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Riemann's Zeta Function
Preface This document grew out of lecture notes for a course taught in Cambridge during Lent 2014 by Adam Harper on the theory of the Riemann zeta function. There are likely to be errors, which are
Sharp conditional bounds for moments of the Riemann zeta function
We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all large T. This is sharp up to the value of the implicit
A note on the maximum of the Riemann zeta function, and log-correlated random variables
In recent work, Fyodorov and Keating conjectured the maximum size of $|\zeta(1/2+it)|$ in a typical interval of length O(1) on the critical line. They did this by modelling the zeta function by the
Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers
We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 +
Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function
We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is
On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
We investigate the ``partition function'' integrals $\int_{-1/2}^{1/2} |\zeta(1/2 + it + ih)|^2 dh$ for the critical exponent 2, and the local maxima $\max_{|h| \leq 1/2} |\zeta(1/2 + it + ih)|$, as
Maxima of a randomized Riemann zeta function, and branching random walks
A recent conjecture of Fyodorov--Hiary--Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is $\exp\{\log \log T
We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$. In the
Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers
We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant
On a paper of K. Soundararajan on smooth numbers in arithmetic progressions
In a recent paper, K. Soundararajan showed, roughly speaking, that the integers smaller than x whose prime factors are less than y are asymptotically equidistributed in arithmetic progressions to