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I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as:

$$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$

for $T^\epsilon < x \ll T^{1/k}$, as $T\rightarrow \infty$

I would appreciate if someone can give me a reference where this calculation is being done (obviously by induction)?

P.S

$$\mathbb{E}_{\omega,T}(F(t)):= \int_{\mathbb{R}} \omega \left(\frac{t}{T}\right)F(t) \frac{dt}{T}$$

where $\omega$ is a non-negative weight function , whose integral over all the real line is 1, its Fourier transform is well defined, smooth and supported by the interval $[-1,1]$.

Thanks in advance.

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1 Answer

Ok, I found where it's proved.

Here's the citation: A. Selberg, On the remainder in the formula for N(T), the number of zeros of \zeta(s) in the strip 0 < t < T, Avh. Nor. Vidensk. -Akad. Oslo I 1944 (1944), 1-27.

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