|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'504'928
Articles rated: 2609
26 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
The Development of a Hybrid Asymptotic Expansion for the Hardy Fuction Z(t), Consisting of Just [2*sqrt(2)-2]*sqrt(t/(2*pi)) Main Sum Terms, some 17% less than the celebrated Riemann-Siegel Formula | | D. M. Lewis
; | | Date: |
17 Feb 2015 | | Abstract: | This paper begins with a re-examination of the Riemann-Siegel Integral, which
first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L.
Siegel on his 1932 account of Riemanns unpublished work on the zeta function.
By application of standard asymptotic methods for integral estimation, and the
use of certain approximations pertaining to special functions, it proves
possible to derive a new zeta-sum for the Hardy function Z(t). In itself this
new zeta-sum (whose terms made up of elementary functions, but are unlike those
that arise from the analytic continuation of the Dirichlet series) proves to be
a computationally inefficient method for calculation of Z(t). However, by
further, independent analysis, it proves possible to correlate the terms the
new zeta-sum with the terms of the Riemann-Siegel formula, thought, since its
discovery by Siegel, to be the most efficient means of calculating Z(t). Closer
examination of this correlation reveals that is possible to formulate a hybrid
asymptotic formula for Z(t) consisting of a sum containing both Riemann-Siegel
terms and terms from the new zeta-sum, in such a way as to reduce the overall
CPU time required by a factor between 14-15 percent. Alongside the obvious
computational benefits of such a result, the very existence of the new zeta-sum
itself highlights new theoretical avenues of study in this field. | | Source: | arXiv, 1502.6903 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER