Analytic and asymptotic properties of non-symmetric Linnik's probability densities

The function, is a characteristic function of a probability distribution iff . This distribution is absolutely continuous; for =0 it is symmetric. The latter case was introduced by Linnik in 1953 13] and several applications were found later. The case 0 was introduced by Klebanov, Maniya, and Melamed in 1984 9], while some special cases were considered previously by Laha 12] and Pillai 18]. In 1994, Kotz, Ostrovskii and Hayfavi 10] carried out a detailed investigation of analytic and asymptotic properties of the density of the distribution for the symmetric case =0. We generalize their results to the non-symmetric case 0. As in the symmetric case, the arithmetical nature of the parameter plays an important role, but several new phenomena appear.