       Mathematica 3.0.0 bug in LerchPhi function

• To: mathgroup at smc.vnet.net
• Subject: [mg9065] Mathematica 3.0.0 bug in LerchPhi function
• From: koehler at REMOVE-THIS.math.uni-bonn.de (Kai Koehler)
• Date: Thu, 9 Oct 1997 01:42:45 -0400
• Organization: RHRZ - University of Bonn (Germany)
• Sender: owner-wri-mathgroup at wolfram.com

```In article <6171ii\$osf at smc.vnet.net>, luca ciotti
<ciotti at boas5.bo.astro.it> wrote:

> I found a problem in symbolic summation using Mathematica 3.0.0.
>
>  A = Sum[ Exp[-k]/(1+k^3), {k,0,Infinity}]
>
> The analitycal result, when evaluated numerically  is
>
>                           N[A,50] = 0.5344386.....
>
>
> This is obviously wrong, in fact each term of the series is
> positive, and the first (k=0) is equal 1.
>
> Performing directly NSum, return 1.20111.....

Hi,

I verified the exact symbolic result which is given by Mathematica and it
is correct (albeit written up in a too complicated way, even after
Simplify...). When you write it as

(2 + (-1)^(2/3)/E*LerchPhi[1/E, 1, (-1)^(1/3)] -
(-1)^(1/3)/E*LerchPhi[1/E, 1, -(-1)^(2/3)] -
E*Log[1 - 1/E])/3

then even Mathematica is able to evaluate it correctly.

The actual problem is a major bug in the numerical calculation of the
LerchPhi function: Try e.g.

z = 0.4; v = -0.5; Plot[NSum[z^n/(n + v), {n, 0, 20}] -
LerchPhi[z, 1, v], {v, -1, 1}]

The numerical sum is the truncated definition of the LerchPhi function
(which converges very well). For negativ v you shall see a huge error.

It took me some time to find this out. Isn't it just wonderfull that we
serve as beta testers for Wolfram and we shall never get a free bug fix
(if there shall ever be any bug fix at all...)? Imagine that I paid money
for this super-buggy product.

Best regards

Kai Koehler

```

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