I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property.
We know that for certain values of 'b' in s = 1/2 + ib , ζ(s) = 0.
When I observed most of the values of 'b' here
I found that most of the primes are related to these values of 'b' in there square forms like:
Property: [b] = p^2 {where 'p' is a prime number and [b] is the Box Function}
example: b = 841.0363...
so, [841.0363...] = 29^2
below 200 there are only 6 primes which are not following the this above property.
I am searching a formula or a program to find out all those primes which follow above property but till now I didn't got any solution. I am also not good in programming please help me in this problem.
The $n$th zeta zero has $b(n) \sim 2\pi n/W(n/e)$, where $W$ is the Lambert function. The Lambert function grows as roughly the logarithm of its argument, so we see that the zeta zeroes get denser as $n$ increases, and at $n \approx 10^4$, their imaginary parts should start hitting every integer. This turns out to be the case, as the last square of a prime it misses is $103^2 = 10609$.
We can also estimate the probability of each square of a prime not having a zeta zero near it. The density of zeta zeroes is $W(n/e)/(2\pi)$, so we expect each square of a prime to not be near a zeta zero with probability $1 - W(p^2/e)/(2\pi)$. Summing this over all primes where this value is positive, we get an expected count of $7.7$, which is reasonably close to the actual value of 6.