Please enter verification code
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
Views 840      Downloads 540
Mei Xiaochun, Department of Theoretic Physics and Pure Mathematics, Institute of Innovative Physics in Fuzhou, Fuzhou, China
Article Tools
Follow on us
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.
Riemann Hypothesis, Riemann Zeta Function, Zeta Function Equation, Jacobi’s Function, Residue Theorem, Cauchy-Riemann Equation
To cite this article
Mei Xiaochun, The Inconsistency Problem of Riemann Zeta Function Equation, Mathematics Letters. Vol. 5, No. 2, 2019, pp. 13-22. doi: 10.11648/
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Riemann G. F. B., Uber die Anzabl der Primahlem unter einer gegebenen Grosse, Monatsberichte der Berliner Akademine, 1859, 2, 671- 680.
Bent E. Petersen, Riemann Zeta Function, https://pan.
Neukirch, J., Algebraic Number Theory, 1999, Springer, Berlin, Heidelberg (The original German edition was published in 1992 under the title Algebraische Zahlentheorie).
Iwaniec H., Lectures on the Riemann Zeta Function, American Mathematical Society, 2014, Providence.
Bender, C. M., Brody, D. C., M¨uller, M. P., Hamiltonian for the zeros of the riemann zeta function, Physical Review Letters, 2017, 118 (13), 130201.
Lagarias, J. C. An elementary problem equivalent to the Riemann hypothesis, The American Mathematical Monthly, 2002, 109 (5), 534–543.
Jacobi, C. G. J., Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Borntraeger, Konigsberg, 1829, (Reprinted by Cambridge University Press, 2012).
Mathematics Handbook, Scientific Publishing House, 1980,p. 144.
Guo Dunreng, The Methods of Mathematics and Physics, Education publishing House, 1965,p. 109.
Felix Rubin, Riemann's First Proof of the Analytic Continuation of Zeta function, http://www2.math. cation/bachelor/seminars/ws0607/modular-forms/Riemanns_first_proof. pdf.
Gerald Tenenbaum, Michel Mendes France, Les Nombres, Premiers, Presses Universitaires de France, 1997, 44-52.
Gelbart, S., Miller, S., Riemann’s zeta function and beyond, Bulletin of the American Mathematical Society, 2004, 41 (1), 59–112.
Jessen, B., Wintner, A., Distribution functions and the Riemann zeta function, Transactions of the American Mathematical Society, 1935, 38 (1), 48–88.
Ru Changhai, The Riemann Hypothesis, Qianghua University Publishing Company, 2016, p. 61, 52, 192.
Gourdon X., The 10^(13) first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004, pdf.
Katz N M, Sarnak P. Zeroes of zeta functions and symmetry [J]. AMS,1999,36 (1) 1-26.
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
Tel: (001)347-983-5186