title: Topics in number theory

authors: Braun, Peter Brendan

abstract: 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.

text: 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.

year: 1979

title: On the doctoral journey: An introduction to articles in this issue

authors: Newcombe, Nicolina Isabel

abstract: We are pleased to contribute to the ongoing work of Waikato Journal of Educationin growingknowledge and understanding of educational research and the experiences of researchers. Wehope you find value in the articles in this special issue.

text: On the doctoral journey: An introduction to articles in this issue We are pleased to contribute to the ongoing work of Waikato Journal of Educationin growingknowledge and understanding of educational research and the experiences of researchers. Wehope you find value in the articles in this special issue.

year: 2022-09-08

title: Anti-racism Allyship Amongst Psychologists and Students in Psychology Training in Aotearoa New Zealand

authors: Tan, Kyle K. H.

abstract: Racism is a well-established social determinant of health, impacting Māori, Pacific, and Asian communities in Aotearoa New Zealand. However, few scholarly studies have focused on racial justice allyship in Aotearoa, particularly within the psychology discipline. This study examines demographic and attitudinal factors associated with anti-racism allyship among psychologists and students in psychology training amidst escalating racist rhetoric and opposition to Māori self-determination in Aotearoa. Drawing data from an online anonymous survey (n = 280), we assessed participants’ responses on three scales: 1) Attitudes towards aspects of Māori (taha Māori); 2) Attitudes towards acknowledging the existence of racism and settler colonialism 3) Anti-racist allyship. Certain demographic groups (Pākehā, Asian, older-aged, male, or those who entered training decades ago) scored lower on these scales, indicating the need for targeted training. Regression analyses showed both ‘Attitudes towards taha Māori’ and ‘Attitudes towards racism and colonisation’ scales were significantly correlated with anti-racist allyship. Our findings call for a sustained investment in training anti-racist psychologists who are competent in providing culturally safe care and capable of challenging racial injustices. Future research is required on the development of a racial justice allyship model across all institutions that train, employ and regulate psychologists in Aotearoa that is informed by Te Tiriti o Waitangi.

text: Anti-racism Allyship Amongst Psychologists and Students in Psychology Training in Aotearoa New Zealand Racism is a well-established social determinant of health, impacting Māori, Pacific, and Asian communities in Aotearoa New Zealand. However, few scholarly studies have focused on racial justice allyship in Aotearoa, particularly within the psychology discipline. This study examines demographic and attitudinal factors associated with anti-racism allyship among psychologists and students in psychology training amidst escalating racist rhetoric and opposition to Māori self-determination in Aotearoa. Drawing data from an online anonymous survey (n = 280), we assessed participants’ responses on three scales: 1) Attitudes towards aspects of Māori (taha Māori); 2) Attitudes towards acknowledging the existence of racism and settler colonialism 3) Anti-racist allyship. Certain demographic groups (Pākehā, Asian, older-aged, male, or those who entered training decades ago) scored lower on these scales, indicating the need for targeted training. Regression analyses showed both ‘Attitudes towards taha Māori’ and ‘Attitudes towards racism and colonisation’ scales were significantly correlated with anti-racist allyship. Our findings call for a sustained investment in training anti-racist psychologists who are competent in providing culturally safe care and capable of challenging racial injustices. Future research is required on the development of a racial justice allyship model across all institutions that train, employ and regulate psychologists in Aotearoa that is informed by Te Tiriti o Waitangi.

year: 2025