There have been spectacular developments in Analytic Number Theory in recent times. The main theme of this school is to introduce emerging themes and progress in Analytic Number Theory. Starting from basic concepts the school aims to cover : Equidistribution and Modular Forms; Maass Forms and Galois Representations; Applications of linear forms in logarithms; Recent developments in sieve theory; Zeta and L-functions. Some open problems will also be presented.
Proposed Syllabus
| Speakers | Duration | Details of the syllabus |
| K. Srinivas | 9 | Introduction to Basic Techniques in Analytic Number Theory : Arithmetical functions, asymptotic estimates, analytical properties of the Riemann zeta-function, prime number theorem. |
| R. Thangadurai | 9 | Introduction to Basics of Modular Forms : Modular group, action on upper half plane, fundamental domains; Modular forms of weight k, dimension formula; Hecke operators and their multiplicative property; Eigen functions of Hecke operators; Estimates of Fourier coefficients of Modular forms. |
| Ritabrata Munshi | 4.5 | Convexity and sub-convexity results of Zeta and L-functions and their applications. |
| Kaneenika Sinha | 4.5 | Equidistribution problems in modular forms: Explicit estimation of the characteristic function of an interval by trigonometric polynomials and their application to the study of equidistribution of Fourier coefficients of modular forms. |
| Shanta Laishram | 4.5 | Linear forms in logarithms and applications: Introduction of Baker’s theory of linear forms in logarithms, application to some diophantine equations and approximations. |
| Michel Waldschmidt | 4.5 | Transcendental numbers: Introduction to some of the main methods and to some of the open problems. |
| Ram Murty | 9 | Recent developments and some open problems in analytic number theory. |
| Kumar Murty | 9 | Modular forms, Artin L-functions, Maass forms and Galois representations, recent progress and some open problems. |
Time Table
| Lecture 1 | Tea | Lecture 1 | Lunch | Tutorial | Tea | ||
| Dates | 9:30-11:00 | 11:30-1:00 | 1:00-2:00 | 2:00-4:00 | 4:00-4:15 | ||
| Week1 | 1/6/2015 | KS | TD | ||||
| 2/6/2015 | KS | TD | |||||
| 3/6/2015 | KS | TD | |||||
| 4/6/2015 | KS | TD | |||||
| 5/6/2015 | KS | TD | |||||
| 6/6/2015 | KS | TD | |||||
| week2 | 8/6/2015 | SK | SL | ||||
| 9/6/2015 | SK | SL | |||||
| 10/6/2015 | SK | SL | |||||
| 11/6/2015 | RM | RI | |||||
| 12/6/2015 | RM | RI | |||||
| 13/06/2015 | RM | RI | |||||
| week3 | 15/06/2015 | RM | KM | ||||
| 16/06/2015 | RM | KM | |||||
| 17/06/2015 | RM | KM | |||||
| 18/06/2015 | MW | KM | |||||
| 19/06/2015 | MW | KM | |||||
| 20/06/2015 | MW | KM |
Speakers
| Name of Speakers | Affiliation |
| K. Srinivas (KS) | IMSc, Chennai |
| Kaneenika Sinha (SK) | IISER, Pune |
| Ram Murty(RM) | Queens Univ, Kingston, Canada |
| R. Thangadurai (TD) | HRI, Allahabad |
| Shanta Laishram (SL) | ISI, Delhi |
| Ritabrata Munshi (RI) | TIFR, Mumbai |
| Michel Waldschmidt (MW) | Université Pierre et Marie Curie (Paris 6) , France |
| Kumar Murty (KM) | University of Toronto, Canada |
| Course associates | |
| Name of Speakers | Affiliation |
| Brundaban Sahoo | NISER Bhubaneswa |
| Jaban Meher | IISc, Bangalore |
| Kamalakshya Mahatab | IMSc, Chennai |
| Usha Sangale | SRTM, Nanded |
| Sudheer Pujahari | IISER, Pune |
| Subramani Muttukrishnan | CMI, Chennai |