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.
Published in | Mathematics Letters (Volume 5, Issue 2) |
DOI | 10.11648/j.ml.20190502.11 |
Page(s) | 13-22 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2019. Published by Science Publishing Group |
Riemann Hypothesis, Riemann Zeta Function, Zeta Function Equation, Jacobi’s Function, Residue Theorem, Cauchy-Riemann Equation
[1] | Riemann G. F. B., Uber die Anzabl der Primahlem unter einer gegebenen Grosse, Monatsberichte der Berliner Akademine, 1859, 2, 671- 680. |
[2] | Bent E. Petersen, Riemann Zeta Function, https://pan. baidu.com/s/1geQsZxL. |
[3] | Neukirch, J., Algebraic Number Theory, 1999, Springer, Berlin, Heidelberg (The original German edition was published in 1992 under the title Algebraische Zahlentheorie). |
[4] | Iwaniec H., Lectures on the Riemann Zeta Function, American Mathematical Society, 2014, Providence. |
[5] | 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. |
[6] | Lagarias, J. C. An elementary problem equivalent to the Riemann hypothesis, The American Mathematical Monthly, 2002, 109 (5), 534–543. |
[7] | Jacobi, C. G. J., Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Borntraeger, Konigsberg, 1829, (Reprinted by Cambridge University Press, 2012). |
[8] | Mathematics Handbook, Scientific Publishing House, 1980，p. 144. |
[9] | Guo Dunreng, The Methods of Mathematics and Physics, Education publishing House, 1965，p. 109. |
[10] | Felix Rubin, Riemann's First Proof of the Analytic Continuation of Zeta function, http://www2.math. ethz.ch/edu cation/bachelor/seminars/ws0607/modular-forms/Riemanns_first_proof. pdf. |
[11] | Gerald Tenenbaum, Michel Mendes France, Les Nombres, Premiers, Presses Universitaires de France, 1997, 44-52. |
[12] | Gelbart, S., Miller, S., Riemann’s zeta function and beyond, Bulletin of the American Mathematical Society, 2004, 41 (1), 59–112. |
[13] | Jessen, B., Wintner, A., Distribution functions and the Riemann zeta function, Transactions of the American Mathematical Society, 1935, 38 (1), 48–88. |
[14] | Ru Changhai, The Riemann Hypothesis, Qianghua University Publishing Company, 2016, p. 61, 52, 192. |
[15] | Gourdon X., The 10^(13) first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004, http://pdfs.semanticscholar.org/6eff/62ff5d98e8ad2ad8757c0faf4bac87546f27. pdf. |
[16] | Katz N M, Sarnak P. Zeroes of zeta functions and symmetry [J]. AMS，1999，36 (1) 1-26. |
APA Style
Mei Xiaochun. (2019). The Inconsistency Problem of Riemann Zeta Function Equation. Mathematics Letters, 5(2), 13-22. https://doi.org/10.11648/j.ml.20190502.11
ACS Style
Mei Xiaochun. The Inconsistency Problem of Riemann Zeta Function Equation. Math. Lett. 2019, 5(2), 13-22. doi: 10.11648/j.ml.20190502.11
AMA Style
Mei Xiaochun. The Inconsistency Problem of Riemann Zeta Function Equation. Math Lett. 2019;5(2):13-22. doi: 10.11648/j.ml.20190502.11
@article{10.11648/j.ml.20190502.11, author = {Mei Xiaochun}, title = {The Inconsistency Problem of Riemann Zeta Function Equation}, journal = {Mathematics Letters}, volume = {5}, number = {2}, pages = {13-22}, doi = {10.11648/j.ml.20190502.11}, url = {https://doi.org/10.11648/j.ml.20190502.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20190502.11}, 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) 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.}, year = {2019} }
TY - JOUR T1 - The Inconsistency Problem of Riemann Zeta Function Equation AU - Mei Xiaochun Y1 - 2019/08/13 PY - 2019 N1 - https://doi.org/10.11648/j.ml.20190502.11 DO - 10.11648/j.ml.20190502.11 T2 - Mathematics Letters JF - Mathematics Letters JO - Mathematics Letters SP - 13 EP - 22 PB - Science Publishing Group SN - 2575-5056 UR - https://doi.org/10.11648/j.ml.20190502.11 AB - 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) 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. VL - 5 IS - 2 ER -