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

  1. 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.
  2. 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:1512.01565Freely accessible. doi:10.4007/annals.2016.184.2.7.
  3. Bourgain, Jean (2016-01-26). "On the Vinogradov mean value". arXiv:1512.01565Freely accessible.
  4. 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.
  5. I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
  6. 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.
  7. Stečkin, Sergeĭ Borisovich (1975). "Mean values of the modulus of a trigonometric sum". Trudy Mat. Inst. Steklov (in Russian). 134: 283–309. MR 0396431.
  8. 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.
  9. 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.
This article is issued from Wikipedia - version of the 7/29/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.