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

**Jorge Jimenez Urroz , Florian Luca, Michel Waldschmidt**

**Abstract:**We show that if \(F_n = 22^n+ 1\) is the nth Fermat number, then the binary digit sum of \(π(F_n)\) tends to infinity with \(n\), where \(π(x)\) is the counting function of the primes \(p ≤ x\). We also show that if \(F_n\) is not prime, then the binary expansion of \(φ(F_n)\) starts with a long string of 1’s, where \(φ\) 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 \(\mathbb{F}_{2}\). Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.