mpmath ascribes a value to divergent Lerch Phi series.

[kspalaiologos]

To quote some code from the mpmath implementation:

    if z == 0:
        return a ** (-s)
    # Faster, but these cases are useful for testing right now
    if z == 1:
        return ctx.zeta(s, a)
    if a == 1:
        return ctx.polylog(s, z) / z

it is apparent that Re(s)>1 condition is not checked. As such, per Wolfram, lerchphi(1,0.5,1) is complex infinity, while mpmath outputs the following:

>>> lerchphi(1, 0.5, 1)
mpf('-1.4603545088095868')

[fredrik-johansson]

This corresponds to the usual analytic continuation of the Riemann zeta function. Is there a strong reason to disallow such input?

[skirpichev]

This corresponds to the usual analytic continuation of the Riemann zeta function.

@fredrik-johansson, are you sure? According to the DLMF 25.14.2 - it's not: zeta(s, a) could be used only for certain conditions. Ditto for the polylog in the next if case.

[fredrik-johansson]

It says that the identity should hold when those conditions are met. It doesn't say that the same identity can't hold when those conditions aren't met.

It's just a matter of convention how you define analytic continuations of functions of several variables; the question is what convention is more useful here.

[kspalaiologos]

Could we at least have this behaviour explicitly mentioned in the documentation then, if it is indeed intended?
I have previously filed a similar issue regarding the Zeta function crashing for arguments in which it should not have been evaluated: #653