Adam Harper

Adam James Harper (* in Lowestoft ) is a British mathematician who studies analytical and probabilistic number theory.

Career

Harper studied at the University of Oxford (Exeter College), where he won the Oxford Junior Mathematics Prize, and received his PhD from the University of Cambridge with Ben Green in 2012 ( Some topics in analytic and probabilistic number theory ). In Cambridge he won the Smith Prize. As a post-doctoral student he was at CRM in Montreal with Andrew Granville and then from 2013 to 2016 Research Fellow at Jesus College, Cambridge University. He is an Assistant Professor at the University of Warwick .

In 2018 he was Simons visiting professor at CRM in Montreal.

plant

He dealt with the Riemann zeta function and its behavior on the critical line, random multiplicative number theoretic functions, equations for S-units , even numbers and the large sieve . As a student, he refuted a long-standing assumption that the sums of random multiplicative number-theoretic functions (the sum of the whole numbers in a large interval from 1 to x) were normally distributed . It was previously known (Bob Hough) that they are normally distributed if the sum is over integers with a fixed number k of prime factors. According to Harper, this also applies if k is small against , but almost all have numbers in the interval [1, x] around the prime factors. If k is of the order of magnitude , the sum is not normally distributed. ${\ displaystyle \ ln \ ln x}$ ${\ displaystyle \ ln \ ln x}$ ${\ displaystyle \ ln \ ln x}$

Assuming the Riemann conjecture, Harper proved an asymptotic formula assumed by Jonathan P. Keating and Nina Snaith for higher moments of the Riemann zeta function on the critical line, i.e. for

{\ displaystyle \ int _ {0} ^ {T} {| \ zeta ({\ frac {1} {2}} + it) |} ^ {2k} dt}

The correctness of the formula for k = 1.2 was already known, but the question for higher k was open. Harper looked at the general case and proved the correct limit (based on preliminary work by K. Soundararajan , who only "almost" proved the limit). He also made significant progress in a conjecture by Hiary, Fyodorov, and Keating about the asymptotic form of the maximum value of the Riemann zeta function on the critical straight line at almost all intervals of length 1.

He also gave deep-seated results on equations for S-units, that is, equations of the form with the prime factors for a finite set S. He especially considered such equations for even numbers. ${\ displaystyle a + 1 = b}$ ${\ displaystyle a, b}$

Harper's strongest results so far with regard to the distribution of even numbers in arithmetic sequences (analogue of Bombieri and Winogradow's theorem for even numbers instead of prime numbers). Further results concern the prime number race by Daniel Shanks and Alfréd Rényi . He contributed to the new access to analytic number theory by Andrew Granville and K. Soundararajan at ( pretentious access , English pretentious approach ) by a new proof of the underlying set of Gábor Halász were over the upper bound of mean values of multiplicative number theoretic functions.

Honors and memberships

In 2019 Harper received the SASTRA Ramanujan Prize . For 2020 he was awarded a Whitehead Prize from the London Mathematical Society .

Web links

Homepage, University of Warwick

Individual evidence

↑ Adam Harper in the Mathematics Genealogy Project (English)
↑ Harper, On the limit distributions of some sums of a random multiplicative function, J. Reine Angew. Math., Volume 678, 2013, Arxiv 2010
↑ Harper, Sharp conditional bounds for moments of the Riemann zeta function, Arxiv
↑ Harper, The Riemann zeta function in short intervals (after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan), Bourbaki Seminar 2019, Arxiv
↑ Harper, Bombieri - Vinogradov and Barban - Davenport - Halberstam type theorems for smooth numbers, Arxiv 2012
↑ Kevin Ford, Adam J. Harper, Youness Lamzouri: Extreme biases in prime number races with many contestants, Mathematische Annalen, Volume 374, 2019, pp. 517–551, Arxiv
↑ Granville, Harper, Soundararajan, A new proof of Halász's Theorem, and its consequences, Compositio Math., Volume 155, 2019, pp. 126-163, Arxiv
↑ Granville, Harper, Soundararajan, A more intuitive proof of a sharp version of Halász's theorem, Proc. AMS 2018, Arxiv 2017
↑ Adam Harper to Receive 2019 SASTRA Ramanujan Prize , AMS, October 15, 2019

personal data
SURNAME	Harper, Adam
ALTERNATIVE NAMES	Harper, Adam James
BRIEF DESCRIPTION	British mathematician
DATE OF BIRTH	20th century
PLACE OF BIRTH	Lowestoft

[1] Adam Harper in the Mathematics Genealogy Project (English)

[2] Harper, On the limit distributions of some sums of a random multiplicative function, J. Reine Angew. Math., Volume 678, 2013, Arxiv 2010

[3] Harper, Sharp conditional bounds for moments of the Riemann zeta function, Arxiv

[4] Harper, The Riemann zeta function in short intervals (after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan), Bourbaki Seminar 2019, Arxiv

[5] Harper, Bombieri - Vinogradov and Barban - Davenport - Halberstam type theorems for smooth numbers, Arxiv 2012

[6] Kevin Ford, Adam J. Harper, Youness Lamzouri: Extreme biases in prime number races with many contestants, Mathematische Annalen, Volume 374, 2019, pp. 517–551, Arxiv

[7] Granville, Harper, Soundararajan, A new proof of Halász's Theorem, and its consequences, Compositio Math., Volume 155, 2019, pp. 126-163, Arxiv

[8] Granville, Harper, Soundararajan, A more intuitive proof of a sharp version of Halász's theorem, Proc. AMS 2018, Arxiv 2017

[9] Adam Harper to Receive 2019 SASTRA Ramanujan Prize , AMS, October 15, 2019