The Prime Number Theorem states that:
where is the prime counting function (number of prime numbers ), and is the natural logarithm of x. The convergence of the abovementioned limit, first conjectured by Legendre in 1798, is now well established.
In this exercise we are more concerned with the difference function Δ, defined as follows:
where the symbol means rounding-off to nearest integer. Δ appears divergent and seems to increase without bound as x increases.
Given a number d, our goal is to find the value of integer x when first exceeds d.
As an example, if , the value of x is since:
,
and for all , .
Solve

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George Berken David Hill GeeTwo Mehdi Dehghan

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