> and came to a conjecture:

However, if the k-tuple conjecture holds, the lim sup will be O[d/log(d)]:

>

> lim sup | pi(n+d) - pi(n) - integral(1/log(t), t from n to n+d) | = O[sqrt(d)]

> where d and n are positive integers and lim sup is taken for n-->inf.

http://mathworld.wolfram.com/k-TupleConjecture.html

http://www.opertech.com/primes/k-tuples.html