This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x) or loge(x).
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev functionϑ (x) or θ (x) is given by
The second Chebyshev functionψ (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:
[1]
By minimizing this function for different values of , one obtains every point on a Pareto front, even in the nonconvex parts.[1] Often the functions to be minimized are not but for some scalars . Then [2]
(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.
Properties
A theorem due to Erhard Schmidt states that, for some explicit positive constant K, there are infinitely many natural numbersx such 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 #.
Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form
The Riemann hypothesis
The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part1/2. In this case, |xρ| = √x, and it can be shown that
By the above, this implies
Smoothing function
The smoothing function is defined as
Obviously
Notes
^ a bJoshua Knowles (2 May 2014). "Multiobjective Optimization Concepts, Algorithms and Performance Measures" (PDF). The University of Manchester. p. 34.
^Ho-Huu, V.; Hartjes, S.; Visser, H. G.; Curran, R. (2018). "An improved MOEA/D algorithm for bi-objective optimization problems with complex Pareto fronts and its application to structural optimization" (PDF). Expert Systems with Applications. Delft University of Technology. Page 6 equation (2). doi:10.1016/j.eswa.2017.09.051.
^Apostol, Tom M. (2010). Introduction to Analytic Number Theory. Springer. pp. 75–76.
^Pierre Dusart, "Estimates of some functions over primes without R.H.". arXiv:1002.0442
^ Pierre Dusart, "Sharper bounds for ψ, θ, π, pk", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(log k + log log k − 1) for k ≥ 2", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.
^ Erhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.
^ G .H. Hardy and J. E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41 (1916) pp. 119–196.
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001