Efficient Prime Counting and the Chebyshev Primes

Table 1

Upper part of the table: maximum error 𝜂_max in the new prime counting function for 𝑥<10⁴ (left-hand part) in comparison to the maximum error using the Riemann prime counting function (right-hand part). Lower part of the table: as above in the range 𝑥<10⁵.

𝑁	𝑥max	𝜂max	𝑥max	(𝑅−𝜋)max
3	6889	1.118	7450	6.174
4	6889	1.118	7450	6.174
5	1330	−1.061	9859	−5.506
6	7	−0.862	7450	5.879
7	1330	−0.936	9949	−5.609
10	7	−0.884	7450	5.661
50	1330	−0.885	9949	−5.557
3	80090	1.840	87888	15.304
10	49727	−1.158	59797	−15.729