Spring 2014Number Theory-106397

Lecturer: Professor Ross PinskyTime: Sunday 12:30-14:30 and Wednesday 10:30-11:30Prerequisites: A basic knowledge of complex function theory

Course OutlineThis course will concentrate on multiplicative number theory and the the-

ory of prime numbers. The first part of the course will be very much discretemathematics, while the second part will be of a very analytic nature.1. (Review of) Some basics: greatest common divisor, congruences, ChineseRemainder Theorem.2. Arithmetic functions: multiplicative arithmetic functions; some impor-tant arithmetic functions in number theory: d(n)–the number of divisors ofn, ω(n)–the number of distinct prime divisors of n, µ(n)–the Mobius func-tion, φ(n)–the Euler totient function; convolution of arithmetic functions,Mobius inversion3. Limiting density for square free numbers and for pairs of relatively primenumbers4. Chebyshev’s theorem on the asymptotic distribution of the primes5. Mertens’ theorems on the asymptotic behavior of the primes6. The Hardy-Ramanujan theorem on the number of distinct prime divisors7. Dirichlet series and Euler products8. Summation techniques—Abel summation,9. Riemann’s zeta function and its basic properties for Re(s) > 1; thefunction ζ′(s)

ζ(s) and the von Mangoldt function Λ(n); the extension of thezeta function to Re(s) ≤ 1; the absence of zeros on the line Re(s) = 1 andin a small strip to the left of it; growth estimates for the zeta function10. Perron inversion of Dirichlet series11. The prime number theorem: π(n) ∼ n

logn , where π(n) is the number ofprimes no greater than n12. Error estimates in the prime number theorem and its connection withthe zero-free region of the zeta-function and the Riemann hyothesis13. Generalization of the prime number theorem: Πk(n) ∼ πk(n) ∼ n

logn(log logn)k−1

(k−1)! ,where Πk(n) (respectively, πk(n)) is the number of integers no greater thann which are the product of exactly k not necessarily distinct primes (respec-tively, k distinct primes)14. Primes in arithmetic progressions—Dirichlet characters, Dirichlet L-functions, Dirichlet’s theorem

Recommended Books1. T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,1976.2. G.J.O. Jameson, The Prime Number Theorem, Cambridge UniversityPress, 2003.3. M. Nathanson, Elementary Methods in Number Theory, Springer-Verlag,2000.

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