# 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 10 step 10-3 step 10-4 step 10-5 Δ(5-0) = -0.3952461978 Δ(1+0) = +1.0000000000 101 to 102 step 10-2 step 10-3 step 10-4 Δ(11-0) = -0.5492343329 Δ(19+0) = +0.5607597113 102 to 103 step 10-1 step 10-2 step 10-3 Δ(223-0) = -0.6051733874 Δ(113+0) = +0.7848341482 103 to 104 step 1 step 10-1 step 10-2 Δ(1423-0) = -0.7542604400 Δ(1627+0) = +0.6754517455 104 to 105 step 101 step 1 step 10-1 Δ(19373-0) = -0.7278356754 Δ(24137+0) = +0.7457431860 105 to 106 step 102 step 101 step 1 Δ(302831-0) = -0.6995719492 Δ(355111+0) = +0.7008073861 106 to 107 step 103 step 102 step 101 Δ(1090697-0) = -0.6389660809 Δ(3445943+0) = +0.6809987397 107 to 108 step 104 step 103 step 102 Δ(36917099-0) = -0.7489165055 Δ(30909673+0) = +0.7157292126 108 to 109 step 105 step 104 step 103 Δ(516128797-0) = -0.6775687236 Δ(110102617+0) = +0.7878100197 109 to 1010 step 106 step 105 step 104 Δ(7712599823-0) = -0.6889577485 Δ(1110072773+0) = +0.6833192028 1010 to 1011 step 107 step 106 step 105 Δ(11467849447-0) = -0.7251609705 Δ(10016844407+0) = +0.6386706267 1011 to 1012 step 108 step 107 step 106 Δ(110486344211-0) = -0.7355462679 Δ(330957852107+0) = +0.7533813432 1012 to 1013 step 109 step 108 step 107 Δ(1635820377397-0) = -0.6892596608 Δ(2047388353069+0) = +0.6808028098 1013 to 1014 step 1010 step 109 step 108 Δ(36219717668609-0) = -0.8360329846 Δ(21105695997889+0) = +0.6896466780 1014 to 1015 step 1011 step 1010 step 109 Δ(348323506633621-0) = -0.6494959371 Δ(117396942462053+0) = +0.6789107425 1015 to 1016 step 1012 step 1011 step 1010 Δ(1212562524413153-0) = -0.7750460589 Δ(1047930291039067+0) = +0.7042622330 1016 to 1017 step 1013 step 1012 step 1011 Δ(18019655286689201-0) = -0.5710665212 Δ(16452596773450399+0) = +0.7144542025 1017 to 1018 step 1014 step 1013 step 1012 Δ(266175790131587543-0) = -0.7599282036 Δ(125546149553907317+0) = +0.6572554320 1018 to 1019 step 1015 Δ(5805523423155128399-0) = -0.6804259482 Δ(1325005986250807813+0) = +0.7839983342 1019 to 1020 step 1016 Δ(55496658217283199013-0) = -0.8042730098 Δ(11538454954199984761+0) = +0.7574646817 1020 to 1021 step 1018 (to do) (to do) 1021 + 214 entries Δ(x) has no global minimum Δ(x) has no global maximum

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 (fully double-checked). 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 1017 ≤ x ≤ 1020 were taken from .

The values of π(x) for 1020 < x < 1021 were computed with primecount written by Kim Walisch.

The values of π(x) for x ≥ 1021 were taken from ,  and from Sloane's A006988, A007097. See also the results of Jan Büthe , David J. Platt , Thomas R. Nicely  and Xavier Gourdon .

# 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 101 max08(01)09.txt 20 001 From 3.309·1011 to 3.310·1011 with a step of 104 max11(04)12.txt 10 001 From 3.309578·1011 to 3.309580·1011 with a step of 101 max11(01)12.txt 20 001 From 3.590·1013 to 3.625·1013 with a step of 107 min13(07)14.txt 35 001 From 3.62194·1013 to 3.62200·1013 with a step of 104 min13(04)14.txt 60 001 From 3.62197176·1013 to 3.62197178·1013 with a step of 101 min13(01)14.txt 20 001 From 1.212·1015 to 1.214·1015 with a step of 108 min15(08)16.txt 20 001 From 1.212556·1015 to 1.212565·1015 with a step of 105 min15(05)16.txt 90 001 From 1.212562517·1015 to 1.212562526·1015 with a step of 102 min15(02)16.txt 90 001 From 3.2949·1015 to 3.2957·1015 with a step of 107 min15(07)16.txt 80 001 From 2.6615·1017 to 2.6635·1017 with a step of 1010 min17(10)18.txt 20 001 From 2.661751·1017 to 2.661760·1017 with a step of 107 min17(07)18.txt 90 001 From 2.6617579011·1017 to 2.6617579017·1017 with a step of 103 min17(03)18.txt 60 001 From 1.3245·1018 to 1.3260·1018 with a step of 1011 max18(11)19.txt 15 001 From 1.3250059·1018 to 1.3250067·1018 with a step of 107 max18(07)19.txt 80 001 From 1.32500598624·1018 to 1.32500598626·1018 with a step of 103 max18(03)19.txt 20 001 From 1.1536·1019 to 1.1542·1019 with a step of 1011 max19(11)20.txt 60 001 From 1.15384544·1019 to 1.15384551·1019 with a step of 107 max19(07)20.txt 70 001 From 1.153845497419·1019 to 1.153845497421·1019 with a step of 103 max19(03)20.txt 20 001 From 5.5496655·1019 to 5.5496662·1019 with a step of 108 min19(08)20.txt 70 001 From 5.54966582172·1019 to 5.54966582174·1019 with a step of 104 min19(04)20.txt 20 001
These results confirm some previously made computations  . See also  and  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 where  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): Here you can see the plot of π(x) (the purple line) compared to the blue line of The difference between these two heuristically oscillates with an amplitude of about so we have the following expression for Δ(x), the function which clearly represents the fluctuations of the distribution of primes: There's a plot of Δ(x) on the log scale: Some estimations of the logarithmic density of Δ(x) are given in  and .

# On the explicit formula for π(x)

Curiously, this formula seems to be never seen in literature , so let's describe its origin. The formula comes from the Möbius inversion of where is so-called Riemann prime-counting function (the first sum runs over the powers of primes). We have the following expression for Π0(x) : 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:    The first two of them are well-known ; the third one comes straightly from (32) in , while the last one is obvious if we allow generalized summation # References

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

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

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

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

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

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

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

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

 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.

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

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

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

 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

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

 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 Saturday, 31st of December, 2016