JP Journal of Algebra, Number Theory and Applications
Volume 26, Issue 1, Pages 75 - 101
(August 2012)
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ON A RAPIDLY CONVERGING SERIES FOR THE RIEMANN’S ZETA FUNCTION
Alois Pichler
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Abstract: To evaluate Riemann’s zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann’s zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allows to reduce Riemann’s hypothesis to similar properties on polynomials. And interestingly, Laguerre’s polynomials form a kind of leitmotif through all sections. |
Keywords and phrases: Riemann zeta function, Kummer function, Laguerre polynomials, Fourier transform, Riemann hypothesis. |
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