Van der Corput's inequality and related bounds

In this post, we prove several bounds for rather general exponential sums, depending on the growth of the derivative of their phase functions. We already proved in this post Vinogradov's bound for such sums under the assumption that some derivative of order $k\geq 4$ is suitably small. Here we prove similar but better bounds when the first, second or third derivative of the phase function is suitably small. We follow Titchmarsh's book The Theory of the Riemann Zeta Function and the book Van der Corput's Method of Exponential Sums by S. W. Graham and G. Kolesnik. The bounds depending on the derivatives enable us to estimate long zeta sums and hence complete the proof of the error term in the prime number theorem from the previous post. As a byproduct, we also get the Hardy-Littlewood bound

\[\begin{eqnarray}\zeta\left(\frac{1}{2}+it\right)\ll t^{\frac{1}{6}}\log t,\end{eqnarray}\]

for the zeta function on the critical line, which will be utilized in the next post in proving Ingham's theorem on primes in short intervals. Two more consequences of the Weyl differencing method, which will be applied to bounding exponential sums based on their derivatives, are the van der Corput inequality and a related equidistribution test. This test easily gives the uniform distribution of the fractional parts of polynomials, as we shall see.

Error term in the prime number theorem

We prove here an improved prime number theorem of the form

\[\begin{eqnarray}\pi(x)=Li(x)+O(x\exp(-c\log^{\frac{4}{7}}x)),\quad Li(x):=\int_{2}^{x}\frac{dt}{\log t},\end{eqnarray}\]

for all $c>0$, a result which is essentially due to Chudakov. The proof is based on bounding the growth of the Riemann zeta function in the critical strip near $\Re(s)=1$, and achieving this in turn builds on bounds for the zeta sums

\[\begin{eqnarray}\sum_{M\leq n\leq N}n^{-it}.\end{eqnarray}\]

To bound these sums, we use a bound for exponential sums with slowly growing phase from this post (Theorem 5 (i)). The proof of that bound was based on Vinogradov's mean value theorem. When $M$ is large enough, however, the theorem from the previous post is no longer helpful, and we must follow a different approach based on Weyl differencing and van der Corput's inequality, which is quite useful by itself. That approach will be postponed to the next post, and here we concentrate on short zeta sums and on deducing the stronger prime number theorem. We follow Chandrasekharan's book Arithmetical functions.

We remark that without relying on exponential sums, one has not been able to prove anything significantly better than $\pi(x)=Li(x)+O(x\exp(-c\log^{\frac{1}{2}}x))$, which was proved by de la Valleé Poussin already in 1899. On the other hand, the best known form of the prime number theorem is due to Vinogradov and Korobov, and it states that 

\[\begin{eqnarray}\pi(x)=Li(x)+O(x\exp(-c_0\log^{\frac{3}{5}}x(\log \log x)^{-\frac{1}{5}}))\end{eqnarray}\] for some $c_0>0$; this was proved nearly 60 years ago in 1958.

Vinogradov's mean value theorem and Weyl sums

In this post, we consider Weyl sums, which are exponential sums with polynomial phases; a Weyl sum is defined by

\[\begin{eqnarray}f(\alpha,N):=\sum_{n\leq N}e(\alpha_kn^k+...+\alpha_1n),\quad \alpha\in \mathbb{R}^k.\end{eqnarray}\]

We also consider the closely related Vinogradov integrals

\[\begin{eqnarray}J_{s,k}(X):=\int_{[0,1]^k}|f(\alpha,X)|^{2s}d\alpha,\end{eqnarray}\]

which are moments of Weyl sums with respect to the coefficient vector $\alpha$. 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 $\sum_{n\leq N}e(F(n))$, 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

\[\begin{eqnarray}\sum_{M\leq n\leq N}n^{-it}.\end{eqnarray}\]

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 $\pi(x)=Li(x)+O(x\exp(-c\log^{\frac{1}{2}} x))$ ($\log^{\frac{1}{2}}x$ will be replaced by $\log^{\frac{4}{7}} x$). 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.

Prime exponential sums and Vaughan's identity

We prove in this post an estimate for the prime exponential sums \[\begin{eqnarray}S(N;\alpha):=\sum_{p\leq N}e(\alpha p)\end{eqnarray}\] using an identity of Vaughan that splits the von Magoldt function $\Lambda(n)$, which is a natural weighted version of the characteristic function of the primes, into pieces that are suited to efficient estimation of oscillating sums $\sum_{n\leq N}f(n)$ over primes. Vinogradov was the first to prove nontrivial cancellation in prime exponential sums in 1937, but the proof remained long and technical until Vaughan discovered in 1977 his formula that simplifies the estimation considerably, giving at the same time a much better bound of $N^{\frac{4}{5}}$ (times logarithmic factors) when $\alpha$ has bounded continued fraction coefficients. This estimate was also required in this post on the ternary Goldbach problem. We will also apply the derived bounds to show that the fractional parts $\{\alpha p\}$ are uniformly distributed for any irrational $\alpha$ as $p$ ranges through the primes.