Chebyshev function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ  (x) or θ (x) is given by

ϑ
        (
        x
        )
        =
        
          ∑
          
            p
            ≤
            x
          
        
        log
        ⁡
        p
      
    
    {\displaystyle \vartheta (x)=\sum _{p\leq x}\log p}

where 
 
 
 
 log
 
 
 {\displaystyle \log }
 
 denotes the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>, with the sum extending over all <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> p that are less than or equal to x.
The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all <a href="/facts/Prime_power/R0vo3uYp">prime powers</a> not exceeding x

ψ
        (
        x
        )
        =
        
          ∑
          
            k
            ∈
            
              N
            
          
        
        
          ∑
          
            
              p
              
                k
              
            
            ≤
            x
          
        
        log
        ⁡
        p
        =
        
          ∑
          
            n
            ≤
            x
          
        
        Λ
        (
        n
        )
        =
        
          ∑
          
            p
            ≤
            x
          
        
        
          ⌊
          
            
              log
              
                p
              
            
            ⁡
            x
          
          ⌋
        
        log
        ⁡
        p
        ,
      
    
    {\displaystyle \psi (x)=\sum _{k\in \mathbb {N} }\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,}

where Λ is the <a href="/facts/Von_Mangoldt_function/wwHKs2bg">von Mangoldt function</a>. The Chebyshev functions, especially the second one ψ (x), are often used in <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a> related to <a href="/facts/Prime_number/L20FLkgn">prime numbers</a>, because it is typically simpler to work with them than with the <a href="/facts/Prime-counting_function/Dk1xJzsA">prime-counting function</a>, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the <a href="/facts/Prime_number_theorem/3KA9egls">prime number theorem</a>.
Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function:

f
 
 T
 c
 h
 b
 
 
 (
 x
 ,
 w
 )
 =
 
 max
 
 i
 
 
 
 w
 
 i
 
 
 
 f
 
 i
 
 
 (
 x
 )
 .
 
 
 {\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}f_{i}(x).}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
By minimizing this function for different values of 
 
 
 
 w
 
 
 {\displaystyle w}
 
, one obtains every point on a <a href="/facts/Pareto_front/nMTdVPqy">Pareto front</a>, even in the nonconvex parts. Often the functions to be minimized are not 
 
 
 
 
 f
 
 i
 
 
 
 
 {\displaystyle f_{i}}
 
 but 
 
 
 
 
 |
 
 
 f
 
 i
 
 
 −
 
 z
 
 i
 
 
 ∗
 
 
 
 |
 
 
 
 {\displaystyle |f_{i}-z_{i}^{*}|}
 
 for some scalars 
 
 
 
 
 z
 
 i
 
 
 ∗
 
 
 
 
 {\displaystyle z_{i}^{*}}
 
. Then 
 
 
 
 
 f
 
 T
 c
 h
 b
 
 
 (
 x
 ,
 w
 )
 =
 
 max
 
 i
 
 
 
 w
 
 i
 
 
 
 |
 
 
 f
 
 i
 
 
 (
 x
 )
 −
 
 z
 
 i
 
 
 ∗
 
 
 
 |
 
 .
 
 
 {\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.}

All three functions are named in honour of <a href="/facts/Pafnuty_Chebyshev/LzRqLqRm">Pafnuty Chebyshev</a>.

Chebyshev function open-in-new

Chebyshev function