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Mathematics:Arithmetical Functions

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

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Mathematics:Arithmetical Functions

  1. 1. ARITHMETICAL FUNCTIONS Sowmya K Assistant Professor Department of Mathematics St.Mary’s College,Thrissur.
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 11. Tom M Apostol, Introduction to Analytic Number Theory, Narosa Publishing House,1990. References Arithmetical Functions, Sowmya K, St.Mary’s College, Thrissur.

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