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Concept
Chebyshev function
Summary
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
where 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
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:
By minimizing this function for different values of , one obtains every point on a Pareto front, even in the nonconvex parts. Often the functions to be minimized are not but for some scalars . Then
All three functions are named in honour of Pafnuty Chebyshev.
The second Chebyshev function can be seen to be related to the first by writing it as
where k is the unique integer such that p k ≤ x and x < p k + 1. The values of k are given in . A more direct relationship is given by
Note that this last sum has only a finite number of non-vanishing terms, as
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.
Values of lcm(1, 2, ..., n) for the integer variable n are given at .
The following theorem relates the two quotients and .
Theorem: For , we have
Note: This inequality implies that
In other words, if one of the or tends to a limit then so does the other, and the two limits are equal.
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