For large X, we have [tex]\pi(X) \approx X/log X[/tex] according to PNT where logX is the average gap size between consecutive primes less than or equal to X.

Now we let,

1. X = s * log X + b * [tex]log^{2}(X)[/tex],

approximately.

We seek to show that equation one leads to a contradiction because of G(X) for some positive integers, s >> b.

Note: The integer constants, s and b, are the number of primes associated with the gap sizes, average and G(X), respectively.

Therefore, equation one implies,

2. [tex]X / log X \approx \pi(X) \approx s + b*log X = s + b * X / \pi(X)[/tex].

Moreover, equation two implies approximately,

3. [tex]\pi^{2}(X) - s* \pi(X) - b* X = 0[/tex].

In turn, equations, three and one, imply with the help of quadratic formula,

4. [tex]\pi(X) = \frac{s + \sqrt{s^{2} + 4bX}}{2} = s + b * log X[/tex].

However, equation four implies approximately,

5. X = s / 2 + b * [tex]log^{2} X[/tex] which contradicts equation one!

Thus, G(X) < [tex]log^{2} X[/tex] for large X.

Dave,

https://www.researchgate.net/profile/David_Cole29/amp