On the prime number theorem for arithmetic progressions (Q1119689)

In 1984 the author gave an ingenious elementary proof of the prime number theorem by showing \[ \overline{\lim}_{x\to \infty}(1/x)\quad | \sum_{n\leq x}\mu (n)| =0 \] [C. R. Acad. Sci., Paris, Sér. I 298, 161-164 (1984; Zbl 0565.10037)]. In the present paper the method is extended to arithmetic progressions. It seems to be most remarkable that the word ``character'' does not appear in the whole article. It has been proved in an elementary way by \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 2, 293-308 (1949; Zbl 0034.172)] that the statement \[ \overline{\lim}_{x\to \infty}(1/x)\quad | \sum_{n\leq x;n\equiv a(q)}\mu (n)| =0\quad ((a,q)=1) \] is equivalent to the prime number theorem in arithmetic progressions.