2. An arithmetical function, also called number theoretic function
is a real or complex valued function defined on the set of
positive integers.
Examples
ο§ Euler totient function
ο§ M πbiusfunction
ο§ Mangoldt function
ο§ Liouvilleβs function
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
Definition
3. ο§ It is the number of positive integers not exceeding π and
relatively prime to π.
ο§ π 1 = 1,π 2 = 1, π 3 = 2, π 4 = 2, π 5 = 4, π 6 = 2.
ο§ π π = π β 1, for a prime π.
ο§ π/π π π = π.
ο§ π π = π π/π(1 β
1
π
)(Product formula for π π )
ο§ π ππ = π π π π
π
π π
where π = gcd(π, π).
Euler totient function,π(π)
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
7. ο§ An arithmetical function which is not identically zero is
multiplicative if π(ππ) = π(π)π(π) for all π, π with
gcd π, π = 1.
ο§ A multiplicative function is completely multiplicative if
π(ππ) = π(π)π(π) for all π, π
ο§ If π is multiplicative ,then π 1 = 1
ο§ Given f with π 1 = 1.Then
a) π is multiplicative if and only if π(π1
π1 β¦ π π
π π)=
π( π1
π1)β¦π( π π
π π) for primes ππ and all integers ππ β₯ 1.
b) If π is multiplicative ,then π is completely multiplicative
if and only if π(π π) = π(π) π for all primes π and
integers π β₯ 1.
Multiplicative functions
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
8. Euler totient function is multiplicative
since
But it is not completely multiplicative
since π 4 β π 2 π 2 .
π(ππ) = π(π)π(π) if gcd (π, π) = 1.
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
9. M πbius function is multiplicative, since
π ππ = π π π(π) if gcd π, π = 1.
But it is not completely multiplicative
since π 4 β π 2 π 2 .
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
10. Mangoldt function is not multiplicative
since Ξ 1 β 1.
Liouvilleβs function is completely
multiplicative.
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.
11. οΆTom M Apostol, Introduction to Analytic Number Theory,
Narosa Publishing House,1990.
References
Arithmetical Functions, Sowmya K, St.Maryβs College, Thrissur.