Uniqueness of $L$-functions and meromorphic functions under sharing of sets

Banerjee, Abhijit; Khoai, Ha Huy; Kundu, Arpita

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 967-985

In [4], the authors proved that the zero set of a uniqueness polynomial, satisfying some additional conditions, becomes a unique range set for $L$-functions. They also determined the conditions under which a polynomial becomes a strong uniqueness polynomial for $L$-functions. These results are the improved version of one result of [3]. In this paper we obtain a number of uniqueness theorems for $L$-functions in the extended Selberg class, which significantly extend the results of [3] and [4] in a new direction and improve them in some cases. From our results we can show some classes of unique range sets for $L$-functions which cannot be found by the results of [3] and [4].

Dans [4], les auteurs ont démontré que l’ensemble des zéros d’un polynôme d’unicité, satisfaisant certaines conditions supplémentaires, est un ensemble d’unicité pour les fonctions $L.$ Ils ont également déterminé les conditions sous lesquelles un polynôme est un polynôme d’unicité forte pour les fonctions $L.$ Ces résultats sont une version améliorée d’un résultat de [3]. Dans cet article, nous obtenons un certain nombre de théorèmes d’unicité pour les fonctions $L$ appartenant à la classe de Selberg étendue, qui étendent, de manière significative, les résultats de [3] et [4] dans une nouvelle direction et les améliorons dans certains cas. En utilisant ces résultats, nous pouvons exhiber certaines classes d’ensembles d’unicité pour les fonctions $L$, qui ne peuvent pas être trouvées avec les résultats de [3] et [4].

Classification : 11M36, 30D35
Keywords: Meromorphic functions, uniqueness, shared sets, small functions, $L$-functions.

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