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.


1*Delange, Hubert*, [**Sur la distribution des entiers ayant certaines propriétés**](http://dx.doi.org/10.24033/asens.1041), Ann. Sci. Éc. Norm. Supér., III. Sér. 73, 15-74 (1956). [ZBL0072.27501](https://zbmath.org/?q=an:0072.27501), [MR0083519](https://mathscinet.ams.org/mathscinet-getitem?mr=0083519).



Reference needed for proof of a Tauberian theorem
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**](http://dx.doi.org/10.24033/asens.1023), Ann. Sci. Éc. Norm. Supér., III. Sér. 71, 213-242 (1954). [ZBL0056.33101](https://zbmath.org/?q=an:0056.33101), [MR0068667](https://mathscinet.ams.org/mathscinet-getitem?mr=0068667).


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](https://en.wikipedia.org/wiki/Annales_Scientifiques_de_l'%C3%89cole_Normale_Sup%C3%A9rieure) is available at NUMDAM.


