Chebyshev function

In mathematics, 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

${\displaystyle \vartheta (x)=\sum _{p\leq x}\log p}$

where ${\displaystyle \log }$ denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x.

The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x

${\displaystyle \psi (x)=\sum _{k\in \mathbb {N} }\sum _{p^{k}\leq x}\log p=\sum _{n\leq x}\Lambda (n)=\sum _{p\leq x}\left\lfloor \log _{p}x\right\rfloor \log p,}$

where Λ is the von Mangoldt function. The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.

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:

${\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}f_{i}(x).}$^[1]

By minimizing this function for different values of ${\displaystyle w}$, one obtains every point on a Pareto front, even in the nonconvex parts.^[1] Often the functions to be minimized are not ${\displaystyle f_{i}}$ but ${\displaystyle |f_{i}-z_{i}^{*}|}$ for some scalars ${\displaystyle z_{i}^{*}}$. Then ${\displaystyle f_{Tchb}(x,w)=\max _{i}w_{i}|f_{i}(x)-z_{i}^{*}|.}$^[2]

All three functions are named in honour of Pafnuty Chebyshev.