IST Number Theory (2015)

 Topics covered by Dr. Thangadurai, HRI

1. b-ary Expansion
   Existence of b-ary expansion of real numbers, non-uniqueness of the representation for rationals, classification of rationals, Motivation to Gauss conjecture about the periods and generalization of Gauss Conjecture by E. Artin.
2. Continued fractions.
   finite, simple continued fractions, properties of the kth convergent, existence of infinite continued fractions for irrational numbers, classification of quadratic irrationals, computation of infinite continued fraction expression for e using Pade approximation method.
3. Well-ordering Principle, weak and strong Induction equivalence.

Topics covered by Prof. B Ramakrishnan, HRI

• Lecture 1: Arithmetical functions; several examples, multiplicative, additive functions, Mobius identity, Dirichlet convolution of arithmetic functions, some properties of $\mu(n)$, $\varphi(n)$, $\Lambda(n)$.
• Lecture 2: Properties of Dirichlet convolution and its applications, viz., proving certain identities, evaluating the convolution and proving multiplicative property; Asymptotic estimates (Arithmetic means; Summatory functions); certain applications of these estimates; big O and little o notation; the logarithmic integral.
• Lecture 3: Euler summation formula and applications; partial sums of $\log n$ and Stirling's formula for $n!$ as an application; integral representation of the Riemann zeta function; Abel summation formula.
• Lecture 4: Relation between asymptotic mean and logarithmic mean; Dirichlet series and summatory functions (Mellin transform representation of a Dirichlet series); finding average orders of certain arithmetical functions using convolution method, especially $\varphi(n)$, $\mu^2(n)$, $d(n)$.duction equivalence.

Topics covered by Prof. S. D. Adhikari, , HRI

• Lecture 1: Congruences modulo an integer, Some results on finite fields, basic congruences modulo a prime.
• Lecture 2: Lagrange theorem for polynomials over Z/pZ, quadratic congruences, there are infinitely many primes of the form 4n+1, 4n-1 - statement of Dirichlet's theorem on primes over an A.P. , solution of some diophantine equations.
• Lecture 3: Chinese Remainder Theorem, some related problems.
• Lecture 4: Quadratic reciprocity law - an elementary proof, related problems.

Topics covered by Prof. S. A. Katre, Savitribai Phule Pune University

• Lecture 1: No. of primesis infinite. No. of primes of the form 4n- 1 is infinite.Statement of Dirichlet's Theorem on primes in A.P.. Large primes and their application in RSA Cryptography. Information about polynomial time algorithm for primality testing by Manindra Agrawal, Statement of prime number theorem and Bertrand's postulate, Introduction to Chebyshev's Lambda, psi and theta functions.
• Lecture 2: Relation between psi and theta functions. Application of Abel's identity to get relations between theta function and \pi(x) function. Equivalence of the asymptotic results for \pi(x), \psi(x) and \theta(x).
• Lecture 3: Proof for the upper bound for theta function and application this upper bound to the Chebyshev bounds: n/6 log n < \pi(n) < 6n/log n.
• Lecture 4: Proof for the upper and lower bounds for psi function and their application to the proof of Bertrand's postulate. Discussion about the relation of PNT with the nonvanishing of the Riemann zeta function on x=1. (Some part also covered in the tutorial time.)Application of Mobius Inversion Formula for getting a formula for the n-th cyclotomic polynomial was discussed in tutorial time.

References:

1. Introduction to Analytic Number Theory by Tom M. Apostol, UTM, Springer, 1976 (Narosa, Indian Edn.) (Chapter 4)
2. Introduction to the Theory of Numbers, I. Niven, H. S. Zuckerman and L. Montgomery, John Wiley & Sons, 1991. (Chapter 8, Section 8.1)
3. An Introduction to the Theory of Numbers, G. H. Hardy and E. M. Wright, sixth edition, Oxford University Press, 2008.

 Topics covered by Prof. M Manickam, KSOM

Existence of finite Fourier series for periodic arithmetic function. The construction of such function like Ramanujan function, the function $s_k(n)$.Gauss sum associated with quadratic character and derive the reciprocity law for quadratic symbol. Quadratic Gauss sum and the reciprocity of
the quadratic Gauss sum using Residue theorem.Primitive roots and their existence.

Topics covered by A. Mukhopadyaya, IMSc, Chennai

Dirichlet Character and Dirichlet Prime Number Theorem

Registration: 9:00 AM - 9.30 AM on 05-10-2015

Time - Table

Speakers

1. A. Mukhopadyaya, IMSc, Chennai - AM
2. S. A. Katre, University of Poona, Pune - SA
3. S. D. Adhikari, HRI, Allahabad - SDA
4. M. Manickam, KSOM, Calicut - MM
5. R. Thangadurai, HRI, Allahabad - RT
6. B. Ramakrishnan, HRI, Allahabad - BR

Tutors

1. Mr. Aravind Kumar, HRI, Allahabad - AK
2. Mr. Balesh Kumar, HRI, Allahabad - BK

 How to reach

Kerala School of Mathematics (KSOM) is located at a distance of about 15 Kms from the city centre (train/bus stations) situated off the Kozhikode-Medical College-Kunnamangalam bus route.Prepaid auto counter is available at Railway Station itself
   (Near 1st number Platform).