We know that the number of prime numbers less than tends to
Using this I have thought of a way of estimation of sum of prime numbers less than "n".
Using the formula above, we get that the number of prime numbers less than is , so the number of prime numbers less than is .
So we can say that the probability of being prime is
As will grow bigger, will become a very small value for it, so we can replace it with . Here is the small quantity we generally use in calculus.
Probability of being prime is
Solving the derivative, we get that the probability of being prime is:-
We can generally say that the sum of prime numbers less than is . Here is the probability of being prime.
So we can rewrite this summation as an integration:-
Solving the integration, we get (Here represents sum of prime numbers less than "n"):-
Ignoring the constant value on the RHS, which would be too small as grows bigger, we have:-
Please read this and comment if there are any mistakes or about anything else regarding this derivation. If you are good at coding, please check if this formula works for large numbers.