Vinogradov's mean-value theorem
Vinogradov's mean value theorem is an important inequality in analytic number theory, named for I. M. Vinogradov. It relates to upper bounds for , the number of solutions to the system of simultaneous Diophantine equations in variables given by
with
- .
In other words, an estimate is provided for the number of equal sums of k-th powers of integers up to X. An alternative analytic expression for is
where
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.[1] 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 .
On December 4, 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth announced a proof of Vinogradov's Mean Value Theorem.[2][3]
The conjectured form
By considering the solutions where
one can see that .
A more careful analysis (see Vaughan [4] equation 7.4) provides the lower bound
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.
Vinogradov's bound
Vinogradov's original theorem of 1935 [5] 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[6] and Stechkin[7] 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[8] improved the range of for which the conjectural form holds. He proved that for
- and
and for any we have
Ford and Wooley[9] have shown that the conjectural form is established for small in terms of . Specifically they show that for
and
for any
we have
References
- ↑ Titchmarsh, Edward Charles (1986). The theory of the Riemann Zeta-function. Edited and with a preface by D. R. Heath-Brown (Second ed.). New York: The Clarendon Press, Oxford University Press. ISBN 0-19-853369-1. MR 0882550.
- ↑ Bourgain, Jean; Demeter, Ciprian; Guth, Larry (2016). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Ann. of Math. 184 (2): 633–682. arXiv:1512.01565. doi:10.4007/annals.2016.184.2.7.
- ↑ Bourgain, Jean (2016-01-26). "On the Vinogradov mean value". arXiv:1512.01565.
- ↑ Vaughan, Robert C. (1997). The Hardy-Littlewood method. Cambridge Tracts in Mathematics. 25 (Second ed.). Cambridge: Cambridge University Press. ISBN 0-521-57347-5. MR 1435742.
- ↑ I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
- ↑ Karatsuba, Anatoly (1973). "Mean value of the modulus of a trigonometric sum". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 37: 1203–1227. MR 0337817.
- ↑ Stečkin, Sergeĭ Borisovich (1975). "Mean values of the modulus of a trigonometric sum". Trudy Mat. Inst. Steklov (in Russian). 134: 283–309. MR 0396431.
- ↑ Wooley, Trevor D. (2012). "Vinogradov's mean value theorem via efficient congruencing". Ann. of Math. 175 (3): 1575–1627. doi:10.4007/annals.2012.175.3.12. MR 2912712.
- ↑ Ford, Kevin; Wooley, Trevor D. (2014). "On Vinogradov's mean value theorem: strong diagonal behaviour via efficient congruencing". Acta Math. 213 (2): 199–236. doi:10.1007/s11511-014-0119-0. MR 3286035.