Let the integers x,y satisfy x>y>4 and let pi(x,y) count the number of primes between x and y.
So the number of primes in the closed interval [y,x] is equal to pi(x,y).
Here I present a lower estimate for the function pi(x,y) :
pi(x,y) >~~ [ (x-y)/x ] [ x/ln(x-1 + ln(3)) - y/ln(y-1 + ln(3)) ] + [ y/x ] [ (x-y)/(ln(x-1 + ln(3))*(ln(y-1 + ln(3))) ]
where ln() is the natural log.
(the square brackets are just the same as usual brackets but used to make it easier to remember)
with lower estimate, I mean an estimate for a lower bound.
This estimate is based on a few ideas.
For instance let pi(x) be the number of primes between 1 and x then for x>1 :
x/(ln(x+2 Pi) - 1 + ln(3)) < pi(x)
where Pi (with capital letter) is the famous constant 3.1415..
Another idea is the gap between primes is believed to be at most around ln(x)^2 within interval [1,x].
What is often known as Cramer's conjecture.
Ofcourse also the prime number theorem (PNT) is used :
pi(x) is close to x/ln(x) and even closer to x/(ln(x) -1) for large x.
PNT :=
Lim pi(x) ln(x) / x = 1
So the estimate is based on proven and unproven things depending mainly on the sizes of x and y and in particular the difference x-y.
There is room for investigating theoretically and numercially.
And in particular to see how it relates to other conjectures.
( firoozbahkt's conjecture for instance )
A generalization can be made for counting prime twins and other fixed gaps.
For instance for prime twins by replacing ln() with ln()^2.
Although it is a lower bound estimate , it relates to upper bound ideas too.
Thank you for your time.