arXiv subjects

Darrell Cox

Publications and source records attributed to Darrell Cox.

4 recordsLinked to original sources

arXiv2021

The Farey Sequence and the Mertens Function

Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a relationship between the Mertens function and the Farey sequence. Functions of subsets of the fractions in Farey sequences that are analogous to the Mertens function are introduced. Mikolas proved that the sum of certain Mertens function values is 1. Results analogous to Mikolas theorem are the defining property of these functions. A relationship between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem is postulated. This conjecture involves a theorem of Mertens and the second Chebyshev function.

math.NT↗

arXiv2021

Collatz Cycles and $3n+c$ Cycles

Halbeisen and Hungerbuhler determined optimal bounds for the length of rational Collatz cycles. Their methods are extended to $3n+c$ cycles. Another sequence having properties similar to those of Riemann zeta function zeros is introduced.

math.GM↗

arXiv2020

Bounds of the Mertens Function

In this paper, we derive new properties of the Mertens function and discuss a likely upper bound of the absolute value of the Mertens function $\sqrt{\log{x!}}>|M(x)|$ when $x>1$. Using this likely bound we show that we have a sufficient condition to prove the Riemann Hypothesis.

math.GM↗

arXiv2020

Sequences and Polynomial Congruence

In this paper, we shall find a new connection between $n$th degree polynomial mod $p$ congruence with $n$ roots and higher-order Fibonacci and Lucas sequences. We shall first discuss the recent work been done in sequences and their connection to polynomial congruence and then find out new relations between particular recurrence relation and the congruence of the sequences.

math.GM↗