Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study...Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if <i>β</i>=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for <i>β</i> > <i>β</i><sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.</i>展开更多
By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show ...By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.展开更多
文摘Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying <i>ζ</i> and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function <img alt="" src="Edit_8fcdfff5-6b95-42a4-8f47-2cabe2723dfc.bmp" />, <img alt="" src="Edit_6ce3a4bd-4c68-49e5-aabe-dec3e904e282.bmp" />, <img alt="" src="Edit_29ea252e-a81e-4b21-a41c-09209c780bb2.bmp" /> by geometric analysis, which has the symmetry: v=0 if <i>β</i>=0, and basic expression <img alt="" src="Edit_bc7a883f-312d-44fd-bcdd-00f25c92f80a.bmp" />. We show that |u| is single peak in each root-interval <img alt="" src="Edit_d7ca54c7-4866-4419-a4bd-cbb808b365af.bmp" /> of <i>u</i> for fixed <em>β</em> ∈(0,1/2]. Using the slope u<sub>t</sub>, we prove that <i>v</i> has opposite signs at two end-points of I<sub>j</sub>. There surely exists an inner point such that , so {|u|,|v|/<em>β</em>} form a local peak-valley structure, and have positive lower bound <img alt="" src="Edit_bac1a5f6-673e-49b6-892c-5adff0141376.bmp" /> in I<sub>j</sub>. Because each <i>t</i> must lie in some I<sub>j</sub>, then ||<em>ξ</em>|| > 0 is valid for any <i>t</i> (<i>i.e.</i> RH is true). Using the positivity <img alt="" src="Edit_83c3d2cf-aa7e-4aba-89f5-0eb44659918a.bmp" /> of Lagarias (1999), we show the strict monotone <img alt="" src="Edit_87eb4e9e-bc7b-43e3-b316-5dcf0efaf0d5.bmp" /> for <i>β</i> > <i>β</i><sub>0</sub> ≥ 0 , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the un-known and method”.</i>
文摘By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.
