Number Theory


On the Values of the Function Zeta and Gamma

Authors: Simon Plouffe

An analysis of the function 1/π Arg ζ((1/2)+in) is presented. This analysis permits to find a general expression for that function using elementary functions of floor and fractional part. These formulas bring light to a remark from Freeman Dyson which relates the values of the ζfunction to quasi-crystals. We find these same values for another function which is very similar, namely 1/π Arg Γ((1/4)+in/2). These 2 sets of formula have a definite pattern, the n’th term is related to values like π,ln⁡(π),ln⁡(2),…,log⁡(p), where p is a prime number. The coefficients are closed related to a certain sequence of numbers which counts the number of 0’s from the right in the binary representation of n. These approximations are regular enough to deduce an asymptotic and precise formula. All results presented here are empirical.

Comments: 13 Pages. The abstract in english and the main text in french

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Submission history

[v1] 2014-08-26 22:24:59

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