Chebyshev function
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.
, one obtains every point on a Pareto front, even in the nonconvex parts.
[2] 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.
A more direct relationship is given by 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 OEIS: A003418.
The following theorem relates the two quotients
, we have This inequality implies that In other words, if one of the
we have the trivial inequality so Lastly, divide by
The following bounds are known for the Chebyshev functions:[1][2] (in these formulas pk is the kth prime number; p1 = 2, p2 = 3, etc.)
Furthermore, under the Riemann hypothesis, for any ε > 0.
Upper bounds exist for both ϑ (x) and ψ (x) such that[4] [3] for any x > 0.
An explanation of the constant 1.03883 is given at OEIS: A206431.
In 1895, Hans Carl Friedrich von Mangoldt proved[4] an explicit expression for ψ (x) as a sum over the nontrivial zeros of the Riemann zeta function: (The numerical value of ζ′ (0)/ζ (0) is log(2π).)
Here ρ runs over the nontrivial zeros of the zeta function, and ψ0 is the same as ψ, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right: From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of xω/ω over the trivial zeros of the zeta function, ω = −2, −4, −6, ..., i.e.
Similarly, the first term, x = x1/1, corresponds to the simple pole of the zeta function at 1.
It being a pole rather than a zero accounts for the opposite sign of the term.
A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many natural numbers x such that and infinitely many natural numbers x such that In little-o notation, one may write the above as Hardy and Littlewood[7] prove the stronger result, that The first Chebyshev function is the logarithm of the primorial of x, denoted x #: This proves that the primorial x # is asymptotically equal to e(1 + o(1))x, where "o" is the little-o notation (see big O notation) and together with the prime number theorem establishes the asymptotic behavior of pn #.
Define Then The transition from Π to the prime-counting function, π, is made through the equation Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2.
In this case, |x ρ| = √x, and it can be shown that By the above, this implies The smoothing function is defined as Obviously
