This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ(<span style="white-space:nowrap;"><s...This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">τ) lie on critical line. Denote <img src="Edit_189dc2b2-73ef-4036-9f06-ecf8a47fe58b.png" width="140" height="16" alt="" />, and <img src="Edit_a8ec55cb-e4c4-4156-ba23-ae01a31d1bc8.png" width="110" height="22" alt="" /> on critical line. We have found two mysteries in Riemann’s paper. The first mystery is the equivalence: <img src="Edit_3c075830-3c6c-4a23-9851-5b7d219e8000.png" width="140" height="21" alt="" /> is uniquely determined by its initial value <span style="white-space:nowrap;">u (t). The second mystery is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;">t<sub>j</sub> of u (t) can uniquely express <img src="Edit_b15d9c18-b55b-49e3-97a1-d2e03ccb6343.png" width="175" height="23" alt="" />. We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation <img src="Edit_f5295ff4-90b2-4465-851a-cad140b181c8.png" width="305" height="20" alt="" />. It is known that on critical line <img src="Edit_b45bff49-6d09-456b-9d1f-4259c66293d3.png" width="310" height="23" alt="" /> and <img src="Edit_4182ba79-0fcb-4f84-b7e7-c7574406596e.png" width="85" height="26" alt="" />, then we have the upper bound of growth <img src="Edit_d3d84d75-cc56-47b8-a9a7-ef8a9a5f07b1.png" width="250" height="33" alt="" /> To prove RC2 (or RC), by contradiction. If <span style="white-space:nowrap;">ξ(τ) has conjugate complex roots t'<span style="white-space:nowrap;">±i<span style="white-space:nowrap;">β'’, <span style="white-space:nowrap;">β'>0, R<sup>2</sup>=t'<sup>2</sup>+<span style="white-space:nowrap;">β'<sup>2</sup>, by symmetry <span style="white-space:nowrap;">ξ(τ)=<span style="white-space:nowrap;">ξ(-τ), then -(t'<span style="white-space:nowrap;">±iβ'') do yet. So ξ must contain four factors. Then u(t) contains a real factor <img src="Edit_ac03c1a5-0480-4efa-aac4-7788852a42a9.png" width="225" height="22" alt="" /> and <span style="white-space:nowrap;">ln|u(t)| contains a term (the lower bound) <img src="Edit_6e94ad71-a310-4717-99ee-90384b0d89ba.png" width="230" height="19" alt="" /> which contradicts to the growth above. So <span style="white-space:nowrap;">ξ can not have the complex roots and u(t) does not have the factor p(t). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional <span style="white-space:nowrap;">u(t) is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.展开更多
文摘This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;">ξ(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;">τ) lie on critical line. Denote <img src="Edit_189dc2b2-73ef-4036-9f06-ecf8a47fe58b.png" width="140" height="16" alt="" />, and <img src="Edit_a8ec55cb-e4c4-4156-ba23-ae01a31d1bc8.png" width="110" height="22" alt="" /> on critical line. We have found two mysteries in Riemann’s paper. The first mystery is the equivalence: <img src="Edit_3c075830-3c6c-4a23-9851-5b7d219e8000.png" width="140" height="21" alt="" /> is uniquely determined by its initial value <span style="white-space:nowrap;">u (t). The second mystery is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;">t<sub>j</sub> of u (t) can uniquely express <img src="Edit_b15d9c18-b55b-49e3-97a1-d2e03ccb6343.png" width="175" height="23" alt="" />. We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation <img src="Edit_f5295ff4-90b2-4465-851a-cad140b181c8.png" width="305" height="20" alt="" />. It is known that on critical line <img src="Edit_b45bff49-6d09-456b-9d1f-4259c66293d3.png" width="310" height="23" alt="" /> and <img src="Edit_4182ba79-0fcb-4f84-b7e7-c7574406596e.png" width="85" height="26" alt="" />, then we have the upper bound of growth <img src="Edit_d3d84d75-cc56-47b8-a9a7-ef8a9a5f07b1.png" width="250" height="33" alt="" /> To prove RC2 (or RC), by contradiction. If <span style="white-space:nowrap;">ξ(τ) has conjugate complex roots t'<span style="white-space:nowrap;">±i<span style="white-space:nowrap;">β'’, <span style="white-space:nowrap;">β'>0, R<sup>2</sup>=t'<sup>2</sup>+<span style="white-space:nowrap;">β'<sup>2</sup>, by symmetry <span style="white-space:nowrap;">ξ(τ)=<span style="white-space:nowrap;">ξ(-τ), then -(t'<span style="white-space:nowrap;">±iβ'') do yet. So ξ must contain four factors. Then u(t) contains a real factor <img src="Edit_ac03c1a5-0480-4efa-aac4-7788852a42a9.png" width="225" height="22" alt="" /> and <span style="white-space:nowrap;">ln|u(t)| contains a term (the lower bound) <img src="Edit_6e94ad71-a310-4717-99ee-90384b0d89ba.png" width="230" height="19" alt="" /> which contradicts to the growth above. So <span style="white-space:nowrap;">ξ can not have the complex roots and u(t) does not have the factor p(t). Therefore both RC2 and RC are proved. We have seen that the two-dimensional problem is reduced to one-dimension and the one-dimensional <span style="white-space:nowrap;">u(t) is reduced to its product expression. Perhaps this is close to the original idea of Riemann. Other results are also discussed by geometric analysis in the last section.
