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: The joy of having a book in your own language: Home language books in a refugee education centre

authors: Daly, Nicola

abstract: In 2018, Aotearoa/New Zealand increased its annual refugee quota to 1000. When refugees arrive in Aotearoa/New Zealand they spend six weeks in a resettlement programme. During this time, children attend an introduction to schooling. First language (L1) literacy support for children experiencing education in a medium that is not their Home Language has been identified as essential for children’s educational success. This knowledge is reflected in Principle 4 of the International Literacy Association’s Children’s Rights to Read campaign, which states that “children have the right to read texts that mirror their experiences and languages…”. In 2018, the International Board on Books for Young People (IBBY)-Yamada Foundation granted funding to IBBY in Aotearoa/New Zealand (IBBYNZ)/Storylines to supply books in the Home Languages of the refugee children in the introduction to school programme. Over 350 books were sourced in a range of languages including Farsi, Arabic, Tamil, Punjabi, Burmese, Karen, Chin, and Spanish. In this article, the sourcing of these books and their introduction to children in a refugee resettlement programme is described. Interviews with five teachers in the resettlement programme concerning the use of the books and how children and their families have been responding are reported. Future programme developments are outlined.

text: The joy of having a book in your own language: Home language books in a refugee education centre In 2018, Aotearoa/New Zealand increased its annual refugee quota to 1000. When refugees arrive in Aotearoa/New Zealand they spend six weeks in a resettlement programme. During this time, children attend an introduction to schooling. First language (L1) literacy support for children experiencing education in a medium that is not their Home Language has been identified as essential for children’s educational success. This knowledge is reflected in Principle 4 of the International Literacy Association’s Children’s Rights to Read campaign, which states that “children have the right to read texts that mirror their experiences and languages…”. In 2018, the International Board on Books for Young People (IBBY)-Yamada Foundation granted funding to IBBY in Aotearoa/New Zealand (IBBYNZ)/Storylines to supply books in the Home Languages of the refugee children in the introduction to school programme. Over 350 books were sourced in a range of languages including Farsi, Arabic, Tamil, Punjabi, Burmese, Karen, Chin, and Spanish. In this article, the sourcing of these books and their introduction to children in a refugee resettlement programme is described. Interviews with five teachers in the resettlement programme concerning the use of the books and how children and their families have been responding are reported. Future programme developments are outlined.

year: 2020

title: Pretty difficult: Implementing kaupapa Māori theory in English-medium secondary schools

authors: Bishop, Russell

subjects: Te Kotahitanga Project

abstract: Developed in New Zealand some twenty years ago, kaupapa Māori has had a successful impact in education, notably in Māori-medium settings such as kōhanga reo, kura kaupapa Māori and wharekura. However, in mainstream educational settings, where the vast majority of Māori children continue to be educated, achievement disparities between Māori and their non-Māori peers persist. This article focuses on Te Kotahitanga, a large-scale kaupapa Māori school reform project that seeks to address educational disparities by improving the educational achievement of Māori students in mainstream schooling. Experiences with implementing Te Kotahitanga would suggest that reforming mainstream educational practices along kaupapa Māori lines is not easy. This article examines three main impediments encountered in attempts to implement the Te Kotahitanga project in mainstream schools: confusion about the culture of the Māori child; uneven implementation of the project; and problems with measuring student progress. For the project’s aims to be realised, professional development needs to be ongoing, iterative and responsive.

text: Pretty difficult: Implementing kaupapa Māori theory in English-medium secondary schools Developed in New Zealand some twenty years ago, kaupapa Māori has had a successful impact in education, notably in Māori-medium settings such as kōhanga reo, kura kaupapa Māori and wharekura. However, in mainstream educational settings, where the vast majority of Māori children continue to be educated, achievement disparities between Māori and their non-Māori peers persist. This article focuses on Te Kotahitanga, a large-scale kaupapa Māori school reform project that seeks to address educational disparities by improving the educational achievement of Māori students in mainstream schooling. Experiences with implementing Te Kotahitanga would suggest that reforming mainstream educational practices along kaupapa Māori lines is not easy. This article examines three main impediments encountered in attempts to implement the Te Kotahitanga project in mainstream schools: confusion about the culture of the Māori child; uneven implementation of the project; and problems with measuring student progress. For the project’s aims to be realised, professional development needs to be ongoing, iterative and responsive.

year: 2012