Squarefree ideals and an assertion of Ramanujan. (Q1860676)

The main result of this paper states that asymptotically one-half of the squarefree ideals in a quadratic number field have an odd (or even) number of prime factors. This means essentially that the analogue of the Möbius function for ideals of a quadratic number field has its mean value equal to zero. Although this result is true for all algebraic number fields with a standard proof (one should apply Ikehara's theorem to \(\zeta_K(s)\) and to \(\zeta_K(s)+1/\zeta_K(s)\) and compare the results), the presented proof is not correct. The error lies in the proof of Lemma B where two series having infinitely many nonzero integral terms are considered to be convergent.