Can we give Mathematica hints for symbolic sums and products?

I have this question at MO and I would like to know if we can give hints to get the symbolic sums and products to come out as $\frac{6}{\pi^{2}}\pm\epsilon$ instead of $0.60792710...$ accurate to $\epsilon$.

ε = 0.000000001; s = 0;
Sum[(MoebiusMu[n] + ε)/n^s, {n, 1, ∞}] /
Product[n^(MoebiusMu[n] ε), {n, 1, ∞}] == 1/Zeta[s] + ε


When $s=0$ we get $-2\pm\epsilon$, but we have to use NSum. When $s=2$ or greater, we get the decimal expansion and I would like to see the zeta values $\pm\epsilon$.

Can we do this?

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1 Answer

I just created a wrapper for the generating function of the LHS.

mcalc[s_,ε_] := {1/Zeta[s], 1/Zeta[s] - (2 π)^(2 ε)
(1 + ε Zeta[s]^2)/Zeta[s]}
mcalc[2, 0.000000000001]

(* {6/π^2, -3.87945*10^-12} *)

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