Problems & Puzzles: Puzzles

Puzzle 454. Number of primes of the form n^2 + 1

Patrick Capelle proposes the following puzzle:

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 this conjecture:
πq(x) ~ C/2 * Int[1/(sqrt(u)*ln(u)), u = 2 to x]


1. Can one consider the following functions as better approximations of
πq(x) than C * sqrt(x)/ln(x)?

a) sqrt(2x)/ln(x)
b) sqrt(2x)/(ln(x) - 1)
c) sqrt(2/x) * Li(x)

2. Can you find a value of x, greater than 10^20, such that sqrt(2x)/ln(x) > πq(x) ?


Contribution came from Enoch Haga.


Enoch Haga has produced for Q1 computations & charts that "illustrate the superiority of 1c over 1b and of 1b over 1a...The chart "Advantage of 1c over 1b", based on data in Wolf's Table I, columns 3 and 5, shows dramatically the great deviation of 1b from the true values at increasing powers of 10. The chart "Advantage of 1b over 1a" is
based on my own calculations



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