.
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 \(1\le s\le \frac14(k+1)^2 fo \) r 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
