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1stProposal SIGNED

An alternative development of analytic number theory and applications

Total Cost €

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EC-Contrib. €

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Partnership

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Project "1stProposal" data sheet

The following table provides information about the project.

Coordinator
UNIVERSITY COLLEGE LONDON 

Organization address
address: GOWER STREET
city: LONDON
postcode: WC1E 6BT
website: n.a.

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country United Kingdom [UK]
 Project website https://www.ucl.ac.uk/Mathematics/
 Total cost 2˙011˙742 €
 EC max contribution 2˙011˙742 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-ADG
 Funding Scheme ERC-ADG
 Starting year 2015
 Duration (year-month-day) from 2015-08-01   to  2020-07-31

 Partnership

Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    UNIVERSITY COLLEGE LONDON UK (LONDON) coordinator 2˙011˙742.00

Map

 Project objective

The traditional (Riemann) approach to analytic number theory uses the zeros of zeta functions. This requires the associated multiplicative function, say f(n), to have special enough properties that the associated Dirichlet series may be analytically continued. In this proposal we continue to develop an approach which requires less of the multiplicative function, linking the original question with the mean value of f. Such techniques have been around for a long time but have generally been regarded as “ad hoc”. In this project we aim to show that one can develop a coherent approach to the whole subject, not only reproving all of the old results, but also many new ones that appear inaccessible to traditional methods.

Our first goal is to complete a monograph yielding a reworking of all the classical theory using these new methods and then to push forward in new directions. The most important is to extend these techniques to GL(n) L-functions, which we hope will now be feasible having found the correct framework in which to proceed. Since we rarely know how to analytically continue such L-functions this could be of great benefit to the subject.

We are developing the large sieve so that it can be used for individual moduli, and will determine a strong form of that. Also a new method to give asymptotics for mean values, when they are not too small.

We wish to incorporate techniques of analytic number theory into our theory, for example recent advances on mean values of Dirichlet polynomials. Also the recent breakthroughs on the sieve suggest strong links that need further exploration.

Additive combinatorics yields important results in many areas. There are strong analogies between its results, and those for multiplicative functions, especially in large value spectrum theory, and its applications. We hope to develop these further. Much of this is joint work with K Soundararajan of Stanford University.

 Publications

year authors and title journal last update
List of publications.
2019 Granville, Andrew; Koukoulopoulos, Dimitris
Beyond the LSD method for the partial sums of multiplicative functions
published pages: , ISSN: 1382-4090, DOI:
Ramanujan Journal 2 2019-04-30
2019 Granville, Andrew; Harper, Adam J; Soundararajan, K.
A new proof of Hal\'asz\'s Theorem, and its consequences
published pages: , ISSN: 0010-437X, DOI:
Compositio Mathematica 5 2019-04-30

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