Proof of Riemann Conjecture Based on Contradiction between Xi-Function and Its Product Expression

Riemann proved three results: analytically continue ζ(s) over the whole complex plane s =σ + it with a pole s =1;(Theorem A) functional equation ξ(t) = G(s0)ζ (s0), s0 =1/2 + it and (Theorem B) product expression ξ1(t) by all roots of ξ(t). He stated Riemann conjecture (RC): All roots of ξ (t) are real. We find a mistake of Riemann: he used the same notation ξ(t) in two theorems. Theorem B must contain complex roots;it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s0 =1/2 + it. Using all real roots rk and (true) complex roots zj = tj + iaj of ξ (z), define product expressions w(t), w(0) =ξ(0) and Q(t) > 0, Q(0) =1 respectively, so ξ1(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s0)| =ω} for small ω > 0. If ξ(t) has complex roots, then ω =ωQ(t) on L(ω). Finally in a large interval of the first module |z1|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.

Geometric Proof of Riemann Conjecture (Continued)

This paper will prove Riemann conjecture(RC): All zeros of ξ(τ) lie on critical line. Denote , and on critical line. We have found two mysteries in Riemann’s paper. The first mystery is the equivalence: is uniquely determined by its initial value u (t). The second mystery is Riemamm conjecture 2 (RC2): Using all zeros tj of u (t) can uniquely express . We find that the proof of RC is hidden in it. Our basic idea as follows. Consider functional equation . It is known that on critical line and , then we have the upper bound of growth To prove RC2 (or RC), by contradiction. If ξ(τ) has conjugate complex roots t'±iβ'’, β'>0, R2=t'2+β'2, by symmetry ξ(τ)=ξ(-τ), then -(t'±iβ'') do yet. So ξ must contain four factors. Then u(t) contains a real factor and ln|u(t)| contains a term (the lower bound) which contradicts to the growth above. So 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 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.

Geometric Proof of Riemann Conjecture

This paper proves Riemann conjecture (RH), i.e., that all the zeros in critical region of Riemann ξ -function lie on symmetric line σ =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for ξ = u + iv . Denote . Riemann proved that u is real and v ≡ 0 for β =0 (the symmetry). We prove that the zeros of u and v for β > 0 are alternative, so u (t,0) is the single peak. A geometric model was proposed. is called the root-interval of u (t,β) , if |u| > 0 is inside Ij and u = 0 is at its two ends. If |u (t,β)| has only one peak on each Ij, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for β > 0 were derived. By , the peak u (t,β) will develop toward its convex direction. Besides, ut (t,β) has opposite signs at two ends t = tj , tj+1 of Ij , also does, then there exists some inner point t′ such that v (t′,β) = 0. Therefore {|u|,|v|/β} in Ij form a peak-valley structure such that has positive lower bound independent of t ∈ Ij (i.e. RH holds in Ij ). As u (t,β) does not have the finite condensation point (unless u = const.), any finite t surely falls in some Ij , then holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.

Local Geometric Proof of Riemann Conjecture

Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function , , by geometric analysis, which has the symmetry: v=0 if β=0, and basic expression . We show that |u| is single peak in each root-interval of u for fixed β ∈(0,1/2]. Using the slope ut, we prove that v has opposite signs at two end-points of Ij. There surely exists an inner point such that , so {|u|,|v|/β} form a local peak-valley structure, and have positive lower bound in Ij. Because each t must lie in some Ij, then ||ξ|| > 0 is valid for any t (i.e. RH is true). Using the positivity of Lagarias (1999), we show the strict monotone for β > β0 ≥ 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”.

Proof of Riemann Conjecture

Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where zj are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots tj of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)4 . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.

Superconvergence of Continuous Finite Elements with Interpolated Coeffcients for Initial Value Problems of Nonlinear Ordinary Differential Equation

In this paper, n-degree continuous finite element method with interpolated coefficients for nonlinear initial value problem of ordinary differential equation is introduced and analyzed. An optimal superconvergence u-uh = O(hn+2), n ≥ 2, at (n + 1)-order Lobatto points in each element respectively is proved. Finally the theoretical results are tested by a numerical example.

The Symmetry of Riemann <i>ξ</i>-Function

To prove RH, studying ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function by geometric analysis, which has the symmetry: v = 0 if β = 0, and Assume that |u| is single peak in each root-interval of u for any fixed β ∈ (0,1/2], using the slope ut of the single peak, we prove that v has opposite signs at two end-points of Ij, there surely is an inner point so that v = 0, so {|u|,|v|/β}form a local peak-valley structure, and have positive lower bound in Ij. Because each t must lie in some Ij , then ||ξ|| > 0 is valid for any t. In this way, the summation process of ξ is avoided. We have proved the main theorem: Assume that u (t, β) is single peak, then RH is valid for any . If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of ξ, which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.

FINITE ELEMENT METHOD WITH SUPERCONVERGENCE FOR NONLINEAR HAMILTONIAN SYSTEMS

这份报纸从三个方面为非线性的 Hamiltonian 系统涉及有限元素方法：精力， symplicity，和全球错误的保存。学习有限元素方法的 symplecticity，我们使用所有分析方法而非通常使用的代数学的方法。我们在节点 t 证明集中的顺序最佳[n ] 在中间长的时间并且表明高精确性的 symplecticity。证明强烈取决于 superconvergence 分析。数字实验证明建议方法能保存精力很好并且能也在长时间使全球轨道错误小。[从作者抽象]

TIME-EXTRAPOLATION ALGORITHM （TEA） FOR LINEAR PARABOLIC PROBLEMS

伪制服上的 Crank-Nicolson 计划的快答案协调因为寓言的问题被讨论。到答案的减少整齐要求，首先，一些新错误估计被证明。第二，我们分析寓言的分离计划的二个特征，并且发现 Multigrid 方法(MG ) 的效率极大地被减少。数字实验比较效率直接结合坡度方法(DCG ) 和推测 Cascadic Multigrid 方法(EXCMG ) 。最后，我们建议一个时间推测算法(茶) ，它作为好起始的价值拿以前的几个水平答案的线性联合加速集中的率。一些典型推测公式数字地被比较。并且我们发现在某些精确性要求下面，为 3 顺序和 7 水平推测公式的 CG 重复计数关于 DCG 的的 1/3。因为 TEA 算法独立于空间尺寸，它为伪制服网孔仍然是有效的。当仅仅最好的格子被需要，建议方法被考虑为非线性的寓言的问题很有效。[从作者抽象]

Superconvergence for triangular cubic elements

A new structure of superconvergence for the cubic triangular finite element approximation u/2 to a second-order elliptic problem Au=f is studied based on some orthogonal expansions in an interval. Suppose that Ω is a convex polygonal domain with boundary L, its triangulation is uniform and Th is a set of vertexes and side midpoints of all elements. Then uh itself has no superconvergence points in Ω, while in any interior subdomain Ω0 the average gradient Duh has superconvergence D（dh-u）=O(hm+1lnh) at z∈Th∩Ω0（no other superconvergence points）. Furthermore, prescribe u=0 on L1.Then the superconvergence near L1 will surely disappear; if αavu+bu=0 on L3, where v is the conormal direction, the numercal experiments show superconvergence up to L3（the case of A=-Δ and b=0 has already been proved）.

Implicit DG Method for Time Domain Maxwell’s Equations Involving Metamaterials

An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

An Extrapolation Cascadic MultigridMethod for Elliptic Problems on Reentrant Domains

This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method.Recursive application of this idea results in the EXCMG method proposed in this paper.Finally,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.

Numerical Integration over Pyramids

Pyramidal elements are often used to connect tetrahedral and hexahedral elements in the finite element method.In this paper we derive three new higher order numerical cubature formulae for pyramidal elements.