Weird sum regularization?

Question

I noticed some weird behavior in "Dirichlet" regularization of infinite sums. Let us first compute

 Sum[i, {i, 1, Infinity}, Regularization -> "Dirichlet"]

Sure enough, the answer is

-1/12

Which can be explained by what the regularization actually does. The reference page states that $\lim_{s\to0}\sum_{i=1}^\infty\frac{i}{i^s}=\lim_{s\to0}\zeta(-1+s)=\zeta(-1)=-1/12$ is what is being computed in this case. So far there are no problems. Let us now shift the summation index down by an integer

Sum[i, {i, 0, Infinity}, Regularization -> "Dirichlet"]

Naively, one would not expect the result to change, since adding zero to a sum should do nothing. Even in the regularized version no problems are expected, since we are interested in the $|s|<1$ case. Yet for some reason the result changes to

5/12

Is this a bug? Or maybe the routine relies on some mathematics that I missed? Would be great if someone could clarify this. Thanks for any suggestion!

Answer 1

The suggested answer to a similar question asked here just explains how adding a zero causes a formal overall shift by an integer in the summation range, which causes the Dirichlet regularization to become ambiguous. Indeed, it is not a problem with Mathematica but with the regularization technique. Luckily, this blog entry gives a much sturdier regularization technique, using which we get

SeriesCoefficient[Sum[i Exp[-q i], {i, 1, Infinity}], {q, 0, 0}]

with result

-1/12

And

SeriesCoefficient[Sum[i Exp[-q i], {i, 0, Infinity}], {q, 0, 0}]

with result

-1/12

in both cases, whether a zero in the summation is included or not. The consensus is:

Avoid "Dirichlet" regularization.