Editing Prime-counting function (section)

==Formulas for prime-counting functions==

Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the [[prime number theorem]]. They stem from the work of Riemann and [[Hans Carl Friedrich von Mangoldt|von Mangoldt]], and are generally known as [[Explicit formulae (L-function)|explicit formulae]].<ref name="Titchmarsh">{{cite book |first=E.C. |last=Titchmarsh |year=1960 |title=The Theory of Functions, 2nd ed. |publisher=Oxford University Press}}</ref>

We have the following expression for the second [[Chebyshev function]] {{mvar|ψ}}:

:<math>\psi_0(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log 2\pi - \frac{1}{2} \log\left(1-x^{-2}\right),</math>

where

: <math>\psi_0(x) = \lim_{\varepsilon \to 0} \frac{\psi(x - \varepsilon) + \psi(x + \varepsilon)}{2}.</math>

Here {{mvar|ρ}} are the zeros of the Riemann zeta function in the critical strip, where the real part of {{mvar|ρ}} is between zero and one. The formula is valid for values of {{mvar|x}} greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last [[subtrahend]] in the formula.

For {{math|''Π''<sub>0</sub>(''x'')}} we have a more complicated formula

:<math>\Pi_0(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}\left(x^\rho\right) - \log 2 + \int_x^\infty \frac{\mathrm{d}t}{t \left(t^2 - 1\right) \log t}.</math>

Again, the formula is valid for {{math|''x'' > 1}}, while {{mvar|ρ}} are the nontrivial zeros of the zeta function ordered according to their absolute value.  The first term {{math|li(''x'')}} is the usual [[logarithmic integral function]]; the expression {{math|li(''x<sup>ρ</sup>'')}} in the second term should be considered as {{math|Ei(''ρ'' log ''x'')}}, where {{math|Ei}} is the [[analytic continuation]] of the [[exponential integral]] function from negative reals to the complex plane with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros:

:<math>\int_x^\infty \frac{\mathrm dt}{t \left(t^2 - 1\right) \log t}=\int_x^\infty \frac{1}{t\log t}
\left(\sum_{m}t^{-2m}\right)\,\mathrm dt=\sum_{m}\int_x^\infty \frac{t^{-2m}}{t\log t}
\,\mathrm dt \,\,\overset{\left(u=t^{-2m}\right)}{=}-\sum_{m} \operatorname{li}\left(x^{-2m}\right)
</math>

Thus, [[Möbius inversion formula]] gives us<ref name="RieselGohl" />

:<math>\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}\left(x^\rho\right) - \sum_{m} \operatorname{R}\left(x^{-2m}\right)</math>

valid for {{math|''x'' > 1}}, where

:<math>\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}\left(x^{1/n}\right) = 1 + \sum_{k=1}^\infty \frac{\left(\log x\right)^k}{k! k \zeta(k+1)}</math>

is Riemann's R-function<ref name="mathworld_r">{{MathWorld |title=Riemann Prime Counting Function |urlname=RiemannPrimeCountingFunction}}</ref> and {{math|''μ''(''n'')}} is the [[Möbius function]]. The latter series for it is known as [[Jørgen Pedersen Gram|Gram]] series.<ref name="Riesel94">{{cite book | title=Prime Numbers and Computer Methods for Factorization | edition=2nd | first=Hans | last=Riesel | author-link=Hans Riesel | series=Progress in Mathematics | volume=126 | publisher=Birkhäuser | year=1994 | isbn=0-8176-3743-5 | pages=50–51 }}</ref><ref name="mathworld_gram">{{MathWorld |title=Gram Series |urlname=GramSeries}}</ref> Because {{math|log ''x'' < ''x''}} for all {{math|''x'' > 0}}, this series converges for all positive {{mvar|x}} by comparison with the series for {{mvar|e<sup>x</sup>}}. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as {{math|''ρ'' log ''x''}} and not {{math|log ''x<sup>ρ</sup>''}}.

Folkmar Bornemann proved,<ref>{{cite web |last=Bornemann | first=Folkmar |title=Solution of a Problem Posed by Jörg Waldvogel |url=http://www-m3.ma.tum.de/bornemann/RiemannRZero.pdf }}</ref> when assuming the [[conjecture]] that all zeros of the Riemann zeta function are simple,<ref group="note">[[Hugh Lowell Montgomery|Montgomery]] showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple.</ref> that
:<math>\operatorname{R}\left(e^{-2\pi t}\right)=\frac{1}{\pi}\sum_{k=1}^\infty\frac{(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta(2k+1)}+\frac12\sum_{\rho}\frac{t^{-\rho}}{\rho\cos\frac{\pi\rho}{2}\zeta'(\rho)}</math>
where {{mvar|ρ}} runs over the non-trivial zeros of the Riemann zeta function and {{math|''t'' > 0}}.

The sum over non-trivial zeta zeros in the formula for {{math|''π''<sub>0</sub>(''x'')}} describes the fluctuations of {{math|''π''<sub>0</sub>(''x'')}} while the remaining terms give the "smooth" part of prime-counting function,<ref name="Watkins">{{cite web |title=The encoding of the prime distribution by the zeta zeros |url=http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm |publisher=Matthew Watkins |access-date=2008-09-14}}</ref> so one can use

:<math>\operatorname{R}(x) - \sum_{m=1}^\infty \operatorname{R}\left(x^{-2m}\right)</math>

as a good estimator of {{math|''π''(''x'')}} for {{math|''x'' > 1}}. In fact, since the second term approaches 0 as {{math|''x'' → ∞}}, while the amplitude of the "noisy" part is heuristically about {{math|{{sfrac|{{sqrt|''x''}}|log ''x''}}}}, estimating {{math|''π''(''x'')}} by {{math|R(''x'')}} alone is just as good, and fluctuations of the [[distribution of primes]] may be clearly represented with the function

:<math>\bigl( \pi_0(x) - \operatorname{R}(x)\bigr) \frac{\log x}{\sqrt x}.</math>