Some remarks on a problem of Chowla (Q2888164)

In 1969, \textit{S. Chowla} [J. Number Theory 2, 120--123 (1970; Zbl 0211.07005)] posed the following question; Does there exist a rational-valued arithmetic function \(f\), periodic with prime period \(p\) such that \(\sum_{n=1}^\infty {f(n) \over n}\) converges and equals zero?NEWLINENEWLINEIn 1973, \textit{A. Baker, B. J. Birch} und \textit{E. Wirsing} [J. Number Theory 5, 224--236 (1973; Zbl 0267.10065)] proved the following stronger assertion. If \(f\) is a non-zero periodic arithmetic function with values in a algebraic number field \(K\) and period \(q\) such that \(f(n)=0\) whenever \(1< (n,q)<q\) and \(K\) is linearly disjoint with the \(q\)-th cyclotomic field, then NEWLINE\[NEWLINE \sum_{n=1}^\infty {f(n) \over n} \neq 0. NEWLINE\]NEWLINE In particular, if \(f\) is rational valued, the second condition holds trivially. If \(q\) is prime, then the first condition is vacuous. Thus it resolves Chowla's question. In their paper, Baker, Birch and Wirsing state that Chowla had also solved his conjecture, but no indication of Chowla's proof was given.NEWLINENEWLINEIn this interesting article the author, based on Chowla's earlier published works, constructs a proof which Chowla might have thought of. Further, the author modifies the arguments to deduce the general result of Baker, Birch and Wirsing for prime moduli. In a subsequent paper, the author along with \textit{V. K. Murty} [J. Number Theory 131, No. 9, 1723--1733 (2011; Zbl 1241.11083)] extended the arguments to composite moduli by exploiting the properties of Ramachandra units, hence giving a new proof of the theorem of Baker-Birch-Wirsing.