| Summary: | 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ζ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ξ</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mi>i</mi> <mi>q</mi> </mrow> </semantics> </math> </inline-formula>, in the critical strip. On the one hand, the two-dimensional surface angle <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mi>tan</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>/</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of the Riemann Zeta function <inline-formula> <math display="inline"> <semantics> <mi>ζ</mi> </semantics> </math> </inline-formula> is related to the semi-angle of the fractional part of <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mi>y</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> <mi>ln</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>y</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and, on the other hand, the Ksi function <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> of the Riemann functional equation is analyzed with respect to the coordinates <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>;</mo> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. The computation of the power series expansion of the <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> function with its symmetry analysis highlights the RH by the underlying ratio of Gamma functions inside the <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> formula. The <inline-formula> <math display="inline"> <semantics> <mi>ξ</mi> </semantics> </math> </inline-formula> power series beside the angle of both surfaces of the <inline-formula> <math display="inline"> <semantics> <mi>ζ</mi> </semantics> </math> </inline-formula> function enables to exhibit a Bézout identity <inline-formula> <math display="inline"> <semantics> <mrow> <mi>a</mi> <mi>u</mi> <mo>+</mo> <mi>b</mi> <mi>v</mi> <mo>≡</mo> <mi>c</mi> </mrow> </semantics> </math> </inline-formula> between the components <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> of the <inline-formula> <math display="inline"> <semantics> <mi>ζ</mi> </semantics> </math> </inline-formula> 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.
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