Recently I was asked about the history of Vinogradov’s Mean Value Theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$-tuples $(x_1, ldots, x_s, y_1, ldots, y_s)$ such that

begin{align*}

x_1 + cdots + x_{s} &= y_{1} + cdots + y_{s}\

x_{1}^{2} + cdots + x_{s}^{2} &= y_{1}^{2} + cdots + y_{s}^{2}\

&vdots\

x_{1}^{k} + cdots + x_{s}^{k} &= y_{1}^{k} + cdots + y_{s}^{k}

end{align*}

for $1 leq x_{i}, y_{i} leq X$. It is not hard to see that $$J_{s, k}(X) gtrsim_{s, k} X^{s} + X^{2s – frac{1}{2}k(k + 1)}.$$

The now proven Main Conjecture in Vinogradov’s Mean Value Theorem is that this lower bound is essentially an upper bound. More precisely, the conjecture was:

Conjecture:For every $epsilon > 0$, $$J_{s, k}(X) lesssim_{epsilon, s, k} X^{epsilon}(X^{s} + X^{2s – frac{1}{2}k(k + 1)}).$$

This conjecture follows from classical methods for $k = 2$, first proven by Wooley for $k = 3$ using efficient congruencing in 2014 and then proven by Bourgain, Demeter, and Guth for $k geq 4$ using decoupling methods in 2015.

My question is: when did this conjecture as stated above first appear in the literature?

Looking through Vinogradov’s 1935 paper “New estimates for Weyl sums”, it seems that this conjecture is not stated. The term “Vinogradov’s Mean Value Theorem” referring to any bound of the form $J_{s, k}(X)lesssim X^{2s – frac{1}{2}k(k + 1) + Delta_{s, k}}$ for some $Delta_{s, k}$ positive and $s gtrsim k^{2}log k$ seems to appear in print as early as 1947 or 1948 in these two works by Hua:

- Page 49 of the Russian version of his Additive Theory of Prime Numbers (http://mi.mathnet.ru/eng/tm1019)
- In Hua’s paper “An Improvement of Vinogradov’s Mean-Value Theorem and Several Applications” (https://doi.org/10.1093/qmath/os-20.1.48)

Though in both, it seems to imply that this term was in use as early as 1940. However neither also state the conjecture as mentioned above.