14 research outputs found
Symmetric matrices related to the Mertens function
In this paper we explore a family of congruences over from which
one builds a sequence of symmetric matrices related to the Mertens function.
From the results of numerical experiments, we formulate a conjecture about
the growth of the quadratic norm of these matrices, which implies the Riemann
hypothesis. This suggests that matrix analysis methods may come to play a more
important role in this classical and difficult problem.Comment: Version submitted to LAA; some new reference
The Distribution of Weighted Sums of the Liouville Function and P\'olya's Conjecture
Under the assumption of the Riemann Hypothesis, the Linear Independence
Hypothesis, and a bound on negative discrete moments of the Riemann zeta
function, we prove the existence of a limiting logarithmic distribution of the
normalisation of the weighted sum of the Liouville function, , for . Using
this, we conditionally show that these weighted sums have a negative bias, but
that for each , the set of all for which
is positive has positive logarithmic density. For ,
this gives a conditional proof that the set of counterexamples to P\'olya's
conjecture has positive logarithmic density. Finally, when , we
conditionally prove that is negative outside a set of
logarithmic density zero, thereby lending support to a conjecture of
Mossinghoff and Trudgian that this weighted sum is nonpositive for all .Comment: 33 pages. Several minor revisions and corrections based on referee
comments, and additional references adde
Sign Patterns of the Liouville Function and Mobius Function over the Integers
Let be a large number, and let $0\leq a_
An annotated bibliography for comparative prime number theory
The goal of this annotated bibliography is to record every publication on the
topic of comparative prime number theory together with a summary of its
results. We use a unified system of notation for the quantities being studied
and for the hypotheses under which results are obtained. We encourage feedback
on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey"
(arXiv:1202.3408
Automated Conjecturing Approach to the Discrete Riemann Hypothesis
This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz\u27s Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded