Chebyshev function

File:ChebyshevPsi.png

The Chebyshev function $\psi (x)$, with x < 50

File:Chebyshev.svg

The function $\psi (x)-x$, for x < 10⁴

File:Chebyshev-big.svg

The function $\psi (x)-x$, for x < 10⁷

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

$\vartheta (x)=\sum _{p\le x}\log p$

where $\log$ denotes the natural logarithm, with the sum extending over all prime numbers Template:Mvar that are less than or equal to Template:Mvar.

The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding Template:Mvar

$\psi (x)=\sum _{k\in \mathbb{\mathbb{N}}}\sum _{p^k\le x}\log p=\sum _{n\le x}\mathrm{\Lambda }(n)=\sum _{p\le x}\lfloor {\log }_{p}x\rfloor \log p,$

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 Template: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:

$f_{Tchb}(x,w)=\underset{i}{\max }{w}_i{f}_i(x).$

By minimizing this function for different values of $w$, one obtains every point on a Pareto front, even in the nonconvex parts. Often the functions to be minimized are not $f_i$ but $|f_i-z_i^{*}|$ for some scalars $z_i^{*}$. Then $f_{Tchb}(x,w)=\underset{i}{\max }{w}_i|f_i(x)-z_i^{*}|.$

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

$\psi (x)=\sum _{p\le x}k\log p$

where Template:Mvar is the unique integer such that p^k ≤ x and x < p^k + 1. The values of Template:Mvar are given in OEIS: A206722. A more direct relationship is given by

$\psi (x)=\sum _{n=1}^{\mathrm{\infty }}\vartheta \left(x^{\frac{1}{n}}\right).$

This last sum has only a finite number of non-vanishing terms, as

$\vartheta \left(x^{\frac{1}{n}}\right)=0\text{for}n>{\log }_{2}x=\frac{\log x}{\log 2}.$

The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to Template:Mvar.

$\mathrm{lcm}(1,2,\dots ,n)=e^{\psi (n)}.$

Values of lcm(1, 2, ..., n) for the integer variable Template:Mvar are given at OEIS: A003418.

Relationships between ψ(x)/x and ϑ(x)/x

The following theorem relates the two quotients $\frac{\psi (x)}{x}$ and $\frac{\vartheta (x)}{x}$ .

Theorem: For $x>0$, we have

$0\le \frac{\psi (x)}{x}-\frac{\vartheta (x)}{x}\le \frac{(\log x{)}^2}{2\sqrt{x}\log 2}.$

This inequality implies that

$\underset{x\to \mathrm{\infty }}{\lim }(\frac{\psi (x)}{x}-\frac{\vartheta (x)}{x})=0.$

In other words, if one of the $\psi (x)/x$ or $\vartheta (x)/x$ tends to a limit then so does the other, and the two limits are equal.

Proof: Since $\psi (x)=\sum _{n\le {\log }_{2}x}\vartheta (x^{1/n})$, we find that

$0\le \psi (x)-\vartheta (x)=\sum _{2\le n\le {\log }_{2}x}\vartheta (x^{1/n}).$

But from the definition of $\vartheta (x)$ we have the trivial inequality

$\vartheta (x)\le \sum _{p\le x}\log x\le x\log x$

so

$\begin{array}{rl}0\le \psi (x)-\vartheta (x)& \le \sum _{2\le n\le {\log }_{2}x}{x}^{1/n}\log (x^{1/n})\\ & \le ({\log }_{2}x)\sqrt{x}\log \sqrt{x}\\ & =\frac{\log x}{\log 2}\frac{\sqrt{x}}{2}\log x\\ & =\frac{\sqrt{x}(\log x{)}^2}{2\log 2}.\end{array}$

Lastly, divide by $x$ to obtain the inequality in the theorem.

Asymptotics and bounds

The following bounds are known for the Chebyshev functions:^[1][2] (in these formulas p_k is the Template:Mvarth prime number; p₁ = 2, p₂ = 3, etc.)

$\begin{array}{rlrl}\vartheta (p_k)& \ge k(\log k+\log \log k-1+\frac{\log \log k-2.050735}{\log k})& & \text{for }k\ge 10^{11},\\ [8px]\vartheta (p_k)& \le k(\log k+\log \log k-1+\frac{\log \log k-2}{\log k})& & \text{for }k\ge 198,\\ [8px]|\vartheta (x)-x|& \le 0.006788\frac{x}{\log x}& & \text{for }x\ge 10544111,\\ [8px]|\psi (x)-x|& \le 0.006409\frac{x}{\log x}& & \text{for }x\ge e^{22},\\ [8px]0.9999\sqrt{x}& <\psi (x)-\vartheta (x)<1.00007\sqrt{x}+1.78\sqrt[3]{x}& & \text{for }x\ge 121.\end{array}$

Furthermore, under the Riemann hypothesis,

$\begin{array}{rl}|\vartheta (x)-x|& =O\left(x^{\frac{1}{2}+\epsilon }\right)\\ |\psi (x)-x|& =O\left(x^{\frac{1}{2}+\epsilon }\right)\end{array}$

for any ε > 0.

Upper bounds exist for both ϑ  (x) and ψ (x) such that ^[3]

$\begin{array}{rl}\vartheta (x)& <1.000028x\\ \psi (x)& <1.03883x\end{array}$

for any x > 0.

An explanation of the constant 1.03883 is given at OEIS: A206431.

The exact formula

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:

${\psi }_0(x)=x-\sum _{\rho }\frac{x^{\rho }}{\rho }-\frac{{\zeta }^{\prime }(0)}{\zeta (0)}-\frac{1}{2}\log (1-x^{-2}).$

(The numerical value of Template:Sfrac is log(2π).) Here Template:Mvar runs over the nontrivial zeros of the zeta function, and ψ₀ is the same as Template:Mvar, except that at its jump discontinuities (the prime powers) it takes the value halfway between the values to the left and the right:

${\psi }_0(x)=\frac{1}{2}(\sum _{n\le x}\mathrm{\Lambda }(n)+\sum _{n<x}\mathrm{\Lambda }(n))=\{\begin{array}{ll}\psi (x)-\frac{1}{2}\mathrm{\Lambda }(x)& x=2,3,4,5,7,8,9,11,13,16,\dots \\ \psi (x)& \text{otherwise.}\end{array}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of Template:Sfrac over the trivial zeros of the zeta function, ω = −2, −4, −6, ..., i.e.

$\sum _{k=1}^{\mathrm{\infty }}\frac{x^{-2k}}{-2k}=\frac{1}{2}\log (1-x^{-2}).$

Similarly, the first term, x = Template:Sfrac, 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 Template:Mvar, there are infinitely many natural numbers Template:Mvar such that

$\psi (x)-x<-K\sqrt{x}$

and infinitely many natural numbers Template:Mvar such that

$\psi (x)-x>K\sqrt{x}.$^[5][6]

In [[big-O notation|little-Template:Mvar notation]], one may write the above as

$\psi (x)-x\ne o\left(\sqrt{x}\right).$

Hardy and Littlewood^[7] prove the stronger result, that

$\psi (x)-x\ne o(\sqrt{x}\log \log \log x).$

Relation to primorials

The first Chebyshev function is the logarithm of the primorial of Template:Mvar, denoted x #:

$\vartheta (x)=\sum _{p\le x}\log p=\log \prod _{p\le x}p=\log \left(x\mathrm{\#}\right).$

This proves that the primorial x # is asymptotically equal to e^{(1  + o(1))x}, where "Template:Mvar" is the little-Template:Mvar notation (see [[Big O notation|big Template:Mvar notation]]) and together with the prime number theorem establishes the asymptotic behavior of p_n #.

Relation to the prime-counting function

The Chebyshev function can be related to the prime-counting function as follows. Define

$\mathrm{\Pi }(x)=\sum _{n\le x}\frac{\mathrm{\Lambda }(n)}{\log n}.$

Then

$\mathrm{\Pi }(x)=\sum _{n\le x}\mathrm{\Lambda }(n)\underset{n}{\overset{x}{\int }}\frac{dt}{t{\log }^{2}t}+\frac{1}{\log x}\sum _{n\le x}\mathrm{\Lambda }(n)=\underset{2}{\overset{x}{\int }}\frac{\psi (t)dt}{t{\log }^{2}t}+\frac{\psi (x)}{\log x}.$

The transition from Π to the prime-counting function, Template:Mvar, is made through the equation

$\mathrm{\Pi }(x)=\pi (x)+\frac{1}{2}\pi \left(\sqrt{x}\right)+\frac{1}{3}\pi \left(\sqrt[3]{x}\right)+\cdots$

Certainly π (x) ≤ x, so for the sake of approximation, this last relation can be recast in the form

$\pi (x)=\mathrm{\Pi }(x)+O\left(\sqrt{x}\right).$

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part Template:Sfrac. In this case, Template:Abs =

1. REDIRECT Template:Radic

Template:Rcat shell, and it can be shown that

$\sum _{\rho }\frac{x^{\rho }}{\rho }=O\left(\sqrt{x}{\log }^{2}x\right).$

By the above, this implies

$\pi (x)=\mathrm{li}(x)+O(\sqrt{x}\log x).$

Smoothing function

File:Chebyshev-smooth.svg

The difference of the smoothed Chebyshev function and Template:Sfrac for x < 10⁶

The smoothing function is defined as

${\psi }_1(x)=\underset{0}{\overset{x}{\int }}\psi (t)dt.$

Obviously ${\psi }_1(x)\sim \frac{x^2}{2}.$

Notes

• ^ Pierre Dusart, "Estimates of some functions over primes without R.H.". arXiv:1002.0442File:Lock-green.svg
• ^ Pierre Dusart, "Sharper bounds for Template:Mvar, Template:Mvar, Template:Mvar, p_k", 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.
• ^ Davenport, Harold (2000). In Multiplicative Number Theory. Springer. p. 104. Template:Isbn. Google Book Search.

References

• Template:Apostol IANT

External links

• Riemann's Explicit Formula, with images and movies