Reference needed for proof of a Tauberian theorem

Question

I was reading the following paper by Hubert Delange: http://www.numdam.org/article/ASENS_1956_3_73_1_15_0.pdf ¹. In page 26, he provides a proof of Theorem b, the bulk of which relies on a result in some reference provided by him in footnote 5. However, despite my best attempts, I have been unable to find the aforementioned reference. Any proof or reference with a complete proof of the theorem would be really appreciated. Thanks a lot.

¹Delange, Hubert, Sur la distribution des entiers ayant certaines propriétés, Ann. Sci. Éc. Norm. Supér., III. Sér. 73, 15-74 (1956). ZBL0072.27501, MR0083519.

Answer 1

As I don't speak French, this is partially a guess. But I think this is paper you're looking for. (Maybe somebody else will be able to suggest other references for the same result.)

Looking at the footnote 5 saying "Loc. cit., p. 235-236 et 239." and the footnote 4 saying "Ann. Se. Ec. Norm. Sup., (3), l. 74, 1934' p. '213-242." - this paper seems as a reasonable guess.

Delange, Hubert, Généralisation du théorème de Ikehara, Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 213-242 (1954). ZBL0056.33101, MR0068667.

The paper you linked says about Theorem b: "Ces deux théorèmes se déduisent immédiatement des théorèmes III et IV de notre travail cité plus haut⁵". The paper "Généralisation du théorème de Ikehara" indeed contains Théorème III on pages 235--236 and Théorème IV on page 239.

Archive with old issues Annales Scientifiques de l'École Normale Supérieure is available at NUMDAM.