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The Inconsistency Problem of Riemann Zeta Function Equation
Mathematics Letters
Volume 5, Issue 2, June 2019, Pages: 13-22
Received: Jul. 8, 2019; Accepted: Jul. 31, 2019; Published: Aug. 13, 2019
Author
Mei Xiaochun, Department of Theoretic Physics and Pure Mathematics, Institute of Innovative Physics in Fuzhou, Fuzhou, China
Article Tools Abstract PDF (485KB)
Abstract
Four basic problems in Riemann’s original paper are found. The Riemann hypothesis becomes meaningless. 1. It is proved that on the real axis of complex plane, the Riemann Zeta function equation holds only at point Re(s)=1/2 (s = a+ib). However, at this point, the Zeta function is infinite, rather than zero. At other points of real axis, the two sides of Zeta function equation are contradictory. When one side is finite, another side may be infinite. 2. An integral item around the original point of coordinate system was neglected when Riemann deduced the integral form of Zeta function. The item was convergent when Re(s) > 1 but divergent when Re(s) < 1. The integral form of Zeta function does not change the divergence of its series form. Two reasons to cause inconsistency and infinite are analyzed. 3. When the integral form of Zeta function was deduced, a summation formula was used. The applicable condition of this formula is x > 0. At point x = 0, the formula is meaningless. However, the lower limit of Zeta function integral is x = 0, so the formula can not be used. 4. A formula of Jacobi function was used to prove the symmetry of Zeta function equation. The applicable condition of this formula was also x > 0. However, the lower limit of integral in the deduction was x=0. So this formula can not be used too. The zero calculation of Riemann Zeta function is discussed at last. It is pointed out that because approximate methods are used, they are not the real zeros of strict Riemann Zeta function.
Keywords
Riemann Hypothesis, Riemann Zeta Function, Zeta Function Equation, Jacobi’s Function, Residue Theorem, Cauchy-Riemann Equation
Mei Xiaochun, The Inconsistency Problem of Riemann Zeta Function Equation, Mathematics Letters. Vol. 5, No. 2, 2019, pp. 13-22. doi: 10.11648/j.ml.20190502.11
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