Chebyshev function

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

:$\vartheta(x)=\sum_ \ln p$

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

The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding 

:$\psi(x) = \sum_\sum_\ln p=\sum_ \Lambda(n) = \sum_\left\lfloor\log_p x\right\rfloor\ln p,$

where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (See the exact formula, below.) Both Chebyshev functions are asymptotic to , 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:

:$f_(x,w) = \max_i w_i f_i(x).$

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 $f_(x,w) = \max_i w_i , f_i(x)-z_i^*, .$

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

:$\psi(x)=\sum_k \log p$

where is the unique integer such that and . The values of are given in . A more direct relationship is given by

:$\psi(x)=\sum_^\infty \vartheta \left(x^\frac\right).$

Note that this last sum has only a finite number of non-vanishing terms, as

:$\vartheta \left(x^\frac\right) = 0\quad \text\quad n>\log_2 x\ = \frac.$

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

:$\operatorname(1,2,\dots, n)=e^.$

Values of for the integer variable is given at .

Asymptotics and bounds

The following bounds are known for the Chebyshev functions: (in these formulas is the th prime number , , etc.)

:\begin
\vartheta(p_k)&\ge k\left( \log k+\log\log k-1+\frac\right)&& \textk\ge10^, \\ px\vartheta(p_k)&\le k\left( \log k+\log\log k-1+\frac\right)&& \textk \ge 198, \\ px, \vartheta(x)-x, &\le0.006788\frac&& \textx \ge 10\,544\,111, \\ px, \psi(x)-x, &\le0.006409\frac&& \text x \ge e^,\\ px0.9999\sqrt x&<\psi(x)-\vartheta(x)<1.00007\sqrt x+1.78\sqrt && \textx\ge121.
\end

Furthermore, under the Riemann hypothesis,

:$\begin , \vartheta(x)-x, &=O\left(x^\right) \\ , \psi(x)-x, &=O\left(x^\right) \end$

for any .

Upper bounds exist for both and such that,

:$\begin \vartheta(x)&<1.000028x \\ \psi(x)&<1.03883x \end$

for any .

An explanation of the constant 1.03883 is given at .

The exact formula

In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for as a sum over the nontrivial zeros of the Riemann zeta function:

: $\psi_0(x) = x - \sum_ \frac - \frac - \tfrac \log (1-x^).$

(The numerical value of is .) Here runs over the nontrivial zeros of the zeta function, and is the same as , 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)
= \tfrac12\left( \sum_ \Lambda(n)+\sum_ \Lambda(n)\right)
=\begin \psi(x) - \tfrac \Lambda(x) & x = 2,3,4,5,7,8,9,11,13,16,\dots \\ px\psi(x) & \mbox \end

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of over the trivial zeros of the zeta function, , i.e.

: $\sum_^ \frac = \tfrac \log \left( 1 - x^ \right).$

Similarly, the first term, , 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 , there are infinitely many natural numbers such that

:$\psi(x)-x < -K\sqrt$

and infinitely many natural numbers such that

:$\psi(x)-x > K\sqrt.$

In little- notation, one may write the above as

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

Hardy and Littlewood prove the stronger result, that

:$\psi(x)-x \ne o\left(\sqrt\log\log\log x\right).$

Relation to primorials

The first Chebyshev function is the logarithm of the primorial of , denoted :

:$\vartheta(x)=\sum_ \log p=\log \prod_ p = \log \left(x\#\right).$

This proves that the primorial is asymptotically equal to , where "" is the little- notation (see big notation) and together with the prime number theorem establishes the asymptotic behavior of .

Relation to the prime-counting function

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

: $\Pi(x) = \sum_ \frac.$

Then

: $\Pi(x) = \sum_ \Lambda(n) \int_n^x \frac + \frac \sum_ \Lambda(n) = \int_2^x \frac + \frac.$

The transition from to the prime-counting function, , is made through the equation

: \Pi(x) = \pi(x) + \tfrac \pi\left(\sqrt x\right) + \tfrac \pi\left(\sqrt x\right) + \cdots

Certainly , so for the sake of approximation, this last relation can be recast in the form

: $\pi(x) = \Pi(x) + O\left(\sqrt x\right).$

The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part . In this case, , and it can be shown that

: $\sum_ \frac = O\left(\sqrt x \log^2 x\right).$

By the above, this implies

: $\pi(x) = \operatorname(x) + O\left(\sqrt x \log x\right).$

Good evidence that the hypothesis could be true comes from the fact proposed by Alain Connes and others, that if we differentiate the von Mangoldt formula with respect to we get . Manipulating, we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

: $\left. \zeta\left(\tfrac12+i \hat H \right)\n \ge \zeta\left(\tfrac12+iE_\right)=0,$
and
:$\sum_n e^=Z(u) = e^\frac-e^ \frac-\frac = \operatorname\left(e^\right),$

where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) , which is only true if .

Using the semiclassical approach the potential of satisfies:

:$\frac\sim \int_^\infty e^\,dx$

with as .

solution to this nonlinear integral equation can be obtained (among others) by
:$V^ (x) \approx \sqrt \cdot \frac N(x)$
in order to obtain the inverse of the potential:
:$\pi N(E) = \operatorname \xi \left(\tfrac12+iE\right).$

Smoothing function

The smoothing function is defined as

:$\psi_1(x)=\int_0^x \psi(t)\,dt.$

Obviously :$\psi_1(x) \sim \frac.$

Variational formulation

The Chebyshev function evaluated at minimizes the functional

: J \int_^\frac\,ds-\int_^\!\!\!\int_^ e^f(s)f(t)\,ds\,dt,

so

:$f(t)= \psi \left(e^t\right)e^ \quad\text c > 0.$

Notes

* Pierre Dusart, "Estimates of some functions over primes without R.H.".
* Pierre Dusart, "Sharper bounds for , , , ", Rapport de recherche no. 1998-06, Université de Limoges. An abbreviated version appeared as "The th prime is greater than for ", ''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. . Google Book Search.

References

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External links

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* {{planetmathref, urlname=ChebyshevFunctions, title=Chebyshev functions


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Arithmetic functions