# The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only known for degree 3 by work of Wooley. See this survey of Wooley for a more discussion.

It seems to be folklore that a proof of this conjecture should imply improved bounds on the Riemann zeta function / an improved zero free region for the zeta function / an improved error term on the prime number theorem (among other things, such as progress on Waring's problem). These, of course, are some of the most highly prized problems in analytic number theory and have been stuck for decades.

That said, to the best of my knowledge there is no blackbox reduction for these applications and one likely also needs more explicit information about the dependence of constants on various parameters in these results to reach these applications.

What is the potential of these methods? In other words, what are the implications of the most optimistic dependencies in the mean value theorem (or possible generalizations)?

(See here and here for additional discussion).

• While I have no idea about the particular case, I think the question for the potential of a certain new approach is usually hard to answer ... . – Stefan Kohl Dec 9 '15 at 9:31
• One definitely needs explicit constants for the applications you describe. Nevertheless, I do expect improved results for the Riemann zeta function in the long run along these lines, e.g. Bourgain himself delivered such improvements in recent years. – GH from MO Dec 9 '15 at 18:43
• The short answer is that there will be no qualitative change to the error term in the prime number theorem. The best possible quantitative bounds for the mean value theorem can only change the constant in the Korobov-Vinogradov zero free region, that is the constant $c$ in $$\pi(x)=\text{li}(x)+O\left(x\exp\left(c\log(x)^{3/5}(\log \log x)^{-1/5}\right)\right).$$ – Eric Naslund Dec 10 '15 at 16:34
• It does not directly address the question, but it's probably worth linking to this blogpost of Terry Tao: terrytao.wordpress.com/2015/12/10/… – Sam Hopkins Dec 11 '15 at 17:06
• Concerning the ZFR for the Zeta function, in his ICM talk Wooley (youtu.be/cwn6KKzX87w?t=44m19s) answers this question. This is essentially E. Naslund's comment above. – Sary Dec 15 '15 at 14:40

With respect to the recent breakthrough, Bourgain states in this preprint:

Concerning applications to the zeta-function, our work as it stands does not lead to further progress. The reason for this is that we did not explore the effect of large $k$ (possibly depending on $N$) and in the present form is likely very poor. A similar comment applies to Wooley’s approach

This is also discussed in Ford's papers Vinogradov's Integral and Bounds for the Riemann Zeta Function, Zero-Free Regions for the Riemann Zeta Function, where he states:

Lastly we indicate what the limit of our method is, i.e. what could be accomplished with [a version of Vinogradov's Mean Value Theorem]... [this] yields Theorem $1$ with a constant $B=\sqrt{2}+\epsilon$ (valid for $\sigma\geq\frac{15}{16}$) where $\epsilon$ can be taken arbitrarily small.

Here Theorem $1$ is the inequality $$\zeta(\sigma+it)\leq A t^{B(1-\sigma)^{3/2}}\log^{2/3}t\ \ \ \ \ \ (t\geq 3,\ \frac{1}{2}\leq \sigma\leq 1)$$ which Ford proves with $A=76.2$ and $B=4.45$.

[For additional discussion on the implications of the Bourgain-Demeter-Guth breakthrough see, this recent preprint of Heath-Brown A New $k^{th}$ Derivative Estimate for Exponential Sums via Vinogradov’s Mean Value]

• These are certainly interesting and relevant comments. What I think would be a great answer is the following: (1) What is the most optimistic quantitative for of VMC that one might conjecture? (2) If you plug this form of the VMC into Vinogradov-Korobov-type machinery what do you get for a zero-free region? – Mark Lewko Feb 1 '16 at 16:46

And a few words about "possible generalizations". The main object here is the sum $$S=\sum_{u\asymp a}e^{2\pi i F(u)},$$ where $F(u)=-\frac{t\log u}{2\pi}$, $t=a^{n-\theta}$, $0\le \theta<1$. It is known that $|S|\ll a^{1-\frac{c}{n^2}}$. It was mentioned by Vinogradov in his book "Trigonometrical sums in number theory" that even the much stronger estimate $|S|\ll a^{1-\frac{c}{n}}$ (unreachable by this method) will give only $$\pi(x)=\text{li}(x)+O\left(x\exp\left(c\log(x)^{2/3}(\log \log x)^{-1/5}\right)\right).$$