# A transfer operator approach to Maass cusp forms and the Selberg zeta function

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

Zeta functions have played a central role in mathematics for many centuries and the most famous example, the Riemann zeta function, plays a pivotal role in analytic number theory and the proof of the famous Prime Number Theorem, and is also the subject of the famous Riemann Hypothesis. The study of zeta functions in number theory and related fields lead very naturally to the introduction, in 1956, of the Selberg zeta function. In particular, this is a function of a single complex variable which is analogous to the Riemann zeta function, but is defined in terms of characteristics of the geometry of Riemann surfaces rather than in terms of prime numbers.

One of the principle advantages of the Selberg zeta function over the Riemann zeta function is the more explicit characterization of its zeros. There is a classical approach to the Selberg zeta function using the Laplace-Beltrami operator, which is a second order differential operator whose eigenvalues describe the zeros of the Selberg zeta function viewed as a complex function. In particular, this is a setting where the Hilbert-Polya operator theoretic approach to the analogue of the Riemann hypothesis has been successful.

On the other hand, there are certain aspects of the study of the Selberg zeta function where the successes of the spectral method can be augmented by a different approach. More precisely, there is a more modern dynamical approach originating in the pioneering work of Ruelle which brings together ingredients from mathematical physics, operator theory and ergodic theory. In the past decade, there has been considerable interest in the interplay between the classical and modern approaches to understanding the Selberg zeta function and its connections with number theory.

Of particular interest is the use of the dynamical approach to prove results which appear inaccessible by more classical methods. For example, a particular theme of this proposal is the definition and investigation of period functions, which are intimately related to the spectrum of the Laplace-Beltrami operator. Whereas these important functions are fairly intractable using existing techniques, but the aim of this proposal is to analyse them by developing new approaches along the lines. We have every hope that this work will contribute significantly to the development of this important area of mathematics.

One of the principle advantages of the Selberg zeta function over the Riemann zeta function is the more explicit characterization of its zeros. There is a classical approach to the Selberg zeta function using the Laplace-Beltrami operator, which is a second order differential operator whose eigenvalues describe the zeros of the Selberg zeta function viewed as a complex function. In particular, this is a setting where the Hilbert-Polya operator theoretic approach to the analogue of the Riemann hypothesis has been successful.

On the other hand, there are certain aspects of the study of the Selberg zeta function where the successes of the spectral method can be augmented by a different approach. More precisely, there is a more modern dynamical approach originating in the pioneering work of Ruelle which brings together ingredients from mathematical physics, operator theory and ergodic theory. In the past decade, there has been considerable interest in the interplay between the classical and modern approaches to understanding the Selberg zeta function and its connections with number theory.

Of particular interest is the use of the dynamical approach to prove results which appear inaccessible by more classical methods. For example, a particular theme of this proposal is the definition and investigation of period functions, which are intimately related to the spectrum of the Laplace-Beltrami operator. Whereas these important functions are fairly intractable using existing techniques, but the aim of this proposal is to analyse them by developing new approaches along the lines. We have every hope that this work will contribute significantly to the development of this important area of mathematics.

### Planned Impact

Although the primary beneficiaries of this project will be researchers working in pure mathematics, in general, and dynamical systems and the theory of zeta functions and automorphic forms, in particular, we envisage that this project may well eventually have an impact in a somewhat broader setting.

Potential applications of this research could be in engineering (Hidden Markov Models, resonance phenomena, growth models, data compression), economics (modelling of stockmarket, overlapping generation models), biology (tumor growth), physics (statistical physics, quantum chaos) and so on. However, this project is not aiming to explore these applications, but has as its immediate aim to:

(i) strengthen the overall knowledge base within the field of dynamical systems (and more generally mathematics) in the UK, and to maintain current leadership in research in this area (as demonstrated by Warwick being the home of the leading international journal in the area);

(ii) promote a positive and strong image of british mathematics abroad, including publishing and promoting high quality mathematical results obtained through this project;

(iii) put the UK at the cutting edge of research in a active and vibrant area, in part by developing our existing technical advantage and encouraging research activity in this area; and

(iv) train young mathematicians for their future careers in science.

In particular, this project will strive to support UK universities' drive to be among the principal generators of top-quality international research. Of course this is not merely a matter of prestige, since no country in the world can expect to maintain long term economic success without a strong strong scientific base and strong universities and, in particular, without a strong mathematical foundation.

It remains a challenge in Mathematics to reach out to a wider audience, and so to widen the impact of individual projects. However, the main themes in the this proposal of dynamical systems, number theory and geometry are all subject areas within mathematics which are probably easier to communicate than those in many other areas. The expectation is that by explaining our results to the widest possible audience they will have the greatest possible impact.

Potential applications of this research could be in engineering (Hidden Markov Models, resonance phenomena, growth models, data compression), economics (modelling of stockmarket, overlapping generation models), biology (tumor growth), physics (statistical physics, quantum chaos) and so on. However, this project is not aiming to explore these applications, but has as its immediate aim to:

(i) strengthen the overall knowledge base within the field of dynamical systems (and more generally mathematics) in the UK, and to maintain current leadership in research in this area (as demonstrated by Warwick being the home of the leading international journal in the area);

(ii) promote a positive and strong image of british mathematics abroad, including publishing and promoting high quality mathematical results obtained through this project;

(iii) put the UK at the cutting edge of research in a active and vibrant area, in part by developing our existing technical advantage and encouraging research activity in this area; and

(iv) train young mathematicians for their future careers in science.

In particular, this project will strive to support UK universities' drive to be among the principal generators of top-quality international research. Of course this is not merely a matter of prestige, since no country in the world can expect to maintain long term economic success without a strong strong scientific base and strong universities and, in particular, without a strong mathematical foundation.

It remains a challenge in Mathematics to reach out to a wider audience, and so to widen the impact of individual projects. However, the main themes in the this proposal of dynamical systems, number theory and geometry are all subject areas within mathematics which are probably easier to communicate than those in many other areas. The expectation is that by explaining our results to the widest possible audience they will have the greatest possible impact.

## People |
## ORCID iD |

Mark Pollicott (Principal Investigator) |

### Publications

Bandtlow O
(2014)

*Ergodic Theory, Open Dynamics, and Coherent Structures*
Giulietti P
(2013)

*Anosov flows and dynamical zeta functions*in Annals of Mathematics
Pollicott M
(2013)

*A Weil-Petersson type metric on spaces of metric graphs*in Geometriae Dedicata
Pollicott M
(2013)

*Length asymptotics in higher Teichmüller theory*in Proceedings of the American Mathematical Society
Pollicott M
(2014)

*Estimating Mahler measures using periodic points for the doubling map*in Indagationes MathematicaeDescription | The RA has developed a new study of L-functions using transfer operator methods. This will be published in book form by Springer. |

Exploitation Route | The study of dynamical zeta functions and L-functions has stimulated a considerable interest in this area |

Sectors | Digital/Communication/Information Technologies (including Software) |

URL | http://homepages.warwick.ac.uk/~masdbl/preprints.html |