Professor Adam Harper
Teaching Responsibilities 2020/21:
Term 1: MA4L6 Analytic Number Theory
Term 2: MA257 Introduction to Number Theory
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, multiplicative chaos, extreme values of Gaussian processes, and applications to moments of character sums and Dirichlet polynomials, and 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.
Lecture notes for courses I am currently teaching in Warwick may be accessed by following the relevant links above.
Here are the notes for some Part III (fourth year) number theory courses that I lectured in Cambridge.
Probabilistic Number Theory (Michaelmas term 2015). Chapter 0. Chapter 1. Chapter 2. Chapter 3.
Elementary Methods in Analytic Number Theory (Lent term 2015). Chapter 0. Chapter 1. Chapter 2. Chapter 3.
The Riemann Zeta Function (Lent term 2014). Chapter 0. Chapter 1. Chapter 2. Chapter 3.
These informal notes aren't intended for publication, so aren't extremely polished, but people have occasionally asked me for copies of them. 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.
Most relevant recent publications:
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