Vinogradov's mean-value theorem


In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers.
It is an important inequality in analytic number theory, named for I. M. Vinogradov.
More specifically, let count the number of solutions to the system of simultaneous Diophantine equations in variables given by
with
That is, it counts the number of equal sums of powers with equal numbers of terms and equal exponents,
up to th powers and up to powers of. An alternative analytic expression for is
where
Vinogradov's mean-value theorem gives an upper bound on the value of.
A strong estimate for is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip. Various bounds have been produced for, valid for different relative ranges of and. The classical form of the theorem applies when is very large in terms of.
An analysis of the proofs of the Vinogradov mean-value conjecture can be found in the Bourbaki Séminaire talk by Lillian Pierce.

Lower bounds

By considering the solutions where
one can see that.
A more careful analysis provides the lower bound

Main conjecture and proof announcement

The main conjecture of Vinogradov's mean value theorem is that the upper bound is close to this lower bound. More specifically that for any we have
If
this is equivalent to the bound
Similarly if the conjectural form is equivalent to the bound
Stronger forms of the theorem lead to an asymptotic expression for, in particular for large relative to the expression
where is a fixed positive number depending on at most and, holds.
On December 4, 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth announced a proof of Vinogradov's Mean Value Theorem.

Vinogradov's bound

Vinogradov's original theorem of 1935 showed that for fixed with
there exists a positive constant such that
Although this was a ground-breaking result, it falls short of the full conjectured form. Instead it demonstrates the conjectured form when

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba and Stechkin who showed that for there exists a positive constant such that
where
Noting that for
we have
this proves that the conjectural form holds for of this size.
The method can be sharpened further to prove the asymptotic estimate
for large in terms of.
In 2012 Wooley improved the range of for which the conjectural form holds. He proved that for
and for any we have
Ford and Wooley have shown that the conjectural form is established for small in terms of. Specifically they show that for
and
for any
we have