Gaps in binary expansions of some arithmetic functions and the irrationality of the Euler constant

Abstract

We show that if Fn=22n+1Fn=22n+1 is the nth Fermat number, then the binary digit sum of π(Fn)π(Fn) tends to infinity with nn, where π(x)π(x) is the counting function of the primes p≤xp≤x. We also show that if FnFn is not prime, then the binary expansion of φ(Fn)φ(Fn) starts with a long string of 1’s, where φφ is the Euler function. We also consider the binary expansion of the counting function of irreducible monic polynomials of degree a given power of 2 over the field F2F2. Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.