Adam Harper's Cambridge web page

About me

From October 2008 until July 2012, I was a PhD student in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, and a member of King's College.
My supervisor was Professor Ben Green.
During the 2012-2013 academic year, I was a CRM-ISM postdoctoral fellow at the Université de Montréal, working with Professor Andrew Granville.
From October 2013 until September 2016, I am a Research Fellow at Jesus College, Cambridge.
Starting in September 2016, I will be an Assistant Professor in the Mathematics Institute at the University of Warwick. See this link for contact details: Warwick page

Research interests

I am a number theorist, and am particularly interested in analytic, combinatorial and probabilistic number theory. I also enjoy learning about more general topics in analysis, combinatorics, probability and statistics.
Thus far, my research has dealt with a selection of problems in probability and probabilistic number theory, including the behaviour of random multiplicative functions, extreme values of Gaussian processes, and applications to the Shanks--Rényi prime number race between residue classes; with the distribution and applications of smooth numbers (that is numbers without large prime factors); with the behaviour of the Riemann zeta function on the critical line, both conjecturally and rigorously; with some additive combinatorics questions connected with sieve theory; and with estimating various sums of general deterministic multiplicative functions.

Teaching

In Michaelmas Term 2015, I lectured a Part III (fourth year) course in Cambridge on Probabilistic Number Theory. In Lent Term 2015, I lectured a Part III course on Elementary Methods in Analytic Number Theory, and in Lent Term 2014 I lectured a Part III course on The Riemann Zeta Function. You can find the lecture notes and problem sheets for all those on my Teaching page.

Papers and preprints

Here is a list of my papers and preprints, in reverse order of when they were written, with links to freely accessible versions (either on the arXiv e-Print archive, or an author version of the journal submission). Links are also provided to the final published versions, but these typically require a subscription to access. Copyright of published papers belongs to the publisher.

(Joint with K. Soundararajan) Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions. http://arxiv.org/abs/1602.01984
To appear in Quarterly Journal of Mathematics.

(Joint with Youness Lamzouri) Orderings of weakly correlated random variables, and prime number races with many contestants. http://arxiv.org/abs/1509.07188

(Joint with Louis-Pierre Arguin and David Belius) Maxima of a randomized Riemann Zeta function, and branching random walks. http://arxiv.org/abs/1506.00629
To appear in Ann. Appl. Probab.

(Joint with Ashkan Nikeghbali and Maksym Radziwiłł) A note on Helson's conjecture on moments of random multiplicative functions. http://arxiv.org/abs/1505.01443
Analytic Number Theory (volume in honour of Helmut Maier's 60th birthday), pp 145-169, Springer, Cham, 2015.

(Joint with Andrew Granville and K. Soundararajan) Mean values of multiplicative functions over function fields. http://arxiv.org/abs/1504.05409
Res. Number Theory (2015), 1:25.

Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers. http://arxiv.org/abs/1408.1662
Compositio Math., 152, no. 6 (2016), pp 1121-1158.

Pickands' constant H_α does not equal 1/Γ(1/α), for small α. http://arxiv.org/abs/1404.5505
To appear in Bernoulli.

(Joint with Ben Green) Inverse questions for the large sieve. http://arxiv.org/abs/1311.6176
Geom. Funct. Anal., 24 (2014), pp 1167-1203.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00039-014-0288-1 .

(Joint with Christian Elsholtz) Additive decompositions of sets with restricted prime factors. http://arxiv.org/abs/1309.0593
Trans. Amer. Math. Soc., 367, no. 10 (2015), pp 7403-7427.
Link to Journal Version .

Sharp conditional bounds for moments of the Riemann zeta function. http://arxiv.org/abs/1305.4618

A note on the maximum of the Riemann zeta function, and log-correlated random variables. http://arxiv.org/abs/1304.0677

Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers. http://arxiv.org/abs/1208.5992

On finding many solutions to S-unit equations by solving linear equations on average. http://arxiv.org/abs/1108.3819

On a paper of K. Soundararajan on smooth numbers in arithmetic progressions. http://arxiv.org/abs/1103.2106
J. Number Theory, 132 (2012), pp 182-199.
Link to Journal Version .

Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. http://arxiv.org/abs/1012.0210
Ann. Appl. Probab., 23 (2013), pp 584-616.
Link to Journal Version .

On the limit distributions of some sums of a random multiplicative function. http://arxiv.org/abs/1012.0207
J. Reine Angew. Math., 678 (2013), pp 95-124.
Link to Journal Version .

Two new proofs of the Erdős--Kac theorem, with bound on the rate of convergence, by Stein's method for distributional approximations. Author version
Math. Proc. Camb. Phil. Soc., 147, part 1 (2009), pp 95-114.
Link to Journal Version . Copyright belongs to the Cambridge Philosophical Society.

Other writings

Here are a few other mathematical things that I have written. These are not intended for publication, so aren't extremely polished, but I hope they are accurate and of some interest.

A different proof of a finite version of Vinogradov's bilinear sum inequality. (pdf link) This is a 3 page note giving a different proof of a bilinear sum inequality from a paper of Bourgain, Sarnak and Ziegler. The new proof exploits a classical kind of result from probabilistic number theory, namely that a certain divisor sum (additive function) is "close to constant on average" (i.e. has small variance). See Terence Tao's blog post for some more discussion of this topic.

A version of Baker's theorem on linear forms in logarithms. (pdf link) These are fairly brief notes that I wrote when giving an expository talk about Baker's results on linear forms in logarithms. The notes should be thought of as giving a moderately detailed sketch proof, where my aim was to motivate the various steps of Baker's argument. When I gave the talk, the consensus was that one should think of the argument (constructing an auxiliary function) in the same spirit as the "polynomial method" from combinatorics.

Contact details

Anyone interested in my work is very welcome to contact me by e-mail. My address is A.J.Harper "at" dpmms "dot" cam "dot" ac "dot" uk , or alternatively ajh228 "at" cam "dot" ac "dot" uk .

This page was last updated on the 26th of August 2016.