Values of π(x) and Δ(x) for various x's

The tables were compiled by Andrey V. Kulsha. See below the explanation of Δ(x).

Values of x
Tables
Local minima of Δ(x)
Local maxima of Δ(x)
1 to 10step 10-3step 10-4step 10-5
Δ(5-0) = -0.3952461978
Δ(1+0) = +1.0000000000
101 to 102step 10-2step 10-3step 10-4
Δ(11-0) = -0.5492343329
Δ(19+0) = +0.5607597113
102 to 103step 10-1step 10-2step 10-3
Δ(223-0) = -0.6051733874
Δ(113+0) = +0.7848341482
103 to 104step 1step 10-1step 10-2
Δ(1423-0) = -0.7542604400
Δ(1627+0) = +0.6754517455
104 to 105step 101step 1step 10-1
Δ(19373-0) = -0.7278356754
Δ(24137+0) = +0.7457431860
105 to 106step 102step 101step 1
Δ(302831-0) = -0.6995719492
Δ(355111+0) = +0.7008073861
106 to 107step 103step 102step 101
Δ(1090697-0) = -0.6389660809
Δ(3445943+0) = +0.6809987397
107 to 108step 104step 103step 102
Δ(36917099-0) = -0.7489165055
Δ(30909673+0) = +0.7157292126
108 to 109step 105step 104step 103
Δ(516128797-0) = -0.6775687236
Δ(110102617+0) = +0.7878100197
109 to 1010step 106step 105step 104
Δ(7712599823-0) = -0.6889577485
Δ(1110072773+0) = +0.6833192028
1010 to 1011step 107step 106step 105
Δ(11467849447-0) = -0.7251609705
Δ(10016844407+0) = +0.6386706267
1011 to 1012step 108step 107step 106
Δ(110486344211-0) = -0.7355462679
Δ(330957852107+0) = +0.7533813432
1012 to 1013step 109step 108step 107
Δ(1635820377397-0) = -0.6892596608
Δ(2047388353069+0) = +0.6808028098
1013 to 1014step 1010step 109step 108
Δ(36219717668609-0) = -0.8360329846
Δ(21105695997889+0) = +0.6896466780
1014 to 1015step 1011step 1010step 109
Δ(348323506633621-0) = -0.6494959371
Δ(117396942462053+0) = +0.6789107425
1015 to 1016step 1012step 1011step 1010
Δ(1212562524413153-0) = -0.7750460589
Δ(1047930291039067+0) = +0.7042622330
1016 to 1017step 1013step 1012step 1011*
Δ(18019655286689201-0) = -0.5710665212
Δ(16452596773450399+0) = +0.7144542025
1017 to 1018step 1014step 1013step 1012
Δ(266175790131587543-0) = -0.7599282036
Δ(125546149553907317+0) = +0.6572554320
1018 to 1019step 1015
Δ(5805523423155128399-0) = -0.6804259482
Δ(1325005986250807813+0) = +0.7839983342
1019 to 1020step 1016
Δ(55496658217283199013-0) = -0.8042730098
Δ(11538454954199984761+0) = +0.7574646817
1020 +308 entries
Δ(x) has no global minimum
Δ(x) has no global maximum

*Double-checking in progress

The values of π(x) for x < 1017 were mostly computed with primesieve written by Kim Walisch and are also available with a step of 109 (double-checking in progress). The data is encoded as a stream of 2-byte differences between the successive rounded values of (π(x)-li(x))·3/2, and a small delphi program is provided to get a plain text file. The multiplier of 3/2 doesn't bring the differences out of 2-byte range [-32768 ... +32767] and allows to compute li(x) with an absolute error of up to 1/6, so the Ramanujan method with extended precision works well at least for x < 1018.

The values of π(x) for x > 1017 were mostly taken from [1]. Some points were taken from [2], [3] and from Sloane's A006988 and A007097. See also the results of David J. Platt [4], Thomas R. Nicely [5] and Xavier Gourdon [6].

Some extreme regions where |Δ(x)| exceeds 0.75

Values of x
Tables
# of entries
From 1.100·108 to 1.102·108 with a step of 101max08(01)09.txt
20 001
From 3.309·1011 to 3.310·1011 with a step of 104max11(04)12.txt
10 001
From 3.309578·1011 to 3.309580·1011 with a step of 101max11(01)12.txt
20 001
From 3.590·1013 to 3.625·1013 with a step of 107min13(07)14.txt
35 001
From 3.62194·1013 to 3.62200·1013 with a step of 104min13(04)14.txt
60 001
From 3.62197176·1013 to 3.62197178·1013 with a step of 101min13(01)14.txt
20 001
From 1.212·1015 to 1.214·1015 with a step of 108min15(08)16.txt
20 001
From 1.212556·1015 to 1.212565·1015 with a step of 105min15(05)16.txt
90 001
From 1.212562517·1015 to 1.212562526·1015 with a step of 102min15(02)16.txt
90 001
From 3.2949·1015 to 3.2957·1015 with a step of 107min15(07)16.txt
80 001
From 2.6615·1017 to 2.6635·1017 with a step of 1010min17(10)18.txt
20 001
From 2.661751·1017 to 2.661760·1017 with a step of 107min17(07)18.txt
90 001
From 2.6617579011·1017 to 2.6617579017·1017 with a step of 103min17(03)18.txt
60 001
From 1.3245·1018 to 1.3260·1018 with a step of 1011max18(11)19.txt
15 001
From 1.3250059·1018 to 1.3250067·1018 with a step of 107max18(07)19.txt
80 001
From 1.32500598624·1018 to 1.32500598626·1018 with a step of 103max18(03)19.txt
20 001
From 1.1536·1019 to 1.1542·1019 with a step of 1011max19(11)20.txt
60 001
From 1.15384544·1019 to 1.15384551·1019 with a step of 107max19(07)20.txt
70 001
From 1.153845497419·1019 to 1.153845497421·1019 with a step of 103max19(03)20.txt
20 001
From 5.5496655·1019 to 5.5496662·1019 with a step of 108min19(08)20.txt
70 001
From 5.54966582172·1019 to 5.54966582174·1019 with a step of 104min19(04)20.txt
20 001
These results confirm some previously made computations [7] [8]. See also [9] and [10] about the oscillations of Δ(x) at larger x's.

Where did Δ(x) come from?

The prime-counting function, π(x), may be computed analytically. The explicit formula for it, valid for x > 1, looks like

\pi_{0}(x) = \mathrm{R}(x) - \sum_{\rho}\mathrm{R}(x^{\rho}) - \frac{1}{\ln x} + \frac{1}{\pi}\arctan\frac{\pi}{\ln x}

where

\pi_{0}(x) = \lim_{\varepsilon \rightarrow 0}\frac{\pi(x-\varepsilon)+\pi(x+\varepsilon)}{2}

\mathrm{R}(x^\rho) = 1 + \sum_{k=1}^\infty \frac{(\rho \ln x)^k}{k! k \zeta(k+1)}

and the sum runs over the non-trivial (i.e. with positive real part) zeros of Riemann ζ-function in order of increasing the absolute value of the imaginary part. This sum describes the fluctuations of π(x), while the remaining terms give the «smooth» part of it and may be used as a very good estimator of π(x):

The smooth part of Pi(x)

Here you can see the plot of π(x) (the purple line) compared to the blue line of

\mathrm{R}(x) - \frac{1}{\ln x} + \frac{1}{\pi}\arctan\frac{\pi}{\ln x}

The difference between these two heuristically oscillates with an amplitude of about

\frac{\sqrt x}{\ln x}

so we have the following expression for Δ(x), the function which clearly represents the fluctuations of the distribution of primes:

\Delta(x) = \left( \pi_{0}(x) - \mathrm{R}(x) + \frac{1}{\ln x} - \frac{1}{\pi}\arctan\frac{\pi}{\ln x} \right) \frac{\ln x}{\sqrt x}

There's a plot of Δ(x) on the log scale:

Delta(x) on the log scale

Some estimations of the logarithmic density of Δ(x) are given in [11] and [12].

On the explicit formula for π(x)

Curiously, this formula seems to be never seen in literature [13], so let's describe its origin. The formula comes from the Möbius inversion

\pi_{0}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \Pi_{0}(x^{\frac{1}{n}})

of

\Pi_{0}(x) = \lim_{\varepsilon \rightarrow 0}\frac{\Pi(x-\varepsilon)+\Pi(x+\varepsilon)}{2}

where

\Pi(x) = \sum_{p^n \le x} \frac{1}{n} = \sum_{n=1}^{\infty} \frac{1}{n} \pi(x^{\frac{1}{n}})

is so-called Riemann prime-counting function (the first sum runs over the powers of primes). We have the following expression for Π0(x) [13]:

\Pi_{0}(x) = \mathrm{li}(x) - \sum_{\rho}\mathrm{li}(x^{\rho}) - \ln 2 + \int\limits_x^\infty \frac{dt}{t(t^2-1) \ln t}

where li is the logarithmic integral; li(xρ) should be considered as Ei(ρlnx), where Ei is the analytic continuation of the exponential integral function from positive reals to the complex plane with branch cut along the negative reals. The sum runs, as before, over the non-trivial zeros of ζ-function in the same manner. Thus, the formula immediately follows from these four equalities:

\sum_{n=1}^{\infty} \frac{\mu(n)}{n} = 0

\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \mathrm{li}(x^{\frac{1}{n}}) = \mathrm{R}(x)

\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \left( - \frac{n}{2 \ln x} + \int\limits_{x^{1/n}}^{\infty} \frac{dt}{t (t^2-1) \ln t} \right) = \frac{1}{\pi}\arctan\frac{\pi}{\ln x}

\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \left( \sum_{\rho}\mathrm{li}(x^{\frac{\rho}{n}}) - \frac{n}{2 \ln x} \right) = \sum_{\rho}\mathrm{R}(x^{\rho}) + \frac{1}{\ln x}

The first two of them are well-known [14]; the third one comes straightly from (32) in [15], while the last one is obvious if we allow generalized summation

\sum_{n=1}^{\infty}\mu(n) = \frac{1}{\zeta(0)} = -2

References

[1] Tomás Oliveira e Silva. Tables of values of pi(x) and of pi2(x). http://sweet.ua.pt/tos/primes.html

[2] Jan Büthe. Analytic computation of pi(x). http://www.math.uni-bonn.de/people/jbuethe/topics/AnalyticPiX.html

[3] Douglas B. Staple. Prime counting function records. http://www.mersenneforum.org/showthread.php?t=19863

[4] David J. Platt. Computing π(x) Analytically. http://arxiv.org/abs/1203.5712

[5] Thomas R. Nicely. A table of prime counts pi(x) to 1e16. http://www.trnicely.net/pi/pix_0000.htm

[6] Xavier Gourdon. Counting the number of primes. http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html

[7] Nuna da Costa Pereira. Computational results on some prime number functions. http://mat.fc.ul.pt/ind/ncpereira/

[8] Tadej Kotnik. The prime-counting function and its analytic approximations. Adv. Comp. Math., Vol. 29, N. 1 (2008), pp. 55-70

[9] Douglas A. Stoll, Patrick Demichel. The impact of ζ(s) complex zeros on π(x)-li(x) for x < 101013. Math. Comp., Vol. 80, N. 276 (2011), pp. 2381-2394.

[10] Patrick Demichel. The prime counting function and related subjects. http://sites.google.com/site/dmlpat2/li_crossover_pi.pdf

[11] Michael Rubinstein, Peter Sarnak. Chebyshev's Bias. Experiment. Math., Vol. 3, Issue 3 (1994), pp. 173-197

[12] Colin Myerscough. Application of an accurate remainder term in the calculation of Residue Class Distributions. To be published in Math. Comp.

[13] Jonathan M. Borwein, David M. Bradley, Richard E. Crandall. Computational strategies for the Riemann zeta function. J. Comp. App. Math., Vol. 121 (2000), pp. 247-296

[14] H.M. Edwards. Riemann's Zeta Function. Academic Press, 1974

[15] Hans Riesel, Gunnar Gohl. Some Calculations Related to Riemann's Prime Number Formula. Math. Comp., Vol. 24, N. 112 (1970), pp. 969-983

updated on Friday, 19th of December, 2014