And surely it is much more likely that a large gap(n)/n exists for some n, than it is that gap(n) does not exist at all for some n.
The former is a lack of primes p,q that sum to 2n within a certain large range from about 3n/5 to 7n/5. The latter is a lack of primes p,q that sum to 2n within...
For any positive integer n>1, define gap(n) as
gap(n) = |n-p| = |n-q| for p,q the two primes closest to n such that p+q = 2n.
The Goldbach Conjecture is equivalent to the existence of p and q, and thus the existence of gap(n), for all n>1. (Or all n>2 if p and q are required to be odd...
You can find where the zeros of the Riemann zeta function are on the critical line Re(s)=1/2 by using the Riemann-Siegel formula. You can perform a good approximation of this formula on a calculator.
The Riemann-Siegel formula is a function that is positive where the Riemann zeta function is...