In Hardy-Littlewood's 1923 paper "Some problems of 'Partitio Numerorum' III" it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\chi)$ have $\Re(s)<3/4-\varepsilon$), that for all $k\geqslant 3$, when $n\to\infty$ through the integers with same parity than $k$: $$ r_{k}(n) \sim \frac{2C_k}{(k-1)!}\frac{n^{k-1}}{\log(n)^k}\prod_{\substack{p\mid n \\ p\geqslant 3}} \left(\frac{(p-1)^k+(-1)^k(p-1)}{(p-1)^k -(-1)^k} \right), $$

where $r_{k}(n):= \{(p_1,\ldots,p_k)\in\mathbb{P}^k:\sum p_i = n \}$ and $C_k$ is a constant given by: $$ C_k := \prod_{p\geqslant 3} \left(1- \frac{(-1)^k}{(p-1)^k}\right). $$

Roughly 15 years later Vinogradov introduced his influencial technique that allowed him to prove this estimate unconditionally for $k=3$. One thing that bothers me is: what about the other $k$?

The wikipedia article on Goldbach's conjecture has (as of today [08/Oct/2016]) the following passage:

This formula has been rigorously proven to be asymptotically valid for $k \geqslant 3$ from the work of Vinogradov, but is still only a conjecture when $k=2$.$^{\text{[citation needed]}}$

I gave a quick look at K. F. Roth & Anne Davenport's translation of Vinogradov's The Method of Trigonometrical Sums in the Theory of Numbers, and in Chapter X. "Goldbach's Problem" it has the following passage at the start:

In the present chapter I give a solution of Goldbach's problem concerning the representability of every sufficiently large odd number $N$ as the sum of three primes, and I establish an asymptotic formula for the number of representations.

The method used here enables one also to solve more general additive problems involving primes, for example the question of representability of large numbers $N$ in the form $$ N = p_1^n + \ldots + p_s^n $$ (Waring's problem for primes). But I do not consider these more general questions here.

Every other source I consulted only talks about the case $k=3$. Does Vinogradov's method for $k=3$ implies the other cases as a corollary? Where can I find more (historical) information about the other $k$ in this Hardy & Littlewood's estimate? Thanks in advance!

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    $\begingroup$ Larger $k$ follow trivially from the case $k=3$. For example, the number of ways of writing $n$ as a sum of $4$ primes is just $\sum_{p\le n} R_3(n-p)$ (maybe divide by $4$ for repeats) where $R_3(n-p)$ is the number of ways of writing $n-p$ as a sum of $3$ primes, and now use Vinogradov. $\endgroup$ – Lucia Oct 8 '16 at 15:38
  • $\begingroup$ @Lucia Hmm, it is not really obvious to me that everything fits perfectly just by looking at this asymptotic expression, but that's a pretty good point, I must confess I had not thought of that! Many thanks. $\endgroup$ – Alufat Oct 8 '16 at 15:53

Your question is concerned with the so-called Waring-Goldbach problem. A classic in this topic is Hua Lo Keng's book, Additive theory of prime numbers (Translations of Mathematical Monographs, 13, American Mathematical Society, Providence, R.I. 1965), which focuses on how large the number of summands $s$ should be in terms of $n$ for an asymptotic formula to hold. It is a characteristic feature of the method that if it works for some value of $s$, then it also works for all larger values of $s$. Hua himself derived an asymptotic formula for $s>cn^2\log(2n)$, with some absolute constant $c>0$, and a more concrete lower bound is available for individual $n$'s. The method has been refined in may ways since, but this book is certainly a good starting point. For later developments, and a good overview in general, I recommend warmly the survey by Kumchev-Tolev: see especially Theorem 3 and the subsequent comments there.

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    $\begingroup$ Wow, I had no idea how general this problem could get. This survey will save me months of being stuck at the library! Thank you very much :) $\endgroup$ – Alufat Oct 8 '16 at 16:45
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    $\begingroup$ @ChrisTáfula: I am glad I could help! $\endgroup$ – GH from MO Oct 8 '16 at 19:03
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    $\begingroup$ Also in The Hardy-Littlewood Method by Robert C. Vaughan. The important part is that the second edition mentions me. $\endgroup$ – Will Jagy Oct 9 '16 at 0:29

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