Hardy and Littlewood
proposed a conjecture about the asymptotic number of primes q of
the form n^2 + 1 less than x.
They suggested that this number is asymptotically given by
πq(x) ~ C * sqrt(x)/ln(x), where C = 1.37281346 ...
A better approximation of
the counting function πq(x) was obtained by a generalization of
πq(x) ~ C/2 * Int[1/(sqrt(u)*ln(u)), u = 2 to x]
1. Can one consider the following functions as better
πq(x) than C * sqrt(x)/ln(x)?
b) sqrt(2x)/(ln(x) - 1)
c) sqrt(2/x) * Li(x)
you find a value of x, greater than 10^20, such that sqrt(2x)/ln(x)
> πq(x) ?