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 1012(in progress)
Δ(18019655286689201-0) = -0.5710665212
Δ(16452596773450799+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
(to be computed)
** 1019 to 1020step 1016 (in progress for x > 6.3·1019)
(to be computed)
(to be computed)
** 1020 +large.txt
Δ(x) has no global minimum
Δ(x) has no global maximum
*The values of π(x) were taken from [1].
**The values of π(x) were taken from [2] except the value of π(1.5·1022) from [3].

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 1011(in progress)
15 001
These results confirm some previously made computations [4] [5]. See also [6] and [7] about the oscillations of π(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) = \operatorname{R}(x) - \sum_{\rho}\operatorname{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}

\operatorname{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) roots 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

\operatorname{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) - \operatorname{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 [8].

On the explicit formula for π(x)

Curiously, this formula seems to be never seen in literature [9], 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) [10]:

\Pi_{0}(x) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^{\rho}) - \ln 2 + \int_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 roots of ζ-function in the same manner. Thus, we have

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

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

\sum_{n=1}^{\infty} \frac{\mu(n)}{n} \int_{\frac{\ln x}{n}}^{\infty} \frac{dt}{(e^{2t}-1)t} = \frac{1}{\pi} \arctan \frac{\pi}{\ln x} - \frac{1}{\ln x}

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

and the formula immediately follows.

P.S. The first two equalities are well-known [10]; the third one comes straightly from the (32) in [11], while the last one is based on the identity

\lim_{t\rightarrow\infty}\sum_{n=1}^{\infty}\mu(n)\left(\frac{u}{n}\sum_{|\operatorname{Im}(\rho)|>t}\operatorname{Ei}\left(\rho\frac{u}{n}\right)-\frac12\right) \equiv 1

which holds for u > 0.

References

[1] Thomas R. Nicely. A table of prime counts pi(x) to 1e16. http://www.trnicely.net/pi/pix_0000.htm and http://sage.math.washington.edu/home/kstueve/T_R_NICELY/

[2] Tomás Oliveira e Silva. Tables of values of pi(x) and of pi2(x). http://www.ieeta.pt/~tos/primes.html and http://sage.math.washington.edu/home/kstueve/T_O_SILVA/

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

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

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

[6] Kuok Fai Chao, Roger Plymen. A new bound for the smallest $x$ with $\pi(x) > li(x)$. http://arxiv.org/abs/math/0509312

[7] Patrick Demichel. The prime counting function and related subjects. http://demichel.net/patrick/li_crossover_pi.pdf

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

[9] 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

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

[11] 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 Sunday, 4th of July, 2010