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)
From 1 to 10 with a step of 0.00011e01.txt
Δ(5-0) = -0.3952461978
Δ(1+0) = +1.0000000000
From 101 to 102 with a step of 0.0011e02.txt
Δ(11-0) = -0.5492343329
Δ(19+0) = +0.5607597113
From 102 to 103 with a step of 0.011e03.txt
Δ(223-0) = -0.6051733874
Δ(113+0) = +0.7848341482
From 103 to 104 with a step of 0.11e04.txt
Δ(1423-0) = -0.7542604400
Δ(1627+0) = +0.6754517455
From 104 to 105 with a step of 11e05.txt
Δ(19373-0) = -0.7278356754
Δ(24137+0) = +0.7457431860
From 105 to 106 with a step of 1011e06.txt
Δ(302831-0) = -0.6995719492
Δ(355111+0) = +0.7008073861
From 106 to 107 with a step of 1021e07.txt
Δ(1090697-0) = -0.6389660809
Δ(3445943+0) = +0.6809987397
From 107 to 108 with a step of 1031e08.txt
Δ(36917099-0) = -0.7489165055
Δ(30909673+0) = +0.7157292126
From 108 to 109 with a step of 1041e09.txt
Δ(516128797-0) = -0.6775687236
Δ(110102617+0) = +0.7878100197
From 109 to 1010 with a step of 1051e10.txt
Δ(7712599823-0) = -0.6889577485
Δ(1110072773+0) = +0.6833192028
From 1010 to 1011 with a step of 1061e11.txt
Δ(11467849447-0) = -0.7251609705
Δ(10016844407+0) = +0.6386706267
From 1011 to 1012 with a step of 1071e12.txt
Δ(110486344211-0) = -0.7355462679
Δ(330957852107+0) = +0.7533813432
From 1012 to 1013 with a step of 1081e13.txt
Δ(1635820377397-0) = -0.6892596608
Δ(2047388353069+0) = +0.6808028098
From 1013 to 1014 with a step of 1091e14.txt
Δ(36219717668609-0) = -0.8360329846
Δ(21105695997889+0) = +0.6896466780
From 1014 to 1015 with a step of 10101e15.txt
Δ(348323506633621-0) = -0.6494959371
Δ(117396942462053+0) = +0.6789107425
From 1015 to 1016 with a step of 10111e16.txt*
Δ(1212562524413153-0) = -0.7750460589
Δ(1047930291039067+0) = +0.7042622330
From 1016 to 1017 with a step of 10131e17.txt**
Δ(18019655286689201-0) = -0.5710665212
Δ(16452596773450799+0) = +0.7144542025
From 1017 to 1018 with a step of 10141e18.txt**
Δ(266175790131587543-0) = -0.7599282036
Δ(125546149553907317+0) = +0.6572554320
From 1018 to 1019 with a step of 10151e19.txt**
(to be computed)
(to be computed)
From 1019 to 1020 with a step of 1016
(should be completed for x > 5.675e19)
1e20.txt**
(to be computed)
(to be computed)
1020 and largerlarge.txt**
Δ(x) has no global minimum
Δ(x) has no global maximum
*The values of π(x) were computed by Anatoly F. Selevich.
**The values of π(x) were taken from [1] except the value of π(1.5·1022) from [2].

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 1011e09_max.txt
20 001
From 3.309·1011 to 3.310·1011 with a step of 1041e12_max.txt
10 001
From 3.309578·1011 to 3.309580·1011 with a step of 1011e12_max2.txt
20 001
From 3.590·1013 to 3.625·1013 with a step of 1071e14_min.txt
35 001
From 3.62194·1013 to 3.62200·1013 with a step of 1041e14_min2.txt
60 001
From 3.62197176·1013 to 3.62197178·1013 with a step of 1011e14_min3.txt
20 001
From 1.212·1015 to 1.214·1015 with a step of 1091e16_min.txt
2 001
From 1.21255·1015 to 1.21257·1015 with a step of 1061e16_min2.txt
20 001
From 1.212562517·1015 to 1.212562526·1015 with a step of 1021e16_min3.txt
90 001
From 3.294·1015 to 3.297·1015 with a step of 1091e16_min_.txt
3 001
From 2.6615·1017 to 2.6635·1017 with a step of 10121e18_min.txt
201
From 2.661751·1017 to 2.661760·1017 with a step of 1071e18_min2.txt
90 001
From 2.6617579011·1017 to 2.6617579017·1017 with a step of 1031e18_min3.txt
60 001
These results confirm some previously made computations [3] [4]. See also [5] and [6] 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

On the explicit formula for π(x)

Curiously, this formula seems to be never seen in literature [7], 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) [8]:

\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 [8]; the third one comes straightly from the (32) in [9], 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] Tomás Oliveira e Silva. Tables of values of pi(x) and of pi2(x). http://www.ieeta.pt/~tos/primes.html

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

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

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

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

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

[7] 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.

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

[9] 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 Tuesday, 7th of April, 2009