Integrals involving $1/|\zeta(1+i t)|^2$: closed expression?

Question

Is there by any chance anything resembling a closed expression for, say, the integral $$I = \frac{1}{2 \pi} \int_{-\infty}^\infty \frac{dt}{|\zeta(1+i t)|^2 t^2} ?$$ It is easy to show (by Plancherel) that $$I = \int_0^\infty |m(x)|^2 \frac{d x}{x}$$ for $m(x):=\sum_{n\leq x} \mu(n)/n$, but I'd like to know whether there is something "more closed" than that.

Answer 1

The following Mathematica program:

(*Mathematica start*)
Clear[a, b, upper, lower, n, integral, t, nn];
nn = 10;
f[t_] := 1/(2*
      Pi)*1/(((Zeta[1 + I*t] + Zeta[1 - I*t])/
          2)^2 + ((Zeta[1 + I*t] - Zeta[1 - I*t])/I/2)^2)/t^2;
integral = Sum[(-1)^(n + 1)/n!*t^n*D[f[t], {t, n - 1}], {n, 1, nn}];
t = -1/2;
lower = %;
a = N[integral, 10];
t = 1/2;
upper = %;
b = N[integral, 10];
b - a
NIntegrate[f[t], {t, lower, upper}, WorkingPrecision -> 10]
(*end*)

solves the integral:

$$I = \frac{1}{2 \pi} \int_{-\frac{1}{2}}^\frac{1}{2} \frac{dt}{|\zeta(1+i t)|^2 t^2} $$

symbolically as b-a = 0.1567161649... by repeated integration by parts resulting in some kind of series expansion (probably Taylor expansion), and then at the end compares it to the numerical integral with NIntegrate[f[t], {t, lower, upper}, WorkingPrecision -> 10] = 0.1567161649...

For better precision I would increse nn = 10; but still the lower and upper integration limits are far from infinity.

Not maybe what you wanted, but it is something.