Subject Code: MA7L029 Subject Name:  ANALYTIC NUMBER THEORY L-T-P: 3-1-0 Credit:4
Pre-requisite(s): Complex Analysis or equivalent
Review of Arithmetical functions: Mobius functions, Euler’s φ-function, Divisor function, the von-Mangoldt function, Liouville’s function, Dirichlet multiplication and Mobius inversion formula, Multiplicative functions, Dirichlet series and Euler product. Averages of arithmetical functions: Summation formulas, Average order of divisor function, Euler φ-function, Mobius function and von-Mangoldt function, Dirichlet hyperbola method.
Distribution of primes without complex analysis: Euclid’s method, Prime counting function π(x), Chebyshev’s psi and theta functions, Partial sums of reciprocal series of primes, Merten’s formula, zeta function and Euler’s proof of infinitely many primes.
Distribution of primes using complex analysis: Characters of finite abelian groups, Dirichlet characters, primitive characters, Gauss sum, Riemann zeta function and Dirichlet L-functions as functions of complex variable.
Functional equations and analytic continuations: Schwarz function on R, Z, R/Z and their Fourier transform, the Poisson summation formula, functional equations for zeta function and L-functions, Bernoulli numbers and its relation to Riemann zeta function. Hadamard factorization theorem (statement), product formulas for completed zeta function and L-functions, Classical zero-free region and Riemann hypothesis, Number of zeros of zeta function and L-functions, Explicit formulas, the Prime Number Theorem (PNT) and Dirichlet’s Theorem on primes in arithmetic progression (PNT in AP). Introduction to Public Key Crypography. If time permits: advance topics like Sieve Methods, the circle method, bounds of exponential sums and Modular forms.
Text/Reference Books:
  1. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
  2. H. Davenport, Multiplicative Number Theory, Springer, 2003.
  3. J-P. Serre, A course in arithmetic, Springer-Verlag, 1973.
  4. E.C. Titchmarsh, The Theory of the Riemann zeta-function, Clarendon press, 1986.
  5. M. Ram Murty, Problems in Analytic Number Theory, GTM 206, Springer, 2008.
  6. H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical Theory, Cambridge University Press, 2006.
  7. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.
  8. A. E. Ingham, The Distribution of Prime Numbers, Cambridge University Press, 1990.
  9. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM 84, Springer-Verlag, 1990.
  10. K. Ramachandra, Theory of Numbers: A Textbook, Narosa Publishing House, 2007.