The Mellin transform of powers of the zeta-function
نویسندگان
چکیده
منابع مشابه
The Modified Mellin Transform of Powers of the Zeta-function
The modified Mellin transform Zk(s) = ∫ ∞ 1 |ζ( 1 2 + ix)|x dx (k ∈ N) is investigated. Analytic continuation and mean square estimates of Zk(s) are discussed, as well as connections with power moments of |ζ( 1 2 +ix)|, with the special emphasis on the cases k = 1, 2.
متن کاملThe Mellin transform of the square of Riemann’s zeta-function
This function, when k = 2, was introduced by Y. Motohashi [15] (see also [16]), and its properties were further studied in [10] and [11]. The latter work also contains some results on the function Z1(s), which is the principal object of the study in this paper. It was shown that Z1(s) is regular for σ > −3/4, except for a double pole at s = 1. The principal part of the Laurent expansion of Z1(s...
متن کاملThe Mellin transform of the square of Riemann's zeta-function
This function, when k = 2, was introduced by Y. Motohashi [15] (see also [16]), and its properties were further studied in [10] and [11]. The latter work also contains some results on the function Z1(s), which is the principal object of the study in this paper. It was shown that Z1(s) is regular for σ > −3/4, except for a double pole at s = 1. The principal part of the Laurent expansion of Z1(s...
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Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the result is naturally split into the same three contributions of very different nature, i.e. the series of Riemann zeta functions and the power and negative e...
متن کاملThe Laplace and Mellin transforms of powers of the Riemann zeta-function
AMS subject classification: Primary 11M06, Secondary 11F72
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2000
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-95-4-305-342