However though the accuracy of this formula improves in relative terms as n increases, ultimately approaching 100% accuracy, it is not really very accurate in absolute terms. In the following table, I give the actual occurrence of primes to various powers of 10 (to 10^{16}). I also give the predicted number using the simple log formula and then its percentage accuracy.
Up to n

No. of primes

Estimated no.

% accuracy (1)

% accuracy (2)

10

4

4

100

57.14

100

25

22

88

88.61

1000

168

145

86.31

96.55

10000

1229

1086

88.36

99.03

100000

9592

8686

90.55

99.46

1000000

78498

72382

92.21

99.55

10000000

664579

620421

93.36

99.65

100000000

5761455

5428681

94.22

99.70

1000000000

50847534

48254942

94.90

99.74

10000000000

455052511

434294482

95.44

99.76

As we can see, though the absolute deviation from the true number of primes significantly increases, the relative percentage of estimated primes (in terms of the actual number) steadily increases so that by 10^{10}, it is over 95% accurate.
However a significant improvement can be achieved in a recursive manner by defining
n_{1}_{ }= n/log n and then obtaining the new modified estimate,
i.e. n/log n + n_{1}/log n_{1 } as in (2) above.
As can be seen from the above the new modified version of the formula quickly becomes a much more accurate predictor of the percentage of primes (over this range). For example, already at 10,000 it has reached 99% accuracy (as opposed to 88% using the traditional method).
One other interesting feature is that whereas the traditional method always underpredicts (over the ranges yet capable of estimation), the modified version always overpredicts.
The largest estimate for actual primes that I dealt with was for 10^{16 }, which is
10,000,000,000,000,000. The actual number of primes to this number is 279,238,341,033,925.
Now the simplest log estimate (1) predicts 271,434,051,189,532 primes (an underestimate) which is 97.21% accurate.
The modified log estimate (2) then predicts 279,601,229,526,797 primes (an overestimate) which is 99.87% accurate.
Now these estimates will still remain quite poor when compared to the Gaussian Li estimate and also Riemann's function.
However its still remains of great interest in pointing to what appears to be strong evidence for the recursive nature of overall prime behaviour.
It is not readily apparent how Littlewood's proof that the Gaussian Li conjecture (i.e. that it would always lead to an underestimate of the primes) was in error, would apply to the much less accurate crude log estimate which in version (1) substantially underestimates the number of actual primes. ^{ }
^{}
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