# On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let

$$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$

where $\zeta$ denotes the Riemann zeta function.

What are the reasonable asymptotic estimates for $I(T)$ ?

Here is what i think:

It is well-known that

$$\int_{-T}^{T} \log \Big|\zeta\Big(\frac{1}{2} + it\Big)\Big|\mathrm{d}t \ll T\log T$$ Therefore, by a dyadic decomposition, it follows from the above that

$$I(T)=\int_{-T}^{T} \frac{\log |\zeta(\frac{1}{2} + it)|}{\frac{1}{4}+ t^2}\mathrm{d}t \ll\frac{\log T}{T}$$

REMARK: An almost similar application of dyadic decomposition can be found on https://mathoverflow.net/a/285215/123305

• Asymptotics as $T\to \infty$? – Amir Sagiv Apr 18 '18 at 6:46
• @AmirSagiv, i mean a relation of the form $I(T) =O(f(T))$, where $f$ is reasonable with respect to $I$. – user123416 Apr 18 '18 at 6:50
• Right, as $T$ goes to $\infty$? – Amir Sagiv Apr 18 '18 at 6:54