The Lerch Phi Function is defined as

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k+\nu)^{\alpha}}$$

Now, in my special case I have $\alpha = 1$, hence it does simply reduce to

$$\Phi(-s, 1, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k + \nu)}$$

Clearly if I put $\nu = 0$ the series shall be infinite.

But when I compute it with Mathematica (Wolfram Serious Mathematica software, not Online Alpha) it says that

$$\Phi(-s, 1, \nu) = -\ln(1+s)$$

How is this possible?


There are also other problems related to that. For example setting $\nu = -1$ and Mathematica says

$$\Phi(-s, 1, -1) = 1+ s\ln(1+s)$$

Which is actually not true, since by its definition (both it starts from $k = 0$ or $k = 1$ there is a point in which the series is infinite).

  • $\begingroup$ It looks as though Mathematica starts at $k=1$. This will give you what you seek. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 13:57
  • $\begingroup$ @SimpleArt The problem is that the definition on Wolfram Documentations says it starts from $k = 0$. Sure $k = 1$ would solve the problem... If it's not a bug, then there is an error in the documentation form. $\endgroup$ – Von Neumann Dec 31 '16 at 13:59
  • $\begingroup$ That is probably the case we are seeing. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:00
  • 2
    $\begingroup$ Be careful not to mix up LerchPhi with HurwitzLerchPhi $\endgroup$ – polfosol Dec 31 '16 at 14:14
  • 1
    $\begingroup$ See the first few lines under details $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:16

According to the reference, we have

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{(k+\nu)^{\alpha}}$$

for $\Re(\nu)>0$. For $\Re(\nu)\le0$, we have

$$\Phi(-s, \alpha, \nu) = \sum_{k = 0}^{+\infty} \frac{(-s)^k}{[(k+\nu)^2]^{\alpha/2}}$$

where any term $k+\nu=0$ is eliminated.



  • $\begingroup$ Huh, you know what's funny? I was reading about this function yesterday, and I actually noticed all this and was like "Wow, what a strange function!" only to forget and rediscover the next day. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:19
  • $\begingroup$ Haha! By the way, try to color the last link in RED, so that people will immediately read the strangeness of the Lerch phi! And thanks again. $\endgroup$ – Von Neumann Dec 31 '16 at 14:21
  • $\begingroup$ @AlanTuring There we go, that should be more obvious. $\endgroup$ – Simply Beautiful Art Dec 31 '16 at 14:22

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