How to show something is "true in mean square"?

Question

I am looking at the conjecture, that for every $\varepsilon,B >0$, then $$\Big| \sum_{\substack{n \in \mathbb{N}\\n \leq x}} n^{-it} - \int_1^x u^{-it} du \, \Big| \leq Cx^{1/2}|t|^{\varepsilon}$$ for $1 \leq x \leq |t|^B$, for some constant $C>0$. This is actually equivalent to the Lindelöf Hypothesis. I want to show that for $1 \leq x \leq T$ and $T \leq t \leq 2T$ this is "true in mean square", however i might have misunderstood what this really means. I would think that it corresponds to showing that $$ \frac{1}{T} \int_T^{2T} \Big| \sum_{\substack{n \in \mathbb{N}\\n \leq x}} n^{-it} - \int_1^x u^{-it} du \, \Big|^2 dt \leq \frac{1}{T} \int_T^{2T} C^2x|t|^{2\varepsilon}dt$$

So something like $$\frac{1}{T} \int_T^{2T} \Big| \sum_{\substack{n \in \mathbb{N}\\n \leq x}} n^{-it} - \int_1^x u^{-it} du \, \Big|^2 dt \leq \frac{T^{2 \varepsilon}}{1+2\varepsilon}C^2x=C_1T^{2\varepsilon}x.$$ But it seems like this is not correct. I am thinking that I might not be allowed to integrate on both sides like I've done. Can someone explain how to show that something is true in mean square? :)

Answer 1

I believe "mean square" is an old term meaning "the $L^2$ norm".

I do not know what it means for an inequality to be "true in mean square". Maybe someone else knows that definition.

For an equation like $\sum_{n=1}^\infty f_n = f$ to be "true in mean square" it means that the partial sums of the series converge to $f$ "in mean square", that is, in the $L^2$ norm.

For stochastic integrals $\int_0^t H_s \; dX_s$, the Riemann sums in the definition usually to not converge pointwise, so the convergence is interpreted "in mean square".