In the Titchmarsh treatise "The Theory of the Riemann zeta function", at page 159 of the first edition, we have a lemma at paragraph 8.7. I joined an image
My question is about the proof of the lemma and on the condition "let every m with b_m <>0 be prime to every n with c_n <>0. What is the signification of the condition ? Is it a normal product ? The text of Bohr and Landau « Nachtrag zu unsere Abhandlungen aus den Jahrgäng 1910 und 1923 », Göttinger Nachrichten 1924, 168-172 Translated from Deutsch to English by Titchmarsh say exactly : "in $g(s)h(s) = \sum_{r=1}^\infty\frac{d_r}{r^s}$ entsteht nämich jedes $d_r \ne 0$ nur aus einem product $b_n c_m$ mit $nm=r$". So "in $g(s)h(s) = \sum_{r=1}^\infty\frac{d_r}{r^s}$, each $d_r \ne 0$ is from only a product $b_n c_m$ with $nm=r$". So there is a slight alteration of the original paper. Can you give me a full proof of the lemma ? Thanks you very much.