Author pages are created from data sourced from our academic publisher partnerships and public sources.

Publications Influence

Share This Author

Riemann's Zeta Function

- Adam J. Harper, Tony Feng
- 2014

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… Expand

Sharp conditional bounds for moments of the Riemann zeta function

- Adam J. Harper
- Mathematics
- 20 May 2013

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… Expand

A note on the maximum of the Riemann zeta function, and log-correlated random variables

- Adam J. Harper
- Mathematics
- 2 April 2013

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… Expand

Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

- Adam J. Harper
- Mathematics
- 29 August 2012

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 +… Expand

Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

- Adam J. Harper
- Mathematics
- 1 December 2010

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… Expand

On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture

- Adam J. Harper
- Mathematics
- 13 June 2019

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… Expand

Maxima of a randomized Riemann zeta function, and branching random walks

- Louis-Pierre Arguin, David Belius, Adam J. Harper
- Mathematics
- 1 June 2015

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… Expand

MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS

- Adam J. Harper
- MathematicsForum of Mathematics, Pi
- 20 March 2017

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… Expand

Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

- Adam J. Harper
- MathematicsCompositio Mathematica
- 7 August 2014

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… Expand

On a paper of K. Soundararajan on smooth numbers in arithmetic progressions

- Adam J. Harper
- Mathematics
- 10 March 2011

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… Expand

...

1

2

3

4

...