AMS Subject Classification 2010: 11M26, 33C45, 42A38. Necessary and sufficient conditions for absence of zeros of ζ(s) in the half-plane σ ... Expansion of holomorphic functions in series of Hermite polynomials ...
Riemann’s memoir is devoted to the function π(x) defined as the number of prime numbers less or equal to the real and positive number x. This is really the fact, but the “main role” in it is played by the already mentioned zeta-function.
Mathematics Subject Classification: 33C45. A more general form for a classical generating function for the Jacobi polynomials is given. * Partially supported by Project MM 1305 - National Science Fund, Bulgarian Ministry of Educ. Sci.
It is proved that if the increasing sequence kn n=0..∞ n=0 of nonnegative integers has density greater than 1/2 and D is an arbitrary simply connected subregion of CRthen the system of Hermite associated functions Gkn(z) n=0..∞ is complete in the space H(D) of complex functions holomorphic in D.
[Rusev Peter; Russev Peter; Russev P.; Русев Петр; Русев Петър]
[Rusev Peter; Russev Peter; Russev P.; Русев Петр; Русев Петър] Necessary and sufficient conditions for absence of zeros of the function ζ(s), s=σ+it, in the half-plane σ>θ, 1/2≤θ<1 are proposed in terms of representations of holomorphic functions by series in Hermite and Laguerre polynomials as well as in terms of Fourier and Hankel integral transforms. 2010 Mathematics Subject Classification: 11M26, 33C45, 42A38.
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