ARTICLE | doi:10.20944/preprints201607.0003.v6
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Riemann hypothesis, Farey sequence, Gauss sum associated with a Dirichlet character
Online: 10 August 2016 (11:35:25 CEST)
Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.
ARTICLE | doi:10.20944/preprints202107.0428.v2
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Mobius function; Mangoldt function; summatory totient function
Online: 17 August 2021 (08:22:04 CEST)
The Mobius function is commonly used to define Euler's totient function and the Mangoldt function. Similarly, the summatory Mobius function (the Mertens function) can be used to define the summatory totient function and the summatory Mangoldt function (the second Chebyshev function).
ARTICLE | doi:10.20944/preprints202203.0025.v6
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: sum of divisors function; superabundant numbers; colossally abundant numbers; Riemann hypothesis
Online: 25 March 2022 (10:05:06 CET)
A generalization of the sum of divisors function involves a recursive definition. This leads to variants of superabundant numbers, colossally abundant numbers, and Gronwall's theorem (relevant to the Riemann hypothesis).