I still have to fix the issues rising from these tests, please do not check. Thanks

I'm afraid, but you need more arguments than "empirical testing".

No additional tests have been implemented; please advise if they are required.

At least you could add tests from the issue.

Probably the precision in polylog should be increased by max(0, -nint_distance(s)[1]) in some of the algorithms.

Edit: specifically, this is required in polylog_general where one cancels out two poles of magnitude $\approx 1/(s-nint(s))$.

To make polylog even more reliable (but potentially much slower), sum_accurately could be used.

Tomorrow I'll work on it. Thanks for the suggestion

Hello @fredrik-johansson, I tried this approach but it doesn't seem to work:

return ctx.gamma(1-s)*(-u)**(s-1) + v + max(0, -(round(s)-s))

I think I am missing something because, of course, the output does not differ of that value. Can you help me? Thank you

@ricor07, I guess you changed last line of the polylog_general(). Could you explain what means your code?

Increasing precision seems to be working for me:

diff --git a/mpmath/functions/zeta.py b/mpmath/functions/zeta.py
index 3e3b542..8944666 100644
--- a/mpmath/functions/zeta.py
+++ b/mpmath/functions/zeta.py
@@ -456,6 +456,8 @@ def polylog_general(ctx, s, z):
         return ctx.gamma(v)*(j**v*ctx.zeta(v,0.5+y) + j**-v*ctx.zeta(v,0.5-y))/(2*ctx.pi)**v
     t = 1
     k = 0
+    prec = ctx.prec
+    ctx.prec += max(0, -ctx.nint_distance(s)[1])
     while 1:
         term = ctx.zeta(s-k) * t
         if not abs(term) >= ctx.eps:
@@ -464,7 +466,9 @@ def polylog_general(ctx, s, z):
         k += 1
         t *= u
         t /= k
-    return ctx.gamma(1-s)*(-u)**(s-1) + v
+    r = ctx.gamma(1-s)*(-u)**(s-1) + v
+    ctx.prec = prec
+    return r
 
 @defun_wrapped
 def polylog(ctx, s, z):

On OP example:

>>> import mpmath as mp
... def check(s, z):
...     pl1 = mp.polylog(s, z)
...     print(pl1.real)
...     with mp.workdps(32):
...         pl2 = mp.polylog(s, z)
...         print(pl2.real)
...         
>>> check(1+1e-15, -2)  # that one was bad
-1.09861228866811
-1.0986122886681101262592914356806
>>> check(1, -2 + 1e-8j)
-1.09861228866811
-1.0986122886681096969508007924781
>>> check(1+1e-15, -0.5 + 1e-8j)
-0.405465108108164
-0.40546510810816446598993214576279

Hello, thanks to your suggestion i could fix the issue. I added a control for inf and nan values because your implementation couldn't fit test_issue_505 in test_function2. The tests fit. Let me know what you think

Thanks.