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Lerch's Phi transcendent is

$$ \Phi(z,s,a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} $$

I am trying to compute this for the following parameters:

  1. $z$ is complex, $|z| \approx 1$ and $|z|$ < 1 (typical value for $|z|$ would be $0.99$ to $0.999$)
  2. $s$ is real and positive, and $0 \leq s \leq 10$ or so, with the most important region being something like $0 \leq s \leq 2$
  3. $a$ is real and positive, and $1 \leq a \leq 3000$ or so, although the most important value is $a=1$.

At least for these parameters, after trying a few simple methods, the basic Aitken/Shanks delta-squared acceleration seems to be the fastest. So far I've compared Aitken delta-squared, Wynn's epsilon, and Levin's u-transformation.

There are a few methods mentioned in the literature, most notably the "Combined Nonlinear Condensation Transformation" which is a Van Wijngaarden transformation followed by a Levin u-transformation. However, this only seems to be for real $z$. Some example papers can be seen here and here.

Question: as of 2019, what is the current fastest method to sum this series?


FWIW, this (perhaps unusual) set of parameters has arisen naturally in the setting of audio synthesis, where it is useful for generating anti-aliased digital waveforms such as sawtooth waves, parabolic waves, etc. For instance, if we look at

$$ f(z,s,h) = z\Phi(z, s, 1) - z^{h+1}\Phi(z, s, h+1) $$

then we have $$ f(z,s,h) = \sum_{k=1}^h \frac{z^k}{k^s} $$

so we get something like a "partial polylogarithm," expressible as the difference of two Lerch's transcendents, each of which can be accelerated with something like Aitken's delta-squared.

As a result, we have $\Re[(-i)^s f(e^{2\pi f i t},s, h)]$ is a waveform with exactly $h$ harmonics, with the $k$'th harmonic at amplitude $1/k^s$. So if we set $s=1$ we get the first $h$ harmonics of a sawtooth wave, if we set $s=2$ we get the first $h$ harmonics of a parabolic wave, etc. This is basically just Hurwitz's theorem, but where we then subtract this second Lerch's transcendent term so as to cause all the harmonics beyond $h$ to cancel.

That first term in $f(z,s,h)$ is also equal to the polylogarithm, so perhaps if anyone has any good insight into that, it might be possible to relate to Lerch's transcendent. The end result is also expressible as a Hurwitz's zeta term minus the Lerch term.

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    $\begingroup$ $a=0$ is impossible. Maybe you mean $a=1$ ? Also, what do you mean by $|z|\approx 1$ ? do you always have $|z|=1$, or sometimes $|z|<1$ ? $\endgroup$ Jul 7, 2019 at 10:35
  • $\begingroup$ Oops, yes, edited to say $a=1$. Typically I'm looking at the circle in the complex plane of radius $r$, where $r$ is something like 0.99 and is less than 1. $\endgroup$ Jul 7, 2019 at 13:10

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Crandall’s computation of the incomplete Gamma function and the Hurwitz Zeta function, with applications to Dirichlet L-series gives an overview of a variety of algorithms. For large positive $a$ and $|z|\leq 1$, $s>1$, a rapidly converging integral representation is $$\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1} \exp (-a t)}{1-z \exp (-t)}\,dt.$$

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  • $\begingroup$ Thanks - this unfortunately won't work as I also need $0 < s < 1$. The paper seems interesting, although it also seems like a very, very strange paper, at least at my level of understanding. The paper starts with the above integral and turns it into a sum of two different series, one involving the incomplete gamma function and the other involving Hurwitz zeta, both of which are just as difficult to evaluate. Hmm.... $\endgroup$ Jul 9, 2019 at 8:06
  • $\begingroup$ FWIW, I tried evaluating this integral directly using Riemann sums - it does converge, even if $0 < s < 1$, but seems to require more terms than the delta-squared thing I wrote of in my post. Series acceleration of the Riemann sum didn't seem to do much. What would be a good way to evaluate this integral numerically? $\endgroup$ Jul 9, 2019 at 8:07
  • $\begingroup$ @MikeBattaglia: Use exp-sinh quadrature. $\endgroup$
    – user14717
    Sep 4, 2019 at 21:34
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Using Crandall's Alg#2 (see http://www.marvinrayburns.com/UniversalTOC25.pdf), I computed

Φ(z=0.999,s=0.1,a=1.1)=  5.35222813845137164e+02
real    0m0.031s
user    0m0.025s
sys 0m0.006s

using eight terms. That used Hurwitz Zeta and Gamma functions, not the ascending series. I picked the z,s,a values that would likely give the ascending series difficulty while avoiding special cases (such as occurs with s≡0 and a≡1). I have no idea if that's the "fastest" but it seems better to me. I also used the program to run the (not accelerated) ascending series and got the same accuracy but it used 31811 terms to do so. YMMV

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  • $\begingroup$ How accurate is that result? And how did you calculate the Hurwitz and Gamma functions? $\endgroup$ Aug 13, 2019 at 3:39
  • $\begingroup$ The computation for Hurwitz Zeta sometimes used the series in Crandall#29 (same paper as mentioned above), sometimes DLMF 25.11.10 (dlmf.nist.gov). Those in turn computed with Riemann zeta, incomplete Gamma, and Bernoulli polynomials. The Lerch transcendent is something of an apex function -- to compute it uses a lot of other functions. The accuracy was 15.2 digits (I used Wolfram Alpha's HurwitzLerchPhi to compare) $\endgroup$ Aug 13, 2019 at 18:53
  • $\begingroup$ It is difficult for me to see how useful this is without knowing how difficult it is to compute these other functions. How many terms did you need of the Hurwitz zeta function series? And then, the Hurwitz zeta series uses the incomplete gamma function: how many terms of that are needed for each term of Hurwitz zeta? It does look like your function is reasonably accurate though. $\endgroup$ Aug 13, 2019 at 18:59

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