Английская Википедия:Chebyshev function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function Шаблон:Math or Шаблон:Math is given by
- <math>\vartheta(x) = \sum_{p \le x} \log p</math>
where <math>\log</math> denotes the natural logarithm, with the sum extending over all prime numbers Шаблон:Mvar that are less than or equal to Шаблон:Mvar.
The second Chebyshev function Шаблон:Math is defined similarly, with the sum extending over all prime powers not exceeding Шаблон:Mvar
- <math>\psi(x) = \sum_{k \in \mathbb{N}}\sum_{p^k \le x}\log p = \sum_{n \leq x} \Lambda(n) = \sum_{p \le x}\left\lfloor\log_p x\right\rfloor\log p,</math>
where Шаблон:Math is the von Mangoldt function. The Chebyshev functions, especially the second one Шаблон:Math, are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, Шаблон:Math (see the exact formula below.) Both Chebyshev functions are asymptotic to Шаблон:Mvar, 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:
- <math>f_{Tchb}(x,w) = \max_i w_i f_i(x).</math>[1]
By minimizing this function for different values of <math>w</math>, one obtains every point on a Pareto front, even in the nonconvex parts.[1] Often the functions to be minimized are not <math>f_i</math> but <math>|f_i-z_i^*|</math> for some scalars <math>z_i^*</math>. Then <math>f_{Tchb}(x,w) = \max_i w_i |f_i(x)-z_i^*|.</math>[2]
All three functions are named in honour of Pafnuty Chebyshev.
Relationships
The second Chebyshev function can be seen to be related to the first by writing it as
- <math>\psi(x) = \sum_{p \le x}k \log p</math>
where Шаблон:Mvar is the unique integer such that Шаблон:Math and Шаблон:Math. The values of Шаблон:Mvar are given in Шаблон:OEIS2C. A more direct relationship is given by
- <math>\psi(x) = \sum_{n=1}^\infty \vartheta\big(x^{\frac{1}{n}}\big).</math>
Note that this last sum has only a finite number of non-vanishing terms, as
- <math>\vartheta\big(x^{\frac{1}{n}}\big) = 0\quad \text{for}\quad n>\log_2 x = \frac{\log x}{\log 2}.</math>
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to Шаблон:Mvar.
- <math>\operatorname{lcm}(1,2,\dots,n) = e^{\psi(n)}.</math>
Values of Шаблон:Math for the integer variable Шаблон:Mvar are given at Шаблон:OEIS2C.
Relationships between ψ(x)/x and ϑ(x)/x
The following theorem relates the two quotients <math>\frac{\psi(x)}{x}</math> and <math>\frac{\vartheta(x)}{x}</math> .[3]
Theorem: For <math>x>0</math>, we have
- <math>0 \leq \frac{\psi(x)}{x}-\frac{\vartheta(x)}{x}\leq \frac{(\log x)^2}{2\sqrt{x}\log 2}.</math>
Note: This inequality implies that
- <math>\lim_{x\to\infty}\!\left(\frac{\psi(x)}{x}-\frac{\vartheta(x)}{x}\right)\! = 0.</math>
In other words, if one of the <math>\psi(x)/x</math> or <math>\vartheta(x)/x</math> tends to a limit then so does the other, and the two limits are equal.
Proof: Since <math>\psi(x)=\sum_{n \leq \log_2 x}\vartheta(x^{1/n})</math>, we find that
- <math>0 \leq \psi(x)-\vartheta(x)=\sum_{2\leq n \leq \log_2 x}\vartheta(x^{1/n}).</math>
But from the definition of <math>\vartheta(x)</math> we have the trivial inequality
- <math>\vartheta(x)\leq \sum_{p\leq x}\log x\leq x\log x</math>
so
- <math>\begin{align}
0\leq\psi(x)-\vartheta(x)&\leq \sum_{2\leq n\leq \log_2 x}x^{1/n}\log(x^{1/n})\\ &\leq(\log_2 x)\sqrt{x}\log\sqrt{x}\\ &=\frac{\log x}{\log 2}\frac{\sqrt{x}}{2}\log x\\ &=\frac{\sqrt{x}\,(\log x)^2}{2\log 2}. \end{align}</math>
Lastly, divide by <math>x</math> to obtain the inequality in the theorem.
Asymptotics and bounds
The following bounds are known for the Chebyshev functions:Шаблон:RefШаблон:Ref (in these formulas Шаблон:Math is the Шаблон:Mvarth prime number; Шаблон:Math, Шаблон:Math, etc.)
- <math>\begin{align}
\vartheta(p_k) &\ge k\left( \log k+\log\log k-1+\frac{\log\log k-2.050735}{\log k}\right)&& \text{for }k\ge10^{11}, \\[8px] \vartheta(p_k) &\le k\left( \log k+\log\log k-1+\frac{\log\log k-2}{\log k}\right)&& \text{for }k \ge 198, \\[8px] |\vartheta(x)-x| &\le 0.006788\,\frac{x}{\log x}&& \text{for }x \ge 10\,544\,111, \\[8px] |\psi(x)-x|&\le0.006409\,\frac{x}{\log x}&& \text{for } x \ge e^{22},\\[8px] 0.9999\sqrt{x} &< \psi(x)-\vartheta(x)<1.00007\sqrt{x}+1.78\sqrt[3]{x}&& \text{for }x\ge121. \end{align}</math>
Furthermore, under the Riemann hypothesis,
- <math>\begin{align}
|\vartheta(x)-x| &= O\Big(x^{\frac12+\varepsilon}\Big) \\ |\psi(x)-x| &= O\Big(x^{\frac12+\varepsilon}\Big) \end{align}</math>
for any Шаблон:Math.
Upper bounds exist for both Шаблон:Math and Шаблон:Math such that[4] Шаблон:Ref
- <math>\begin{align} \vartheta(x)&<1.000028x \\ \psi(x)&<1.03883x \end{align}</math>
for any Шаблон:Math.
An explanation of the constant 1.03883 is given at Шаблон:OEIS2C.
The exact formula
In 1895, Hans Carl Friedrich von Mangoldt provedШаблон:Ref an explicit expression for Шаблон:Math as a sum over the nontrivial zeros of the Riemann zeta function:
- <math>\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \tfrac{1}{2} \log (1-x^{-2}).</math>
(The numerical value of Шаблон:Math is Шаблон:Math.) Here Шаблон:Mvar runs over the nontrivial zeros of the zeta function, and Шаблон:Math is the same as Шаблон:Mvar, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:
- <math>\psi_0(x)
= \frac{1}{2}\!\left( \sum_{n \leq x} \Lambda(n)+\sum_{n < x} \Lambda(n)\right) =\begin{cases} \psi(x) - \tfrac{1}{2} \Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\dots \\ [5px] \psi(x) & \mbox{otherwise.} \end{cases}</math>
From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of Шаблон:Math over the trivial zeros of the zeta function, Шаблон:Math, i.e.
- <math>\sum_{k=1}^{\infty} \frac{x^{-2k}}{-2k} = \tfrac{1}{2} \log \left( 1 - x^{-2} \right).</math>
Similarly, the first term, Шаблон:Math, 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 Шаблон:Mvar, there are infinitely many natural numbers Шаблон:Mvar such that
- <math>\psi(x)-x < -K\sqrt{x}</math>
and infinitely many natural numbers Шаблон:Mvar such that
- <math>\psi(x)-x > K\sqrt{x}.</math>Шаблон:RefШаблон:Ref
In [[big-O notation|little-Шаблон:Mvar notation]], one may write the above as
- <math>\psi(x)-x \ne o\left(\sqrt{x}\,\right).</math>
Hardy and LittlewoodШаблон:Ref prove the stronger result, that
- <math>\psi(x)-x \ne o\left(\sqrt{x}\,\log\log\log x\right).</math>
Relation to primorials
The first Chebyshev function is the logarithm of the primorial of Шаблон:Mvar, denoted Шаблон:Math:
- <math>\vartheta(x) = \sum_{p \le x} \log p = \log \prod_{p\le x} p = \log\left(x\#\right).</math>
This proves that the primorial Шаблон:Math is asymptotically equal to Шаблон:Math, where "Шаблон:Mvar" is the little-Шаблон:Mvar notation (see [[Big O notation|big Шаблон:Mvar notation]]) and together with the prime number theorem establishes the asymptotic behavior of Шаблон:Math.
Relation to the prime-counting function
The Chebyshev function can be related to the prime-counting function as follows. Define
- <math>\Pi(x) = \sum_{n \leq x} \frac{\Lambda(n)}{\log n}.</math>
Then
- <math>\Pi(x) = \sum_{n \leq x} \Lambda(n) \int_n^x \frac{dt}{t \log^2 t} + \frac{1}{\log x} \sum_{n \leq x} \Lambda(n) = \int_2^x \frac{\psi(t)\, dt}{t \log^2 t} + \frac{\psi(x)}{\log x}.</math>
The transition from Шаблон:Math to the prime-counting function, Шаблон:Mvar, is made through the equation
- <math>\Pi(x) = \pi(x) + \tfrac{1}{2} \pi\left(\sqrt{x}\,\right) + \tfrac{1}{3} \pi\left(\sqrt[3]{x}\,\right) + \cdots</math>
Certainly Шаблон:Math, so for the sake of approximation, this last relation can be recast in the form
- <math>\pi(x) = \Pi(x) + O\left(\sqrt{x}\,\right).</math>
The Riemann hypothesis
The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part Шаблон:Sfrac. In this case, Шаблон:Math, and it can be shown that
- <math>\sum_{\rho} \frac{x^{\rho}}{\rho} = O\!\left(\sqrt{x}\, \log^2 x\right).</math>
By the above, this implies
- <math>\pi(x) = \operatorname{li}(x) + O\!\left(\sqrt{x}\, \log x\right).</math>
Smoothing function
The smoothing function is defined as
- <math>\psi_1(x) = \int_0^x \psi(t)\,dt.</math>
Obviously <math>\psi_1(x) \sim \frac{x^2}{2}.</math>
Notes
- Шаблон:Note Pierre Dusart, "Estimates of some functions over primes without R.H.". Шаблон:Arxiv
- Шаблон:Note Pierre Dusart, "Sharper bounds for Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar, Шаблон:Math", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The Шаблон:Mathth prime is greater than Шаблон:Math for Шаблон:Math", Mathematics of Computation, Vol. 68, No. 225 (1999), pp. 411–415.
- Шаблон:NoteErhard Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp. 195–204.
- Шаблон:NoteG .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.
- Шаблон:NoteDavenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. Шаблон:Isbn. Google Book Search.
References
External links
- Шаблон:Mathworld
- Шаблон:Planetmathref
- Шаблон:Planetmathref
- Riemann's Explicit Formula, with images and movies
