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Chebyshev’s functions

There are two different functions which are known as the Chebyshev’s functions:

The first Chebyshev function

θ(x) = Underoverscript[∑, p <= x, arg3] log p ,(1)

where the summation is over all positive primes typeset structure , and

the second Chebyshev’s function

ψ(x) = Underoverscript[∑, p, k, arg3] Underoverscript[∑, p^k <= x, arg3] log p = Underoverscript[∑, n <= x, arg3] Λ(n)(2)

where the summation runs over the all prime powers typeset structure with typeset structure (typeset structure is the von Mangoldt’s function) .

Clearly, the first function is nothing else as the logarithm of the product of all primes typeset structure, while the second one is the logarithm of the least common multiple of all positive integers typeset structure. We therefore have

ψ(x) = Underoverscript[∑, m >= 1, arg3] θ(x^(1/m)) = Underoverscript[∑, m <= (log x)/(log 2), arg3] θ(x^(1/m)) .(3)

Theorem. For typeset structure we have

0 <= ψ(x)/x - θ(x)/x <= (log^2 x)/((2 log 2) x^(1/2)) .(4)

Corollary.

Underscript[lim, x -> ∞] (ψ(x)/x - θ(x)/x) = 0 .(5)

Theorem. If typeset structure we have

θ(x) = π(x) log x - Underoverscript[∫, 2, arg3] π(t)/t d t ,(6)

π(x) = θ(x)/(log x) + Underoverscript[∫, 2, arg3] θ(t)/(t log^2 t) d t .(7)

Theorem. The following relations are equivalent

Underscript[lim, x -> ∞] (π(x) log x)/x = 1(8)
Underscript[lim, x -> ∞] θ(x)/x = 1(9)
Underscript[lim, x -> ∞] ψ(x)/x = 1(10)

Cite this web-page as:

Štefan Porubský: Chebyshev’s functions.

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