simon-git: spigot (master): Simon Tatham

[Commits to Tartarus hosted VCS]

TL;DR:
  d92301d Improve description of Riemann zeta in manual and comments.

Repository:     https://git.tartarus.org/simon/spigot.git
On the web:     https://git.tartarus.org/?p=simon/spigot.git
Branch updated: master
Committer:      Simon Tatham <anakin at pobox.com>
Date:           2018-01-06 12:26:00

commit d92301d2340965a403b843234cfe7a438239935b
web diff https://git.tartarus.org/?p=simon/spigot.git;a=commitdiff;h=d92301d2340965a403b843234cfe7a438239935b;hp=baea20a6cf8bad89e8bd0e62f660724dea26b89b
Author: Simon Tatham <anakin at pobox.com>
Date:   Sat Jan 6 12:13:40 2018 +0000

    Improve description of Riemann zeta in manual and comments.
    
    In the main spigot manual, I had previously described zeta by saying
    that for s>1 it is defined by the standard series, and for s<=1 by
    analytic continuation. I didn't really like that because it's very
    vague about _what_ the definition turns out to be, and also because
    analytic continuation is the most mathematically advanced concept in
    the entire spigot manual and I was trying to keep the descriptions of
    functions simpler than that.
    
    This morning I ran across an article that mentioned in passing a
    better approach (well known in general, just not to me until now),
    which is to observe that taking just the even terms of the series for
    zeta(s) is equivalent to calculating zeta(s) 2^{-s}, and therefore,
    subtracting twice the latter from the former tells you that
    (1-2^{1-s})zeta(s) has an _alternating_ series which converges for s>0
    rather than merely s>1. So this allows me to describe zeta in full for
    the whole complex plane, using that and the reflection formula.
    (Though of course these convergent-in-theory sequences converge too
    slowly to be useful in practice.)
    
    While I'm at it, I've also edited the comments in zeta.cpp, which in
    the light of this trick I now realise were also a bit inaccurate. I
    had claimed that the error factor of 1-2^{1-s} in Gourdon and Sebah's
    practical zeta-computation formula arose from the binomial-coefficient
    tapering of the (n+1)th to (2n)th terms of their truncated series. Now
    it's clear to me that that's an error; that factor is the same one as
    above, and comes not from the _truncation_ (cleverly tapered or
    otherwise) of the series, but from making it alternating.

 manual.but | 10 +++++++---
 zeta.cpp   | 38 ++++++++++++++++++++++++--------------
 2 files changed, 31 insertions(+), 17 deletions(-)