Metric number theory, lacunary series and systems of dilated functions

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena. Without any growth conditions on $(n_k)_{k \geq 1}$ the situation becomes much more complicated, and the system $(f(n_k x))_{k \geq 1}$ will typically fail to satisfy probabilistic limit theorems. An important problem which remains is to study the almost everywhere convergence of series $\sum_{k=1}^\infty c_k f(k x)$, which is closely related to finding upper bounds for maximal $L^2$-norms of the form $$\int_0^1 (\max_{1 \leq M \leq N}| \sum_{k=1}^M c_k f(kx)|^2 dx.$$ The most striking example of this connection is the equivalence of the Carleson convergence theorem and the Carleson--Hunt inequality for maximal partial sums of Fourier series. For general functions $f$ this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors.Comment: Survey paper for the RICAM workshop on "Uniform Distribution and Quasi-Monte Carlo Methods", held from October 14-18, 2013, in Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

Lower bounds for the maximum of the Riemann zeta function along vertical lines

Let $\alpha \in (1/2,1)$ be fixed. We prove that $$\max_{0 \leq t \leq T} |\zeta(\alpha+it)| \geq \exp\left(\frac{c_\alpha (\log T)^{1-\alpha}}{(\log \log T)^\alpha}\right)$$ for all sufficiently large $T$, where we can choose $c_\alpha = 0.18 (2\alpha-1)^{1-\alpha}$. The same result has already been obtained by Montgomery, with a smaller value for $c_\alpha$. However, our proof, which uses a modified version of Soundararajan's "resonance method" together with ideas of Hilberdink, is completely different from Montgomery's. This new proof also allows us to obtain lower bounds for the measure of those $t \in [0,T]$ for which $|\zeta(\alpha+it)|$ is of the order mentioned above.Comment: 23 pages. Version 2: removed a footnote concerning an alleged error in a paper of Titus Hilberdink (actually Hilberdink's proof is correct, and I myself was mistaken - sorry). Version 3: Some minor corrections and additions. The manuscript has been accepted for publication in Mathematische Annale