In this post, we consider Weyl sums, which are exponential sums with polynomial phases; a Weyl sum is defined by
f(α,N):=∑n≤Ne(αknk+...+α1n),α∈Rk.
We also consider the closely related Vinogradov integrals
Js,k(X):=∫[0,1]k|f(α,X)|2sdα,
which are moments of Weyl sums with respect to the coefficient vector
α. We derive an efficient estimate for the Vinogradov integrals, known as Vinogradov's mean value theorem and essentially due to Vinogradov, following a
simple proof by Wooley. We then utilize this estimate to achieve significant cancellation in general exponential sums
∑n≤Ne(F(n)), whose phase function
F is smooth and grows very slowly. In proving that estimate, we follow Chandrasekharan's book
Arithmetical functions. In a
later post, this general estimate, also due to Vinogradov, will be employed to bound the zeta sums
∑M≤n≤Nn−it.
These estimates in turn give bounds for growth of the Riemann zeta function, and further a zero-free region, and eventually a much better error term for the prime number theorem than the classical π(x)=Li(x)+O(xexp(−clog12x)) (log12x will be replaced by log47x). With some effort, one could use the Vinogradov integrals to bound Weyl sums, but we instead put the method of Weyl differencing into use, obtaining usually weaker bounds but with less effort.