Average order of an arithmetic function

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".
Let 
 
 
 
 f
 
 
 {\displaystyle f}
 
 be an <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a>. We say that an average order of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is 
 
 
 
 g
 
 
 {\displaystyle g}
 
 if

∑
          
            n
            ≤
            x
          
        
        f
        (
        n
        )
        ∼
        
          ∑
          
            n
            ≤
            x
          
        
        g
        (
        n
        )
      
    
    {\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)}

as 
 
 
 
 x
 
 
 {\displaystyle x}
 
 tends to infinity.
It is conventional to choose an approximating function 
 
 
 
 g
 
 
 {\displaystyle g}
 
 that is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> and <a href="/facts/Monotonic_function/MjQK0YL2">monotone</a>. But even so an average order is of course not unique.
In cases where the limit

lim
          
            N
            →
            ∞
          
        
        
          
            1
            N
          
        
        
          ∑
          
            n
            ≤
            N
          
        
        f
        (
        n
        )
        =
        c
      
    
    {\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\sum _{n\leq N}f(n)=c}

exists, it is said that 
 
 
 
 f
 
 
 {\displaystyle f}
 
 has a mean value (average value) 
 
 
 
 c
 
 
 {\displaystyle c}
 
. If in addition the constant 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is not zero, then the constant function 
 
 
 
 g
 (
 x
 )
 =
 c
 
 
 {\displaystyle g(x)=c}
 
 is an average order of 
 
 
 
 f
 
 
 {\displaystyle f}
 
.

Average order of an arithmetic function open-in-new

Average order of an arithmetic function