General Mathematics

   

Euler Already Knew that the ζ Function Would not be Zero Unless X = 0.5 for Complex Numbers

Authors: Toshiro Takami

\ zeta (s) = \ zeta (1 - s) This is a formula that does not hold unless the real part (x) is 0.5. For example, when the real part (x) is 0.6 (that is, x = 0.6), if 1-s, then x = 0.4, but the formula above is not satisfied. However, no matter what value x takes, the imaginary part becomes symmetric (same value) on the x axis (that is, y = 0). The above formula shows that the ζ function does not become zero unless x = 0.5. That is, Euler already knows that the ζ function does not become zero unless x = 0.5 for complex numbers.

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[v1] 2018-10-09 02:18:45

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