Topics in number theory

The thesis is concerned mainly with topics in the theory of Riemann’s zeta function, but it also includes some contributions to prime number theory and the study of the Möbius function. New conditions are stated for the validity of the quasi-Riemann hypothesis RH (σ₀), that ζ(s) ≠ O for σ > σ₀. The orders and oscillatory behaviour of a variety of summatory functions are considered in this context. Particular study is made of the sums defined by [formula] And [formula] (k = 0, 1, 2 ...), incomplete sums of the form [formula] and summatory functions associated with the coefficients of the Dirichlet series representations for [formula] On the same topic results are proven about connections between RH (σ₀) and the distribution of the set HN of Farey numbers of order N. Some general theorems concerning the sums [formula] are established which allow known results in the cases r = 2, 3 to be extended to r = 4, 5, 6. Other analytic studies include a series representation for Riemann’s function, and a theorem improving earlier results concerning the number of zeros of f(λ + i) with 0 < t < T, and fixed λ between 0 and 1, where f (s) = π⁻ˢ/² г(s/2) ζ (s). The remaining topics are more arithmetic in nature. An attempt is made to show by elementary methods that the order of the Tchebychef difference ψ(x) - x is not greater than that of the Möbius sum M(x), and although only partially successful, the attempt improves on previously published elementary results. Some theorems are proved which relate the order of J (x), the maximum number of consecutive integers each of which is divisible by at least one prime ≤x, to the problems of the least prime in an arithmetic progression and the order of the difference between consecutive primes. Two sections contain a direct attack by elementary methods on the problem of getting estimates of A₀(T) and B₀(T) which would be equivalent to RH(σ₀) for some σ₀ < 1, and the problem is reduced, roughly speaking, to finding sufficiently large N such that the solutions x₁, x₂, ..., xᴛ of the system [formula] are predominantly of one sign. The closing sections in the thesis deal with Möbius functions of ordered semigroups of a certain type, and lead to a conjecture that the Möbius functions in a way characterize the ordered multiplicative structure. Some evidence supporting the conjecture is given in the case of the ordinary Möbius function μ.

Grantor dc:publisher.institution

The University of Waikato

Year dc:date.issued

1979

Author dc:creator

• Braun, Peter Brendan

Advisor dc:contributor.advisor

• Zulauf, A.

Statement dc:rights

• All items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.

Licence

• https://researchcommons.waikato.ac.nz/bitstreams/407655d7-c9dd-46b9-8272-01804e22bd3d/download