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I've tried to calculate few classic sums using Dirichlet regularization:

Sum[n, {n, 1, ∞}, Regularization -> "Dirichlet"]
Sum[n^2, {n, 1, ∞}, Regularization -> "Dirichlet"]

I get expected results: -1/12 and 0.

But when I start sum from the 0, I notice an interesting pattern:

$$ \sum_0^\infty n^k - \sum_1^\infty n^k = (-1)^{k+1}\frac{1}{k + 1} $$

I'm having trouble understanding this. Is this some misuse of Dirichlet regularization in Mathematica, or some interesting (or not) thing in the math itself?

m_goldberg
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m0nhawk
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    You might want to read this Wolfram Blog article. – m_goldberg Sep 11 '14 at 14:52
  • Actually, after this article I tried this. I just can't understand what method Mathematica use; if we add zero, the Maclaurin series shouldn't change, but the answer is different. – m0nhawk Sep 11 '14 at 14:56
  • Dirichlet regularization is not stable. What that means is $\sum_{n=1}^\infty a_n \neq a_1 + \sum_{n=2}^\infty a_n$, where the sums here are Dirichlet regularized. See https://brilliant.org/wiki/sums-of-divergent-series/#dirichlet-series-regularization – Greg Hurst Jan 28 '18 at 21:21
  • This bug has been fixed since then – Anixx Nov 16 '19 at 15:39

1 Answers1

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Since Dirichlet series do not make sense when the index starts at zero, to Zeta-regularize $\sum_{n=0}^\infty f(n),$ Mathematica reasonably replaces it with $\sum_{n=1}^\infty f(n-1),$ so you are saying that $$\sum_{l=0}^{k-1} (-1)^l S_l \binom{k}{l} = (-1)^{k+1}/(k+1),$$ where $S_l$ is the regularized sum of $n^l,$ which is interesting, but is a mathematical (rather than a Mathematica) fact.

Igor Rivin
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