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# Chebyshev function

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

$vartheta\left(x\right)=sum_\left\{ple x\right\} log p$

with the sum extending over all prime numbers p that are less than x. The second Chebyshev function $psi\left(x\right)$ is defined by

$psi\left(x\right) = sum_\left\{n leq x\right\} Lambda\left(n\right),$

where $Lambda$ is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the prime-counting function, $pi\left(x\right)$. Both functions are asymptotic to $x$, a statement equivalent to the prime number theorem.

Both functions are named in honour of Pafnuty Lvovich Chebyshev.

## Relationships

The second Chebyshev function can be seen to be related to the first by writing it as
$psi\left(x\right)=sum_\left\{ple x\right\} k log p$

where k is the unique integer such that $p^kle x$ but $p^\left\{k+1\right\}>x$. A more direct relationship is given by

$psi\left(x\right)=sum_\left\{n=1\right\}^infty vartheta left\left(x^\left\{1/n\right\}right\right).$

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

$vartheta left\left(x^\left\{1/n\right\}right\right) = 0$ for $n>log_2 x.$

The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.
$operatorname\left\{lcm\right\}\left(1,2,dots n\right)=e^\left\{psi\left(n\right)\right\}.$

## Asymptotics and bounds

Pierre Dusart proved the following bounds for the Chebyshev functions:

$vartheta\left(p_k\right)ge kleft\left(ln k+lnln k-1+frac\left\{lnln k-2.0553\right\}\left\{ln k\right\}right\right)$ for k' ≥ exp(22)

$vartheta\left(p_k\right)le kleft\left(ln k+lnln k-1+frac\left\{lnln k-2\right\}\left\{ln k\right\}right\right)$ for k ≥ 198

$psi\left(p_k\right)le kleft\left(ln k+lnln k-1+frac\left\{lnln k-2\right\}\left\{ln k\right\}right\right) + 1.43sqrt x$ for k ≥ 198

$|vartheta\left(x\right)-x|le0.006788frac\left\{x\right\}\left\{ln x\right\}$ for x ≥ 10,544,111

$|psi\left(x\right)-x|le0.006409frac\left\{x\right\}\left\{ln x\right\}$ for x ≥ exp(22)

$psi\left(x\right)-vartheta\left(x\right)<0.0000132frac\left\{x\right\}\left\{ln x\right\}$ for x ≥ exp(30)

Along with $psi\left(x\right)ge vartheta\left(x\right)$, this gives a good characterization of the behavior of these two functions.

## The exact formula

In 1895, Hans Carl Friedrich von Mangoldt proved an explicit expression for $psi\left(x\right)$ as a sum over the nontrivial zeros of the Riemann zeta function:

$psi_0\left(x\right) = x - sum_\left\{rho\right\} frac\left\{x^\left\{rho\right\}\right\}\left\{rho\right\} - frac\left\{zeta\text{'}\left(0\right)\right\}\left\{zeta\left(0\right)\right\} - frac\left\{1\right\}\left\{2\right\} log \left(1-x^\left\{-2\right\}\right).$

Here $rho$ runs over the nontrivial zeros of the zeta function, and

$psi_0\left(x\right) = begin\left\{cases\right\} psi\left(x\right) - frac\left\{1\right\}\left\{2\right\} Lambda\left(x\right) & x = p^m mbox\left\{, p prime, m an integer\right\} psi\left(x\right) & mbox\left\{otherwise.\right\} end\left\{cases\right\}$

From the Taylor series for the logarithm, the last term in the explicit formula can be understood as a summation of $-x^\left\{omega\right\}/\left\{omega\right\}$ over the trivial zeros of the zeta function, $omega = -2, -4, -6, ldots$, i.e.

$sum_\left\{k=1\right\}^\left\{infty\right\} frac\left\{x^\left\{2k\right\}\right\}\left\{2k\right\} = frac\left\{1\right\}\left\{2\right\} log \left(1 - x^\left\{-2\right\} \right).$

## Properties

A theorem due to Erhard Schmidt states that, for any real, positive K, there are values of x such that

$psi\left(x\right)-x < -Ksqrt\left\{x\right\}$

and

$psi\left(x\right)-x > Ksqrt\left\{x\right\}$

infinitely often. On big-O notation, one may write the above as

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

Hardy and Littlewood prove the stronger result, that

$psi\left(x\right)-x ne Oleft\left(sqrt\left\{x\right\}logloglog xright\right).$

## Relation to primorials

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

$vartheta\left(x\right)=sum_\left\{ple x\right\} log p=log prod_\left\{ple x\right\} p = log x#.$

This proves that the primorial x# is asymptotically equal to exp((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

$Pi\left(x\right) = sum_\left\{n leq x\right\} frac\left\{Lambda\left(n\right)\right\}\left\{log n\right\}.$

Then

$Pi\left(x\right) = sum_\left\{n leq x\right\} Lambda\left(n\right) int_n^x frac\left\{dt\right\}\left\{t log^2 t\right\} + frac\left\{1\right\}\left\{log x\right\} sum_\left\{n leq x\right\} Lambda\left(n\right) = int_2^x frac\left\{psi\left(t\right), dt\right\}\left\{t log^2 t\right\} + frac\left\{psi\left(x\right)\right\}\left\{log x\right\}.$

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

$Pi\left(x\right) = pi\left(x\right) + frac\left\{1\right\}\left\{2\right\} pi\left(x^\left\{1/2\right\}\right) + frac\left\{1\right\}\left\{3\right\} pi\left(x^\left\{1/3\right\}\right) + cdots.$

Certainly $pi\left(x\right) leq x$, so for the sake of approximation, this last relation can be recast in the form

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

## The Riemann hypothesis

The Riemann hypothesis states that all nontrivial zeros of the zeta function have real part 1/2. In this case, $|x^\left\{rho\right\}|=sqrt x$, and it can be shown that

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

By the above, this implies

$pi\left(x\right) = operatorname\left\{li\right\}\left(x\right) + O\left(sqrt x log x\right).$

A good evidence that RH could be true comes from the fact proposed by Alain Connes and others, that if we differentiate V. Mangoldt formula respect to x make x = exp(u) manipulating we have the formula we have the "Trace formula" for the exponential of the Hamiltonian operator satisfying

$zeta\left(1/2+i hat H \right)|n ge zeta\left(1/2+iE_\left\{n\right\}\right)=0,$

$sum_\left\{n\right\}e^\left\{iu E_\left\{n\right\}\right\}=Z\left(u\right)=e^\left\{u/2\right\}-e^\left\{-u/2\right\} frac\left\{dpsi _\left\{0\right\}\right\}\left\{du\right\}-frac\left\{e^\left\{u/2\right\}\right\}\left\{e^\left\{3u\right\}-e^\left\{u\right\}\right\} = operatorname\left\{Tr\right\}\left(e^\left\{iuhat H \right\}\right).$

Where the "trigonometric sum" can be considered to be the trace of the operator (statistical mechanics) $e^\left\{iu hat H\right\}$,which is only true if $rho =1/2+iE\left(n\right)$ .

Using the semiclassical approach the potential of H=T+V satisfies:

$frac\left\{Z\left(u\right)u^\left\{1/2\right\}\right\}\left\{sqrt pi \right\}sim int_\left\{-infty\right\}^\left\{infty\right\}e^\left\{i \left(uV\left(x\right)+ pi /4 \right)\right\},dx$

with Z(u) → 0 as u → ∞.

## Smoothing function

The smoothing function is defined as

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

It can be shown that

$psi_1\left(x\right) sim frac\left\{x^2\right\}\left\{2\right\}.$

## Variational formulation

The Chebyshev function evaluated at x = exp(t) minimizes the functional

$J\left[f\right]=int_\left\{0\right\}^\left\{infty\right\}frac\left\{f\left(s\right)zeta\text{'} \left(s+c\right)\right\}\left\{zeta\left(s+c\right)\left(s+c\right)\right\},ds-int_\left\{0\right\}^\left\{infty\right\}!!!int_\left\{0\right\}^\left\{infty\right\} e^\left\{-st\right\}f\left(s\right)f\left(t\right),ds,dt,$

so

$f\left(t\right)= psi \left(e^t\right)e^\left\{-ct\right\},,$

for c > 0.

## References

• Pierre Dusart, "Sharper bounds for ψ, θ, π, $p_k$", Rapport de recherche n° 1998-06, Université de Limoges. An abbreviated version appeared as "The kth prime is greater than k(ln k + ln ln 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 . Springer. p. 104. ISBN 0-387-95097-4. Google Book Search.