values of the prime counting function and estimates for selected inputs

In illustrating the degree of error of various estimates of the prime counting function given in connection to the prime number theorem, it is customary to select powers of 10 as the inputs. These are given in the table below, mixed in with primes beginning and ending prime gaps, the fourth primes of selected prime quadruplets and selected record lows of the Mertens function. The values of the logarithmic integral and the division of $n$ by its natural logarithm have been rounded off to the nearest integer.

$n$	$\pi (n)$	${\int }_2^n\frac{dt}{\log t}$	$\frac{n}{\log n}$
2	1	1	3
3	2	2	3
5	3	4	3
7	4	5	4
10	4	6	4
11	5	7	5
23	9	11	7
29	10	13	9
89	24	28	20
97	25	29	21
100	25	30	22
110	29	32	23
113	30	33	24
127	31	36	26
523	99	105	84
541	100	108	86
829	145	153	123
887	154	161	131
907	155	164	133
1000	168	178	145
1105	185	193	158
1129	189	196	161
1151	190	199	163
1327	217	224	185
1361	218	229	189
1489	237	246	204
1879	289	299	249
9551	1183	1197	1042
9587	1184	1201	1046
10000	1229	1246	1086
15683	1831	1847	1623
15727	1832	1852	1628
19609	2225	2249	1984
19661	2226	2254	1989
23833	2652	2672	2365
31397	3385	3412	3032
31469	3386	3419	3038
99139	9520	9555	8618
100000	9592	9630	8686
1000000	78498	78628	72382
10000000	664579	664918	620421
100000000	5761455	5762209	5428681
1000000000	50847534	50849235	48254942
10000000000	455052511	455055615	434294482

The smaller values (up to $n=2000$) have been verified by hand. Above that, I have trusted Mathematica 4.2 completely.

Title	values of the prime counting function and estimates for selected inputs
Canonical name	ValuesOfThePrimeCountingFunctionAndEstimatesForSelectedInputs
Date of creation	2013-03-22 16:38:53
Last modified on	2013-03-22 16:38:53
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 11A41