Due to work of Ingham, it is known that $$\int_1^T|\zeta^{(m)}(1/2+it)|^2\;dt\sim\frac{1}{2m+1}T(\log T)^{2m+1}.$$ Is there a similar result for the fourth moment? That is, is there an explicit result of the form $$\int_1^T|\zeta^{(m)}(1/2+it)|^4\;dt\sim f(m)T(\log T)^{4m+4}$$ for a function $f(m)$? The answer is surely yes, but I have so far been unable to locate a reference for this except in the case $m=0$. Any comments would be much appreciated.
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This has been studied by J.B. Corney, The fourth moment of derivatives of the Riemann zeta-function (1987). The result for $k\in\{1,2\}$ is $$\lim_{T\rightarrow\infty}T^{-1}\left(\log\frac{T}{2\pi}\right)^{-k^2-2km}\int_1^\infty |\zeta^{(m)}(1/2+it)|^{2k}\,dt=C_{k,m},$$ with $$C_{1,m}=\frac{1}{2m+1},\;\;C_{2,0}=\frac{1}{2\pi^2},\;\;C_{2,1}=\frac{61}{1680\pi^2},\;\;C_{2,m}\simeq \frac{3}{8m^4 \pi^2 }\;\;\text{for}\;\;m\gg 1.$$
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$\begingroup$ The limit is not known to exist for $k\geq 3$. In fact Conrey only defines $C_{k,m}$ for $k\in\{1,2\}$. $\endgroup$GH from MO– GH from MO2026-02-27 14:52:17 +00:00Commented yesterday
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1$\begingroup$ @GHfromMO --- indeed, thanks; I have added $k\in\{1,2\}$ to the answer; without this restriction, a conjecture for $C_{k,0}$ and $C_{k,1}$ is given, respectively, in arxiv.org/abs/math/0511182 and in arxiv.org/abs/math/0508378; $\endgroup$Carlo Beenakker– Carlo Beenakker2026-02-27 17:51:33 +00:00Commented yesterday