Vinogradov's mean-value theorem

Vinogradov's mean-value theorem is a result in additive number theory that concerns the distribution of the values of additive functions. It has significant implications for the study of Diophantine equations and is particularly important in the field of analytic number theory. The theorem essentially states that for a certain class of additive functions (typically of the type that can be exhibited as sums of integers), the average number of representations of a number as a sum of other integers can be understood in a mean-value sense.