Arithmetical Functions-Definition,Euler totient function,Mobius function,Mangoldt function,Liouville function,Nultiplicative function

1. ARITHMETICAL FUNCTIONS
Sowmya K
Assistant Professor
Department of Mathematics
St.Mary’s College,Thrissur.

2. An arithmetical function, also called number theoretic function
is a real or complex valued function defined on the set of
positive integers.
Examples
 Euler totient function
 M 𝑜biusfunction
 Mangoldt function
 Liouville’s function
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.
Definition

3.  It is the number of positive integers not exceeding 𝑛 and
relatively prime to 𝑛.
 𝜑 1 = 1,𝜑 2 = 1, 𝜑 3 = 2, 𝜑 4 = 2, 𝜑 5 = 4, 𝜑 6 = 2.
 𝜑 𝑝 = 𝑝 − 1, for a prime 𝑝.
 𝑑/𝑛 𝜑 𝑑 = 𝑛.
 𝜑 𝑛 = 𝑛 𝑝/𝑛(1 −
1
𝑝
)(Product formula for 𝜑 𝑛 )
 𝜑 𝑚𝑛 = 𝜑 𝑚 𝜑 𝑛
𝑑
𝜑 𝑑
where 𝑑 = gcd(𝑚, 𝑛).
Euler totient function,𝜑(𝑛)
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

4.  𝜇 1 = 1
 𝜇 𝑛 = (−1) 𝑘
, 𝑖𝑓 𝑛 = 𝑝1
𝑎1 … 𝑝 𝑘
𝑎 𝑘 and 𝑎1 = 𝑎2 = … 𝑎 𝑘= 1
 𝜇 𝑛 = 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 𝜇 𝑛 = 0, if 𝑛has a square factor.
 𝑑/𝑛 𝜇 𝑑 =
1
𝑛
 𝑑/𝑛 𝜇 𝑑
𝑛
𝑑
= 𝜑(𝑛)
M 𝑜biusfunction,𝜇 𝑛
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

5.  Λ 1 = 0
 Λ 𝑛 = log p, if 𝑛 = 𝑝 𝑚
for some prime 𝑝 and 𝑚 ≥ 1.
 Λ 2 = log 2, Λ 3 = log 3 , Λ 4 = log 2 , Λ 5 = log 5 , Λ 6 = 0.
 𝑑/𝑛 Λ 𝑑 = log 𝑛
 Λ 𝑛 = 𝑑/𝑛 𝜇 𝑑 𝑙𝑜𝑔
𝑛
𝑑
= − 𝑑/𝑛 𝜇 𝑑 log 𝑑
Mangoldtfunction,Λ 𝑛
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

6.  𝜆(1) = 1
 𝜆(𝑛)=(−1) 𝑎1+⋯+𝑎 𝑘 if 𝑛 = 𝑝1
𝑎1… 𝑝 𝑘
𝑎 𝑘
 𝑑/𝑛 𝜆(𝑑) =
1, 𝑖𝑓 𝑛 𝑖𝑠 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Liouville’s function,𝜆(𝑛)
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

7.  An arithmetical function which is not identically zero is
multiplicative if 𝑓(𝑚𝑛) = 𝑓(𝑚)𝑓(𝑛) for all 𝑚, 𝑛 with
gcd 𝑚, 𝑛 = 1.
 A multiplicative function is completely multiplicative if
𝑓(𝑚𝑛) = 𝑓(𝑚)𝑓(𝑛) for all 𝑚, 𝑛
 If 𝑓 is multiplicative ,then 𝑓 1 = 1
 Given f with 𝑓 1 = 1.Then
a) 𝑓 is multiplicative if and only if 𝑓(𝑝1
𝑎1 … 𝑝 𝑟
𝑎 𝑟)=
𝑓( 𝑝1
𝑎1)…𝑓( 𝑝 𝑟
𝑎 𝑟) for primes 𝑝𝑖 and all integers 𝑎𝑖 ≥ 1.
b) If 𝑓 is multiplicative ,then 𝑓 is completely multiplicative
if and only if 𝑓(𝑝 𝑎) = 𝑓(𝑝) 𝑎 for all primes 𝑝 and
integers 𝑎 ≥ 1.
Multiplicative functions
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

8. Euler totient function is multiplicative
since
But it is not completely multiplicative
since 𝜑 4 ≠ 𝜑 2 𝜑 2 .
𝜑(𝑚𝑛) = 𝜑(𝑚)𝜑(𝑛) if gcd (𝑚, 𝑛) = 1.
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

9. M 𝑜bius function is multiplicative, since
𝜇 𝑚𝑛 = 𝜇 𝑚 𝜇(𝑛) if gcd 𝑚, 𝑛 = 1.
But it is not completely multiplicative
since 𝜇 4 ≠ 𝜇 2 𝜇 2 .
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

10. Mangoldt function is not multiplicative
since Λ 1 ≠ 1.
Liouville’s function is completely
multiplicative.
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

11. Tom M Apostol, Introduction to Analytic Number Theory,
Narosa Publishing House,1990.
References
Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.