DR.P.JANAKI

 1 Divisibility Number Theory concerns itself mostly with the study of the natural numbers (N) and the integers (Z). As a consequence, it deals a lot with prime numbers and sometimes with rational numbers (Q). Recall: Definition. The natural numbers are the numbers N = {1, 2, 3, . . . }. The integers are the numbers Z = {. . . , −2, −1, 0, 1, 2, . . . }. The rational numbers are Q = { a b | a, b ∈ Z, b 6= 0}. There is significant debate about whether the naturals include 0 or not. It’s probably easier to consider the naturals to be just the positive integers. If you want to specify the non-negative integers you may write N0 or Z≥0. Notation. If an integer a divides an integer b we write a|b. Definition. A prime number is a positive integer p 6= 1 such that if p divides ab then p divides a or p divides b. Mathematically, we write this as p|ab =⇒ p|a or p|b Remark. Note, when you get to university and learn about more advanced number theory, negatives of primes will also be included ...

  Analytic number theory [ edit ] Main article:  Analytic number theory Riemann zeta function  ζ( s ) in the  complex plane . The color of a point  s  gives the value of ζ( s ): dark colors denote values close to zero and hue gives the value's  argument . The action of the  modular group  on the  upper half plane . The region in grey is the standard  fundamental domain . Analytic number theory  may be defined in terms of its tools, as the study of the integers by means of tools from  real  and  complex  analysis; [70]  or in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities. [79] Some subjects generally considered to be part of analytic number theory, for example,  sieve theory , [note 10]  are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, [note 11] ...