Vinogradov's mean-value theorem

Vinogradov's mean value theorem is an upper bound for $J_{s,k}(X)$ , the number of solutions to the system of $k$ simultaneous Diophantine equations in $2s$ variables given by

$x_1^j+x_2^j+\cdots+x_s^j=y_1^j+y_2^j+\cdots+y_s^j\quad (1\le j\le k)$

with $1\le x_i,y_i\le X, (1\le i\le s)$ . An analytic expression for $J_{s,k}(X)$ is

$J_{s,k}(X)=\int_{[0,1)^k}|f_k(\mathbf\alpha;X)|^{2s}d\mathbf\alpha$

where

$f_k(\mathbf\alpha;X)=\sum_{1\le x\le X}\exp(2\pi i(\alpha_1x+\cdots+\alpha_kx^k)).$

A strong estimate for $J_{s,k}(X)$ 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 $J_{s,k}(X)$ , valid for different relative ranges of $s$ and $k$ . The classical form of the theorem applies when $s$ is very large in terms of $k$ .

The conjectured form

By considering the $X^s$ solutions where $x_i=y_i, (1\le i\le s)$ we can see that $J_{s,k}(X)\gg X^s$ . A more careful analysis (see Vaughan ^[2] equation 7.4) provides the lower bound

$J_{s,k}\gg X^s+X^{2s-\frac12k(k+1)}.$

The main conjectural form of Vinogradov's mean value theorem is that the upper bound is close to this lower bound. More specifically that for any $\epsilon>0$ we have

$J_{s,k}(X)\ll X^{s+\epsilon}+X^{2s-\frac12k(k+1)+\epsilon}.$

If $s\ge k(k+1)$ this is equivalent to the bound

$J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}.$

Similarly if $s\le k(k+1)$ the conjectural form is equivalent to the bound

$J_{s,k}(X)\ll X^{s+\epsilon}.$

Stronger forms of the theorem lead to an asymptotic expression for $J_{s,k}$ , in particular for large $s$ relative to $k$ the expression $J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)},$ where $\mathcal C(s,k)$ is a fixed positive number depending on at most $s$ and $k$ , holds.

Vinogradov's bound

Vinogradov's original theorem^[3] showed that for fixed $s,k$ with $s\ge k^2\log (k^2+k)+\frac14k^2+\frac54 k+1$ there exists a positive constant $D(s,k)$ such that

$J_{s,k}(X)\le D(s,k)(\log X)^{2s}X^{2s-\frac12k(k+1)+\frac12},$

although a ground-breaking result, this falls short of the full conjectured form. Instead this demonstrates the conjectured form for $\epsilon>\frac12$ .

Subsequent improvements

Vinogradov's approach was improved upon by Karatsuba ^[4] and Stechkin ^[5] who showed that for $s\ge k$ there exists a positive constant $D(s,k)$ such that

$J_{s,k}(X)\le D(s,k)X^{2s-\frac12k(k+1)+\eta_{s,k}},$

where

$\eta_{s,k}=\frac12 k^2\left(1-\frac1k\right)^{\left[\frac sk\right]}\le k^2e^{-s/k^2}.$

Note that for $s>k^2(2\log k-\log\epsilon)$ we have $\eta_{s,k}<\epsilon$ and so this proves that the conjectural form holds for $s$ of this size.

The method can be sharpened further to prove the asymptotic estimate

$J_{s,k}\sim \mathcal C(s,k)X^{2s-\frac12k(k+1)},$

for large $s$ in terms of $k$ .

In 2012 Wooley ^[6] improved the range of $s$ for which the conjectural form holds. He proved that for $k\ge 2$ and $s\ge k(k+1)$ and for any $\epsilon>0$ we have

$J_{s,k}(X)\ll X^{2s-\frac12k(k+1)+\epsilon}.$

Ford and Wooley ^[7] have shown that the conjectural form is established for small $s$ in terms of $k$ . Specifically they show that for $k\ge 4$ and $1\le s\le \frac14(k+1)^2$ for any $\epsilon>0$ we have

$J_{s,k}(X)\ll X^{s+\epsilon}.$

References

↑ E. C. Titchmarsh (rev. D. R. Heath-Brown): The theory of the Riemann Zeta-function, OUP
↑ R.C. Vaughan: The Hardy-Littlewood method, CUP
↑ I. M. Vinogradov, New estimates for Weyl sums, Dokl. Akad. Nauk SSSR 8 (1935), 195–198
↑ A. A. Karatsuba, The mean value of the modulus of a trigonometric sum, Izv. Akad. Nauk SSSR 37 (1973), 1203–1227.
↑ S. B. Stechkin, On mean values of the modulus of a trigonometric sum, Trudy Mat. Inst. Steklov 134 (1975), 283–309.
↑ T. D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Annals of Math. 175 (2012), 1575–1627.
↑ Kevin Ford and Trevor D. Wooley: On Vinogradov’s mean value theorem: strong diagonal behaviour via efficient congruencing http://www.math.uiuc.edu/~ford/wwwpapers/ec3vindiag.pdf