Chebyshev function

In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by

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.

Both 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 k of 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

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