# Controlling High Moments with Short Low Moments

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.

• 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. Jul 19 at 20:50