How to Prove Riemann Conjecture by Riemann’s Four Theorems

Riemann (1859) had proved four theorems: analytic continuation ζ(s), functional equation ξ(z)=G(s)ζ(s)(s=1/2+iz, z=t−i(σ−1/2)), product expression ξ1(z)and Riemann-Siegel formula Z(z), and proposed Riemann conjecture (RC): All roots of ξ(z)are real. We have calculated ξand ζ, and found that ξ(z)is alternative oscillation, which intuitively implies RC, and the property of ζ(s)is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ(t)=ξ1(t)to confuse two concepts. So the product expression only can be used in contraction. We find that if ξhas complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζhave real part 1/2. Of course, RH also holds, but can not be proved directly by ζ(s).

Presentation of the Berry-Tabor conjecture in Levy plates

The Berry-Tabor(BT)conjecture is a famous statistical inference in quantum chaos,which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena.In this paper,the BT conjecture has been extended to Lévy plates.As predicted by the BT conjecture,level clustering is present in the spectra of Lévy plates.The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P(r),which is calculated through the analytical solution obtained by the Hamiltonian approach.Our work investigates the impact of varying foundation parameters,rotary inertia,and boundary conditions on the frequency spectra,and we find that P(r)conforms to a Poisson distribution in all cases.The reason for the occurrence of the Poisson distribution in the Lévy plates is the independence between modal frequencies,which can be understood through mode functions.

Natural Numbers and the Strong Goldbach Conjecture

This study introduces the representation of natural number sets as row vectors and pretends to offer a new perspective on the strong Goldbach conjecture. The natural numbers are restructured and expanded with the inclusion of the zero element as the source of a strong Goldbach conjecture reformulation. A prime Boolean vector is defined, pinpointing the positions of prime numbers within the odd number sequence. The natural unit primality is discussed in this context and transformed into a source of quantum-like indetermination. This approach allows for rephrasing the strong Goldbach conjecture, framed within a Boolean scalar product between the prime Boolean vector and its reverse. Throughout the discussion, other intriguing topics emerge and are thoroughly analyzed. A final description of two empirical algorithms is provided to prove the strong Goldbach conjecture.

The Erdös-Faber-Lovász Conjecture for Gap-Restricted Hypergraphs

An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.

The Proofs of Legendre’s Conjecture and Three Related Conjectures

In this paper, we prove Legendre’s conjecture: There is a prime number between n2 and (n +1)2 for every positive integer n. We also prove three related conjectures. The method that we use is to analyze binomial coefficients. It is developed by the author from the method of analyzing binomial central coefficients, that was used by Paul Erdős in his proof of Bertrand’s postulate - Chebyshev’s theorem.

A Comparative Analysis of the New -3(-n) - 1 Remer Conjecture and a Proof of the 3n + 1 Collatz Conjecture

This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true.

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.

Collatz Conjecture Redefinition on Prime Numbers

The definition of Collatz Operator, the mathematical avatar of the Collatz Algorithm, permits the transformation of the Collatz conjecture, which is delineated over the whole natural number set, into an equivalent inference restricted to the odd prime number set only. Based on this redefinition, one can describe an empirical-heuristic proof of the Collatz conjecture.

关于商高数的Jesmanowicz猜想

本研究主要利用简单同余、二次剩余、κ次剩余、四次剩余特征理论及因式分解法,对关于不定方程a^(x)+b^(y)=c^(z)的Jesmanowicz猜想的一类特殊情形进行证明,并得到如下结论:定理对于商高数组a=n^(2)-4,b=4n,c=n^(2)+4,2×n,当n+2含有素因子p■-1(mod 16)时,Jesmanowicz猜想成立.特别地,有推论对于上述商高数组,当n■-1(mod 16)时,Jesmanowicz猜想成立.

YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH

The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed, by G. Liu in [23]. In the first part, we will give a survey on thc progress. In the second part, we will consider Yau＇s conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number Cn1 is an essential condition to solve Yau＇s conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, Cn1 is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau＇s uniformization conjecture on Kahler manifolds with minimal volume growth.

A New Method to Study Goldbach Conjecture

This paper does not claim to prove the Goldbach conjecture, but it does provide a new way of proof (LiKe sequence);And in detailed introduces the proof process of this method: by indirect transformation, Goldbach conjecture is transformed to prove that, for any odd prime sequence (3, 5, 7, …, Pn), there must have no LiKe sequence when the terms must be less than 3 × Pn. This method only studies prime numbers and corresponding composite numbers, replaced the relationship between even numbers and indeterminate prime numbers. In order to illustrate the importance of the idea of transforming the addition problem into the multiplication problem, we take the twin prime conjecture as an example and know there must exist twin primes in the interval [3Pn, P2n]. This idea is very important for the study of Goldbach conjecture and twin prime conjecture. It’s worth further study.

THE WEINSTEIN CONJECTURE IN PRODUCT OF SYMPLECTIC MANIFOLDS

In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product P;×P;of two strongly geometrically bounded symplectic manifolds under some conditions with P;. In particular, if N is a closed manifold or a noncompact manifold of finite topological type, our result implies that the Weinstein conjecture in CP;×T*N holds.

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”.

Inequalities relating to L_p-version of Petty's conjectured projection inequality

Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.

THE WEINSTEIN CONJECTURE ON C^1-SMOOTH HYPERSURFACE OF CONTACT TYPE

In this note a symplectic capacity of Hofer-Zehnder type that is only invariant under C-1-symplectomorphisms is defined and all computation formulae for Hofer-Zehnder symplectic capacity obtained at present are proved still holding for it. As a consequence some results on Weinstein conjecture are generalized to C-1-smooth hypersurface of contact type.

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.

AN EQUIVALENT PROPOSITION TO THE CONJECTURE OF GOLDBACH

In this paper,a formula is given. The formula gives the number of prime number solutions of the indefinite equation p 1+p 2=2n , and based on it, an equivalent proposition to the conjecture of Goldbach is obtained.

On Kendall's Conjecture for an Extended Birth-death Q-processes with Instantaneous State and Catastrophes

A new structure with the special property that instantaneous state and catas-trophes is imposed to ordinary birth-death processes is considered. Kendall's conjecture forthe processes is proved to be right.

Helliwell-Konkowski Conjecture and the Singularities of Cauchy Horizons in 1 + 1-Dimensional Black Holes

The stability conjecture of Cauchy horizons of black holes suggested by Helliwell and Konkowski is used to investigate the 1+1-dimensional(2D)black holes under perturbations of infalling null dust and of both infalling and outgoing null dust.The result given by this conjecture agrees with that from mass inflation scenario for 2D charged dilaton black hole.For 2D black holes,we show that the Cauchy horizons are unstable and the corresponding singularities exist.

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.