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An algorithm for computing the riemann zeta function based on an analysis of Backlund’s remainder estimate Menz, Petra Margarete
Abstract
The Riemann zeta function, Ϛ(s) with complex argument s, is a widely used special function in mathematics. This thesis is motivated by the need of a cost reducing algorithm for the computation of Ϛ (s) using its Euler-Maclaurin series. The difficulty lies in finding small upper bounds, call them n and k, for the two sums in the Euler-Maclaurin series of Ϛ (s) which will compute Ϛ (s) to within any given accuracy for any complex argument s, and provide optimal computational cost in the use of the Euler-Maclaurin series. This work is based on Backlund’s remainder estimate for the Euler-Maclaurin remain- der, since it provides a close enough relationship between n, k, s, and е. We assumed that the cost of computing the Bernoulli numbers, which appear in the series, is fixed, and briefly discuss how this may influence high precision calculation. Based on our study of the behavior of Backlund’s remainder estimate, we define the ‘best’ pair (n, k), and present a reliable method of computing the best pair. Furthermore, based on a compu- tational analysis, we conjecture that there is a relationship between n and k which does not depend on s. We present two algorithms, one based on our method and the other on the conjecture, and compare their costs of finding n and k as well as computing the Euler-Maclaurin series with an algorithm presented by Cohen and Olivier. We conclude that our algorithm reduces the cost of computing Ϛ(s) drastically, and that good numerical techniques need to be applied to our method and conjecture for finding n and k in order to keep this computational cost low as well.
Item Metadata
| Title |
An algorithm for computing the riemann zeta function based on an analysis of Backlund’s remainder estimate
|
| Creator | |
| Publisher |
University of British Columbia
|
| Date Issued |
1994
|
| Description |
The Riemann zeta function, Ϛ(s) with complex argument s, is a widely used special
function in mathematics. This thesis is motivated by the need of a cost reducing algorithm
for the computation of Ϛ (s) using its Euler-Maclaurin series. The difficulty lies in finding
small upper bounds, call them n and k, for the two sums in the Euler-Maclaurin series of
Ϛ (s) which will compute Ϛ (s) to within any given accuracy for any complex argument s,
and provide optimal computational cost in the use of the Euler-Maclaurin series.
This work is based on Backlund’s remainder estimate for the Euler-Maclaurin remain-
der, since it provides a close enough relationship between n, k, s, and е. We assumed that
the cost of computing the Bernoulli numbers, which appear in the series, is fixed, and
briefly discuss how this may influence high precision calculation. Based on our study
of the behavior of Backlund’s remainder estimate, we define the ‘best’ pair (n, k), and
present a reliable method of computing the best pair. Furthermore, based on a compu-
tational analysis, we conjecture that there is a relationship between n and k which does
not depend on s. We present two algorithms, one based on our method and the other
on the conjecture, and compare their costs of finding n and k as well as computing the
Euler-Maclaurin series with an algorithm presented by Cohen and Olivier. We conclude
that our algorithm reduces the cost of computing Ϛ(s) drastically, and that good numerical techniques need to be applied to our method and conjecture for finding n and k in
order to keep this computational cost low as well.
|
| Extent |
1093612 bytes
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| Genre | |
| Type | |
| File Format |
application/pdf
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| Language |
eng
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| Date Available |
2009-03-03
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
|
| DOI |
10.14288/1.0080016
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| URI | |
| Degree | |
| Program | |
| Affiliation | |
| Degree Grantor |
University of British Columbia
|
| Graduation Date |
1994-11
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| Campus | |
| Scholarly Level |
Graduate
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| Aggregated Source Repository |
DSpace
|
Item Media
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.