TITLE: Bound on $L^2$ norm of $1/\zeta(1+i t)$?
QUESTION [20 upvotes]: What sort of bounds (explicit of preference) can one give for
$$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$
Some obvious points:

One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \leq C \log t$ (with $C\leq 42.9$ for $t\geq 2$) and deduce a bound of the form $\leq K T (\log T)^2$ on the integral above. If possible, we would like to do better (asymptotically and also as far as the constants involved are concerned).
To know that the integral is always finite, one needs to know that $\zeta(1+it)\ne 0$ for all $t$. Thus, it is not enought to just apply a mean-value theorem.

REPLY [7 votes]: Balasubramanian, Ivić and Ramachandra ("An application of the Hooley-Huxley contour", Acta Arith. 65 (1993), no. 1, 45–51) prove the asymptotic result found by Lucia and Terry Tao. Their error term is $O(\log T)$, so slightly better than Lucia's error, but they make use of zero-density results.
More generally, their equation (1) gives an analogous result for $\int_{1}^{T} |\zeta(1+it)^k|^2 dt$ for any complex $k$. In fact, this is itself a special case of their equation (4),
$$\int_{1}^{T} |F(1+it)|^2 dt = T\sum_{n \ge 1}|a_n|^2 n^{-2} + O\left( \log \log T + \sum_{n \le T^C} |a_n|^2 n^{-1}\right)$$
where $F(s) = \sum_{n \ge 1} a_n/n^s$ is a Dirichlet series 'arising' from L-functions (concretely, a product of powers of L-functions, their logarithms and their derivatives) and $C$ is a constant depending only on $F$.
Let me also mention that a quick conditional (on RH) proof of your required result is also given in Titchmarsh (2nd edition, p. 338).