Recently I've been working on a project for which I've needed to reference Iwaniec's paper Fourier coefficients of cusp forms and the Riemann zeta function, where the short fourth moment estimate $$ \tag{1} \int\limits_{T}^{T+T^{2/3}} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^4 dt\ll T^{2/3+\epsilon} $$ is proved. It is claimed that this implies Heath-Brown's classic twelfth moment estimate $$ \tag{2} \int\limits_{T}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} dt\ll T^{2+\epsilon}, $$ but no details are given. I have worked for quite some time, but cannot seem to work out how this computation goes. Thus my question:

How does one deduce the estimate (2) from (1)?

Here are some thoughts/ideas I've had:

  1. It seems that inevitably, one has to apply the Weyl subconvexity estimate $$ \zeta\left(\textstyle{\frac{1}{2}}+it\right) \ll t^{1/6+\epsilon}, $$ however no matter how I try to do this, I can't seem to push this through (I'm aware that this is not the strongest subconvexity, but if I need to apply a subconvexity estimate, it should certainly be sufficient).

  2. It suffices to bound the integral $$ \int\limits_{\substack{T\\ \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| > T^{1/8}}}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} dt, $$ since the remaining integral can be bounded using the fourth moment estimate: $$ \int\limits_{\substack{T\\ \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| \leq T^{1/8}}}^{2T} \left| \zeta\left(\textstyle{\frac{1}{2}}+it\right)\right|^{12} dt\leq T \int\limits_{T}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^4 dt\ll T^{2+\epsilon}. $$

  3. One can estimate the number of large values of $\zeta$ on short intervals via $$ \left|\left\{ t\in[T,T+T^{2/3}] : \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| > T^{1/8}\right\} \right| \leq \int\limits_{T}^{T+T^{2/3}} \left(\frac{\left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|}{T^{1/8}} \right)^4 dt\ll T^{1/6+\epsilon}. $$

  4. It seems that one needs some sort of clever decomposition of the interval $[T,2T]$ into intervals of size $T^{2/3}$, perhaps by distinguishing those intervals on which the twelfth moment is large, i.e. intervals $[T_0,T_0+T^{2/3}]$ such that $$ \int\limits_{T_0}^{T_0+T^{2/3}} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} > W $$ for some parameter $W > 0$ to be optimized.

In the language of $L_p$ norms, in general, if $p < q$, then one cannot hope to control an $L_q$-norm by an $L_p$-norm (and the moments are essentially the $L_4$ and $L_{12}$ norms). However, if one understands the $L_p$ on a "local" level (i.e. on a short interval), then the $L_q$-norm should be able to be controlled by the $L_p$-norm in some way.

Any ideas or help is most appreciated.

  • $\begingroup$ Have you read the argument in Heath-Brown's paper? You might also want to look at a paper of Jutila, "The twelfth moment of central values of Hecke series", for similar ideas for how one bounds these high moments using bounds for low moments. $\endgroup$ Jul 19 at 20:50

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