Generalized zeta function cutoff causes discontinuity
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Assessment
- Difficulty
- 4/5
- Estimated time
- 3-5 days
- Newbie friendliness
- 48/100
Research direction
Start by reproducing the Julia 1.11.4 example with SpecialFunctions v2.5.0 and trace the zeta(s, z) path where the Euler-Maclaurin cutoff determines the initial sum. Compare values across the reported boundary; done means the discontinuity is removed and zeta(-13+7im, 0.5) is consistent with the expected value.
Written by the indexing model from the issue text.
Description
Version Information:
Julia 1.11.4
SpecialFunctions v2.5.0
zeta(s, z) gives incorrect values in the region $\Re (z) \geq 7 + \Re (s - 1) + |\Im (s) |$. For example:
julia> zeta(-13+7im, 0.5) # should be ~350.218774132674 + 259.188383123717im
-3371.4397140865763 + 2764.0891272966132im
The function jumps sharply at the boundary of the affected region and no longer behaves as expected. You can see the effects in these plots (using z=0.5):
If my understanding is correct, the Euler-Maclaurin expansion used should be well-defined for $\Re(s) \gt -17$ or so. However, I think the initial sum is prematurely truncated to zero terms when the cutoff is checked.
- Dominant language
- Julia
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- Merged PRs (30d)
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