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A very short proof of the functional equation for zeta
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Ball, Keith M. (2023) A very short proof of the functional equation for zeta. Mathematika, 69 (1). pp. 17-19. doi:10.1112/mtk.12172 ISSN 0025-5793.
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Official URL: https://doi.org/10.1112/mtk.12172
Abstract
This note contains a short proof of Riemann’s [R] functional equation for the zeta function. The argument is computationally similar to the Hankel contour argument that was one of Riemann’s own, and that is called the “second method” in Titchmarsh [T]. But it is made simpler to follow by the symmetries of sine: the fact that sine is a periodic and an odd function. In this form the argument seems to be new or at least not widely known. We begin with a simple lemma which uses integration by parts to introduce the sine (or hyperbolic sine).
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Functions, Zeta, Functional equations, Riemann integral | ||||||||
| Journal or Publication Title: | Mathematika | ||||||||
| Publisher: | London Mathematical Society | ||||||||
| ISSN: | 0025-5793 | ||||||||
| Official Date: | January 2023 | ||||||||
| Dates: |
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| Volume: | 69 | ||||||||
| Number: | 1 | ||||||||
| Page Range: | pp. 17-19 | ||||||||
| DOI: | 10.1112/mtk.12172 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 28 September 2022 | ||||||||
| Date of first compliant Open Access: | 7 December 2022 | ||||||||
| Related URLs: |
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