Studies in multiplicative number theory
This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n). In Chapter One we give a lower estimate for the number of distinct values...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
1980
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| Online Access: | https://eprints.nottingham.ac.uk/28304/ |
| _version_ | 1848793548576522240 |
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| author | Shiu, Peter Man-Kit |
| author_facet | Shiu, Peter Man-Kit |
| author_sort | Shiu, Peter Man-Kit |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n).
In Chapter One we give a lower estimate for the number of distinct values assumed by the divisor function d(n) in 1 <n <x .We also identify the smallest positive integer which is a product of triangular numbers and not equal to d3(n) for 1 <n <x .
In Chapter Two we show that if f(n) satisfies some conditions and if M=max {f(2a)}1/a, if a> or = 1
then the maximum value of f(n) in 1<n< x is about log x / Mloglog x.
We also show that a function which has a finite mean value cannot be large too often.
In Chapter Three we give an upper estimate to the average value of f(n) as n runs through a short interval in an arithmetic progression with a large modulus . As an application of our general theorem we show, for example, that if f(n) is the characteristic function of the set of integers which are the sum of two squares, then as x -> infinity.
We call a positive integer n a k-full integer if pk divides n whenever p is a prime divisor of n, and in Chapter Four we give an asymptotic formula for the number of k-full integers not exceeding x. In Chapter Five we give an asymptotic formula for the number of 2-full integers in an interval. We also study the problem of the distribution of the perfect squares among the sequence of 2-full integers.
The materials in the first three chapters have been accepted for publications and will appear [31], [22], [33] and [32]. |
| first_indexed | 2025-11-14T19:02:03Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-28304 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T19:02:03Z |
| publishDate | 1980 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-283042025-02-28T11:33:21Z https://eprints.nottingham.ac.uk/28304/ Studies in multiplicative number theory Shiu, Peter Man-Kit This thesis gives some order estimates and asymptotic formulae associated with general classes of non-negative multiplicative functions as well as some particular multiplicative functions such as the divisor functions dk(n). In Chapter One we give a lower estimate for the number of distinct values assumed by the divisor function d(n) in 1 <n <x .We also identify the smallest positive integer which is a product of triangular numbers and not equal to d3(n) for 1 <n <x . In Chapter Two we show that if f(n) satisfies some conditions and if M=max {f(2a)}1/a, if a> or = 1 then the maximum value of f(n) in 1<n< x is about log x / Mloglog x. We also show that a function which has a finite mean value cannot be large too often. In Chapter Three we give an upper estimate to the average value of f(n) as n runs through a short interval in an arithmetic progression with a large modulus . As an application of our general theorem we show, for example, that if f(n) is the characteristic function of the set of integers which are the sum of two squares, then as x -> infinity. We call a positive integer n a k-full integer if pk divides n whenever p is a prime divisor of n, and in Chapter Four we give an asymptotic formula for the number of k-full integers not exceeding x. In Chapter Five we give an asymptotic formula for the number of 2-full integers in an interval. We also study the problem of the distribution of the perfect squares among the sequence of 2-full integers. The materials in the first three chapters have been accepted for publications and will appear [31], [22], [33] and [32]. 1980 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/28304/1/281269.pdf Shiu, Peter Man-Kit (1980) Studies in multiplicative number theory. PhD thesis, University of Nottingham. |
| spellingShingle | Shiu, Peter Man-Kit Studies in multiplicative number theory |
| title | Studies in multiplicative number theory |
| title_full | Studies in multiplicative number theory |
| title_fullStr | Studies in multiplicative number theory |
| title_full_unstemmed | Studies in multiplicative number theory |
| title_short | Studies in multiplicative number theory |
| title_sort | studies in multiplicative number theory |
| url | https://eprints.nottingham.ac.uk/28304/ |