Efficient Prime Counting and the Chebyshev Primes

Table 2

Gauss’s and Riemann’s approximation and the approximation 𝜂₃(𝑥). Compare to table III, page 35 in [1].

𝑥	Planat and Solé error	Riemann’s error	Gauss’s error
1,000,000	0.79	30	130
2,000,000	−0.13	−8.0	121
3,000,000	1.83	1.8	121
4,000,000	1.28	35	130
5,000,000	0.36	−62	121
6,000,000	2.91	25	121
7,000,000	0.03	−36	130
8,000,000	2.99	−4.7	121
9,000,000	1.73	−51	121
10,000,000	−0.37	90	339