A Strict Proof That the Riemann Zeta Function Equation Has No Non-trivial Zeros
DOI:
https://doi.org/10.9734/bpi/nramcs/v2/6049FKeywords:
Riemann hypothesis, Riemann zeta function, Riemann zeta function equation, Jacobi’s function, Residue theorem, Cauchy-Riemann equationAbstract
A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros on the whole complex plain. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Let \(\zeta(a+i b)=\zeta_{1}(a, b)+i \zeta_{2}(a, b)\) and comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is that \(\zeta_{1}(a, b)\) and \(\zeta_{2}(a, b)\) are equal to zero simultaneously. However, by using the compassion method of infinite series, it is proved that \(\zeta_{1}(a, b)\) and \(\zeta_{2}(a, b)\) can not be equal to zero simultaneously. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
