devotion & mathematics

                                                                                                               Congruences :                  The property of congruence provides a way of classifying integers according to the remainder      obtained  upon division by a fixed positive integer. In fact the remainder is the only thing of  interest.  In this section we study a relation on the integers that is defined in terms of remainders. Definition :                    Let m be a fixed positive integer and a,b 𝛜 Z. 'a'  is said to be " congruent to          ...

                           E. T. Bell used formal power series to study properties of multiplicative arithmetical functions . Definition : [ Bell Series ]    Given an arithmetical function f and a prime p , we denote by f p (x) the formal power series         f p (x) = 𝜮 f(p n ) x n and call this the Bell series of f modulo p. Note :           Bell series are especially useful when f is multiplicative. Examples : ·          The Bell series for the Mobius function 𝛍 is given by 𝛍 p (x) = 1-x. ·          The Bell series for the Euler ‘s totient function ⲫ is given by                 ⲫ p (x)  =   (1-x) / (1-px) ·          The Bell series ...