IST Complex Analysis and Analytic Number Theory (2018)

syllabus covered in the IST programme
There were two themes running parallel in this programme: Analytic Number Theory and Complex Analysis. The following is the actual syllabus covered in this school.

• Analytic number theory course covered by Usha Sangale (4), Kasi Viswanad-ham (2).
• The Möbius function μ(n), Euler totient function φ(n), Mangoldt function Λ(n), Lioville’s function λ(n) and a few other counting functions were defined, some important identities P satisfied by these functions were derived. Elementary ways of finding a sum formula (e.g., n≤x d(n) = x log x + O(x)) for these arithmetical functions were discussed. Abel summation formula, Euler’s summation formula were derived. Emphasis was given on the application of these formulas to obtain a better error term in the sum formulas for arithmetical functions. Chebyschev type estimates were derived. Prime counting functions and their equivalence were discussed. Riemann zeta-function, its analytic continuation. Dirichlet characters were introduced, their orthogonality relations were derived.
• Complex analysis (I) course covered by K. Srinivas (6).
• Power series, its radius of convergence, uniqueness of power series, differentiability of power series, analytic functions, Cauchy-Riemann conditions, complex integration,  auchy’s theorem (statement only), Cauchy’s integral formula, analytic functions expressible as power series (Taylor series), Laurent series expansion, discussion of zeros and singularities of analytic functions.
• Complex analysis (II) course covered by Gautam Bharali (6).
• The second week of lectures on Complex Analysis were devoted to the following broad themes: Applications of the Residue Theorem, with an emphasis on computing/estimating improper Riemann integrals via contour integration (which also supplements some of the material being presented in the Number Theory lectures). Geometric aspects of holomorphic functions.
• A description of the topics covered follows: How to compute residues; the winding number; a few examples of improper Riemann integrals. The meaning of the logarithm; the significance of branch points and branch cuts; the holomorphicity of (local) inverses of a holomorphic function; the principal branch of the logarithm. Integration around a branch point; transformations of improper integrals involving integration around a branch point to equivalent forms. The stereographic projection and the Riemann sphere; Möbius transformations; the action of Möbius transformations on circles. Biholomorphic maps and conformality; the automatic holomorphicity of the inverse of an injective holomorphic map; examples. The (holomorphic) automorphism group of a domain; computation of the automorphic group for the open unit disc and the complex plane; the Riemann Mapping Theorem (statement only).
• Analytic number theory (II) course covered by K. Srinivas (6).
• The main theme of the course in the second week was to give a self contained proof of prime number theorem and Dirichlet’s theorem on primes in arithmetical progression.
• The following topics were covered: Euclid’s proof of infinitude of primes, its extension to certain arithmetic progressions and failure to some other arithmetical progressions, Perron’s formula and its importance. Discussion on the zeros of Riemann zeta-function, its order in the critical strip, zero-free regions, non-vanishing of ζ(1 + it), proof of prime number theorem. Statement of Dirichlet theorem and its proof.
• Tutorial sessions.
• The tutors Kasi Viswanadham, Jaykumar and Karthick Babu proved to be indispensable. They were always there, in the lectures and in the tutorials. They prepared problem sheets for each day and ensured that each and every one does the problems given to them. The discussions went beyond the schedule. Participants showed no sign of leaving the lecture hall and going back to their room. The tutors did a splendid job.

 
Time Table

First Week : 29th Jan to 3rd Feb 2018

 Second Week : 5th Feb to 10th Feb 2018

How to reach

Swami Ramanand Teerth Marathwada University (SRTM) is located at Vishnupuri ,Nanded, Maharashtra (www.srtmun.ac.in).

• By Air:  Shri Guru Gobind Singh Ji Airport, Nanded
• Nanded is connected by air service to Hyderabad. Daily one flight(Trujet) is available from Hyderbad – Nanded - Hyderabad.
• By Train: Hazur Sahib Nanded  Railway  Station
• SRTM University  is about 10-12 kms away from Nanded railway station. This station has two sides namely `Gokul Nagar Side' and `Main Side'. Main side is just beside to Platform no.1. You ask for Main side (do not go to Gokul Nagar side). If you come out of railway station to the main side, you will get auto rickshaw stand. From there one can take an  auto for university.
• By Road:
• Nanded is well connected via road to rest of India.

Google map:
https://goo.gl/maps/gEayegWpG792