This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>ξ</em></span>(<span style="...This paper will prove Riemann conjecture(RC): All zeros of <span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>ξ</em></span>(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>τ</em></span></span></span></span>)</span> 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. <em>The first mystery</em> 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;"><em>u</em> (<em>t</em>)</span>. <em>The second mystery</em> is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;"><em>t<sub>j</sub> </em></span>of <em>u</em> (<em>t</em>) 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;"><em>ξ</em>(<em>τ</em>)</span> has conjugate complex roots <em>t</em>'<span style="white-space:nowrap;">±<em>i</em><span style="white-space:nowrap;"><em>β</em></span>'’</span>, <span style="white-space:nowrap;"><em>β</em>'>0</span>, <em>R</em><sup>2</sup>=t'<sup>2</sup>+<span s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;"><em>ξ</em></span>(<span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><span style="white-space:nowrap;"><em>τ</em></span></span></span></span>)</span> 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. <em>The first mystery</em> 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;"><em>u</em> (<em>t</em>)</span>. <em>The second mystery</em> is Riemamm conjecture 2 (RC2): Using all zeros <span style="white-space:nowrap;"><em>t<sub>j</sub> </em></span>of <em>u</em> (<em>t</em>) 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;"><em>ξ</em>(<em>τ</em>)</span> has conjugate complex roots <em>t</em>'<span style="white-space:nowrap;">±<em>i</em><span style="white-space:nowrap;"><em>β</em></span>'’</span>, <span style="white-space:nowrap;"><em>β</em>'>0</span>, <em>R</em><sup>2</sup>=t'<sup>2</sup>+<span s