Worked solution: Hilbert's existence proof for Waring's problem (1909)

Step 7 of 7: Aftermath: Hardy–Littlewood, Vinogradov, and today's near-exact

g(k)

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In plain words

A decade after Hilbert's proof, Hardy and Littlewood found a completely different way to attack Waring's problem, developing a technique — the circle method — that Hardy and Ramanujan had pioneered a few years earlier for a seemingly unrelated question about counting partitions of integers. It reformulates 'how many ways can $N$ be written as a sum of $s$ $k$ -th powers' as counting how a certain complex function winds around a circle, then carefully separates the 'big contributions' near simple fractions from everything else.

This machine is far more powerful than Hilbert's identity: it not only reproves finiteness of $g(k)$ , but gives concrete numbers, and moreover computes the closely related quantity $G(k)$ (the number of $k$ -th powers needed to represent all sufficiently large integers, which can be much smaller than $g(k)$ ). Over the following century, Vinogradov, and more recently Wooley (using efficient congruencing and later decoupling methods), sharpened these bounds dramatically, to the point where the value of $g(k)$ is now known exactly for almost every $k$ .

\mathfrak{S}(N) = \sum_{q\le N} \sum_{(a,q)=1} \Big(\tfrac1q\sum_{r\bmod q} e^{2\pi i a r^k/q}\Big)^k e^{-2\pi i aN/q}

Detailed analysis

Hardy and Littlewood's circle method (1920), building on Hardy and Ramanujan's 1918 work on the partition function, studies $r_s(N)$ , the number of ways to write $N=a_1^k+\cdots+a_s^k$ with $a_i\ge0$ , via the generating function $F(z)=\sum_{a\ge0}z^{a^k}$ for $|z|<1$ : Cauchy's integral formula gives $r_s(N)=\frac{1}{2\pi i}\oint F(z)^sz^{-N-1}\,dz$ around a circle inside the unit disk. Splitting the circle into 'major arcs' near rationals $p/q$ with small $q$ (where the integrand is large and well-approximated by an explicit 'singular series' $\mathfrak{S}(N)$ ) and 'minor arcs' (where cancellation makes the contribution small), Hardy and Littlewood showed $r_s(N)\sim C\,\mathfrak{S}(N)N^{s/k-1}$ for $s$ large enough relative to $k$ , in particular $r_s(N)>0$ for all large $N$ once $s\ge s_0(k)$ for an explicit $s_0(k)$ .

This gives an explicit, if not always optimal, bound on $g(k)$ and also on $G(k)\le g(k)$ (the number of $k$ -th powers needed for all sufficiently large $N$ , ignoring finitely many small exceptions) — already a qualitative improvement over Hilbert's method, which produced no numbers at all. Subsequent refinements (I.M. Vinogradov's mean value method in the 1930s, and Trevor Wooley's efficient congruencing and decoupling methods from the 2010s) dramatically sharpened these bounds; Wooley proved $G(k)\le k(\log k+O(\log\log k))$ , close to the truth for large $k$ .

For $g(k)$ itself, a formula conjectured by J.A. Euler (son of Leonhard Euler), $g(k)=2^k+\lfloor(3/2)^k\rfloor-2$ , is now known to hold for all $k$ up to very large computationally verified bounds, and is proven unconditionally for all sufficiently large $k$ (via a criterion of K. Mahler on how well $3/2$ can be approximated by rationals, combined with explicit computation for the remaining finite range) — so Waring's original 1770 question is today essentially completely resolved, a far cry from Hilbert's purely qualitative starting point in 1909.

Terms in this step

$G(k)$: The smallest number of non-negative $k$ -th powers needed to represent every sufficiently large integer (allowing finitely many small exceptions) — typically much smaller than $g(k)$ , which must also account for a handful of awkward small integers requiring unusually many terms.