Exercise 67, § 1

Let $k\geqslant 2$ and $\ell > k$ be natural numbers. For every integer $N\geqslant 1$, let $r_{k,\ell}(N)$ denote the number of families $(n_1, . . . , n_{\ell})\in \mathbf{N}^{\ell}$ such that

The object of this exercise is to prove that, if $\ell$ is sufficiently large as a function of $k$, then $r_{k,\ell}(N)\geqslant 1$ ("Waring's problem") for every $N\geqslant 1$.

Let $e:\mathbf{T}\rightarrow \mathbf{C}^*$ denote the character defined by passing to the quotient from the character $x\mapsto e^{2i\pi x}$ of $\mathbf{R}$.

Let $P\geqslant 1$ be an integer. Let $S_P$ denote the function on $\mathbf{T}$ such that

Let $dx$ denote the normalized Haar measure on $\mathbf{T}$.

a) If $P>N^{1/k}$, then one has

b) Let $\delta$ be a real number such that $0< \delta <1/3$ and let $P = [N^{1/k}]$. Suppose that N is sufficiently large, so that $P\geqslant 1$. Let $a$ and $q$ be relatively prime integers such that $1\leqslant q\leqslant P^{\delta}$. Let $M_{a,q}$ denote the image in $\mathbf{T}$ of the open interval

For $x\in M_{a,q}$, one has

as $N\rightarrow+\infty$, where

c) Let M denote the union of the sets $M_{a,q}$ when $q$ runs through the set of integers such that $1\leqslant q\leqslant P^{\delta}$ and $a$ the set of integers prime to $q$ such that $1\leqslant a < q$. If $\delta <1/5$, then there exists a real number $\delta '>0$ such that

as $N\rightarrow +\infty$, where one puts, for $Q\geqslant 1$

$q$

1 $\ell aN$ $\mathfrak{S}(Q,N) =\sum\sum S(a;q)e-$

$1\leqslant q\leqslant Qa=1$

J(Q) = $e(yt^k)dte(-y)dy$.

$-Q$ 0

d) One has

J(Q) = + O(Q )

as $Q\rightarrow +\infty$. (First prove that

J(Q) = $(y_1\cdots y_{\ell-1}(1-y_1- \cdots -y_{\ell-1}))dy+ O(Q$ )

where

then evaluate this integral; cf. FVR VII, p. 8, Prop. 4.)

e) Suppose that $\ell > \delta^{-1}k2^k$. There exists $\delta '>0$ such that

when $N\rightarrow +\infty$. (If N is sufficiently large and if $x\in \mathbf{R}$ has image in $\mathbf{T}-$ M, there exist integers $a$ and $q$ relatively prime such that

Then use Exercise 66.)

f) For every prime number $p$ and every integer $m\geqslant 1$, let us put

$a=1$ $(a,q)=1$

and

$N(p^m) =$ Card($\{(x_1, . . . , x_{\ell})\in (\mathbf{Z}/p^m\mathbf{Z})^{\ell}|x^k_1+\cdots +x^k_{\ell}= N$ mod$. p^m\}$). When $\ell >2^k$, one has

$+\infty m$

$1 +\sum A(p^m) =$ lim$_{m\rightarrow+\infty}\frac{N(p )}{p^{m(\ell-1)}}$.

$m=1$

g) When $\ell >2^k$, one has $\mathfrak{S}(Q,N) =\mathfrak{S}(N) +o(1)$ when $Q\rightarrow +\infty$, where

the product being taken over the prime numbers is absolutely convergent. h) Let $p$ be a prime number. Let $\nu \geqslant 0$ be the integer such that $k=p^{\nu}r$ where $p$ does not divide $r$. Let us denote by $\gamma$ the integer

$\{\nu + 1$ if $p\geqslant 3$

$\nu + 2$ otherwise.

If there exist integers $(n_1, . . . , n_{\ell})$ such that $p$ does not divide all the $n_i$ and such that

$x^k_1+\cdots +x^k_{\ell}\equiv 0$ mod$. p^{\gamma}$,

then

lim$_{m\rightarrow+\infty}\frac{N(p^m)}{p^{m(\ell-1)}}>0$.

i) There exists $\ell > k$ such that, if N is sufficiently large, then $\mathfrak{S}(N)>0$. (Use the preceding question and question h) of Exercise 33.)

j) There exists an integer $\ell > k$ such that $r_{k,\ell}(N)\geqslant 1$ for every integer $N\geqslant 1$.