title: Turán–Kubilius inequality
text: # Turán–Kubilius inequality
  
  The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic $function.^{:305–308}$ The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas $Kubilius.^{:316}$
  
  ## Statement of the theorem
  
  This formulation is from $Tenenbaum.^{:302}$ Other formulations are in $Narkiewicz^{:243}$ and in Cojocaru & $Murty.^{:45–46}$
  
  Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and ν for an arbitrary positive integer. Write
  
      $A(x)=\sum _{p^{\nu }\leq x}f(p^{\nu })p^{-\nu }(1-p^{-1})$
  
  and
  
      $B(x)^{2}=\sum _{p^{\nu }\leq x}\left|f(p^{\nu })\right|^{2}p^{-\nu }.$
  
  Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have
  
      ${\frac {1}{x}}\sum _{n\leq x}|f(n)-A(x)|^{2}\leq (2+\varepsilon (x))B(x)^{2}.$
  
  ## Applications of the theorem
  
  Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order of the number ω(n) of distinct prime divisors of an integer $n.^{:316}$ There is an exposition of Turán's proof in Hardy & Wright, §22.11. $Tenenbaum^{:305–308}$ gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.
  
  