truth of the Goldbach conjecture, the infinitude of prime twins, and more general of prime pairs with a fixed even difference.

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arXiv2017

Algorithmic concepts for the computation of Jacobsthal's function

The Jacobsthal function has aroused interest in various contexts in the past decades. We review several algorithmic ideas for the computation of Jacobsthal's function for primorial numbers and discuss their practicability regarding computational effort. The respective function values were computed for primes up to 251. In addition to the results including previously unknown data, we provide exhaustive lists of all sequences of the appropriate maximum lengths in ancillary files.

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