Numerical Calculations to Grasp a Mathematical Issue Such as the Riemann Hypothesis

Abstract

This article presents the use of data processing to apprehend mathematical questions such as the Riemann Hypothesis (RH) by numerical calculation. Calculations are performed alongside graphs of the argument of the complex numbers ζ ( x + i y ) = a + i b and ξ ( x + i y ) = p + i q , in the critical strip. On the one hand, the two-dimensional surface angle tan − 1 ( b / a ) of the Riemann Zeta function ζ is related to the semi-angle of the fractional part of y 2 π ln ( y 2 π e ) and, on the other hand, the Ksi function ξ of the Riemann functional equation is analyzed with respect to the coordinates ( x , 1 − x ; y ) . The computation of the power series expansion of the ξ function with its symmetry analysis highlights the RH by the underlying ratio of Gamma functions inside the ξ formula. The ξ power series beside the angle of both surfaces of the ζ function enables to exhibit a Bézout identity a u + b v ≡ c between the components ( a , b ) of the ζ function, which illustrates the RH. The geometric transformations in complex space of the Zeta and Ksi functions, illustrated graphically, as well as series expansions, calculated by computer, make it possible to elucidate this mathematical problem numerically. A final theoretical outlook gives deeper insights on the functional equation’s mechanisms, by adopting a computer–scientific perspective.