devotion & mathematics

                           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 ...

Formal power series: In calculus an infinite series of the form 𝜮 a(n)  = a(0) + a(1) x + a(2) x² +...+ a(n) x +... is called a power series in x. Here both x and a(n) are real or complex numbers. To each power series there corresponds a radius of convergence r > 0 such that the series converges absolutely if | x |<r and diverges if | x | > r. Note:          Here the radius r can be +∞ Here in this, we consider power series from a different point of view. We call these power series as FORMAL power series to distinguish them from the ordinary power series of calculus. In the formal power series, x is never assigned a numerical value. In power series 𝜮 c(n) x", the symbol x" is simply a device for locating the position of the nth coefficient a(n). The coefficient a(0) is called the constant coefficient of the series. Let A(x)=a(n) x"; B(x) = b(n) x. Then 1. A(x)+B(x) iff a(n) = b(n) for all n > 0 2. A(x)+B(x)=(a(x)+b(x)) x". 3. A(x) B(x) = c(n) ...

                Liouville’s function λ (n) :                    We define λ(n) =   1                             if n = 1                                          = ( -1 ) a1+a2+…+ak        if n = p 1 a1 . p 2 a2 . … . p k ak Note :            Liouville's function Λ (n) is an important example of a completely multiplicative            function. Theorem :                  For every n ≥ 1 , we have 𝜮  λ(d)  =   1    if n is a s...

  Multiplicative functions   :              Definition :   ( Multiplicative function )                                                                An arithmetical function f is called multiplicative if f is not identically zero and if f (mn) = f(m)f(n) whenever (m,n) = 1.         Definition :   ( completely multiplicative function )                                       ...