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.

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).

协同拟凸函数的Riemann-Liouville分数阶积分不等式

基于Riemann-Liouville分数阶积分,对协同拟凸函数的Hermite-Hadamard分数阶积分不等式进行了研究。剖析Riemann-Liouville分数阶积分的运算特点,深入探究SAR?KAYA给出的Riemann-Liouville分数阶积分恒等式。在该Riemann-Liouville分数阶积分恒等式的基础上,利用二元函数的单调性和协同拟凸性,巧妙应用三角不等式和H?lder不等式等经典不等式,建立了若干个协同拟凸函数的Hermite-Hadamard分数阶积分不等式。

基于Riemann-Liouville分数阶积分Bernstein-Kantorovich算子的逼近性质

文章构造一种基于Riemann-Liouville分数阶积分的Bernstein-Kantorovich算子。利用一阶、二阶光滑模、Peetre’-K泛函和Lipschitz型极大函数等工具研究该算子的近似性质。然后利用一阶、二阶和四阶中心矩对该算子建立Vorononskaja型渐进公式。该算子的构建使得在曲线曲面造型方面对于给定的函数将会有更小的近似误差。

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.

On the Absence of Zeros of Riemann Zeta-Function Out of ℜ(z) = 1/2

This work shows, after a brief introduction to Riemann zeta function , the demonstration that all non-trivial zeros of this function lies on the so-called “critical line”,, the one Hardy demonstrated in his famous work that infinite countable zeros of the above function can be found on it. Thus, out of this strip, the only remaining zeros of this function are the so-called “trivial ones” . After an analytical introduction reminding the existence of a germ from a generic zero lying in , we show through a Weierstrass-Hadamard representation approach of the above germ that non-trivial zeros out of cannot be found.

Fast Converging Series for Riemann Zeta Function

Riemann zeta function has a key role in number theory and in its applications. In this paper we present a new fast converging series for . Applications of the series include the computation of the and recursive computation of , and generally . We discuss on the production of irrational number sequences e.g. for encryption coding and zeta function maps for analysis and synthesis of log-time sampled signals.

RIEMANN BOUNDARY VALUE PROBLEMS FOR SOME K-REGULAR FUNCTIONS IN CLIFFORD ANALYSIS

In this paper,we study the R m（m〉0） Riemann boundary value problems for regular functions,harmonic functions and bi-harmonic functions with values in a universal clifford algebra C（Vn,n）.By using Plemelj formula,we get the solutions of R m（m〉0） Riemann boundary value problems for regular functions.Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions,we obtain the solutions of R m（m〉0） Riemann boundary value problems for harmonic functions and bi-harmonic functions.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

The behavior of the zeros in finite Taylor series approximations of the Riemann Xi function (to the zeta function), of modified Bessel functions and of the Gaussian (bell) function is investigated and illustrated in the complex domain by pictures. It can be seen how the zeros in finite approximations approach to the genuine zeros in the transition to higher-order approximation and in case of the Gaussian (bell) function that they go with great uniformity to infinity in the complex plane. A limiting transition from the modified Bessel functions to a Gaussian function is discussed and represented in pictures. In an Appendix a new building stone to a full proof of the Riemann hypothesis using the Second mean-value theorem is presented.

Special Values for the Riemann Zeta Function

The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression for representing the zeta function at the odd integer values 2n+1 for n a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers E2n and the polygamma functions ψ(2n)(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.

A Standard Method to Prove That the Riemann Zeta Function Equation Has No Non-Trivial Zeros

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.

Approach to Riemann Hypothesis by Combined Commensurable Step Function Approximation with Bonnet Method

To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.

Accelerated Series for Riemann Zeta Function at Odd Integer Arguments

Riemann zeta function is an important tool in signal analysis and number theory. Applications of the zeta function include e.g. the generation of irrational and prime numbers. In this work we present a new accelerated series for Riemann zeta function. As an application we describe the recursive algorithm for computation of the zeta function at odd integer arguments.

关于Riemann-Lebesgue引理的一个推广

通过“把Riemann-Lebesgue引理看做是在一个周期上可积的周期函数的一种性质”的这个角度,提出了亚周期函数的概念,并得到有界亚周期函数的刻画.

Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

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.

Computational Prooving of Riemann’s Hypothesis

Formulated in 1859 by the mathematician Bernhard Riemann, the Riemann hypothesis is a conjecture. She says that the Riemann’s Zeta function non-trivial zeros of all have real part . This demonstration would improve the prime numbers distribution knowledge. This conjecture constitutes one of the most important mathematics unsolved problems of the 21st century: it is one of the famous Hilbert problems proposed in 1900. In this article, a method for solving this conjecture is given. This work has been started by finding an analytical function which gives a best accurate 10-8 of particular zeros sample that this number has increased gradually and finally prooving that this function is always irrational. This demonstration is important as allows Riemann’s zeta function to be a model function in the Dirichlet series theory and be at the crossroads of many other theories. Also, it is going to serve as a motivation and guideline for new studies.

A Direct Proof for Riemann Hypothesis Based on Jacobi Functional Equation and Schwarz Reflection Principle

Using the properties of theta-series and Schwarz reflection principle, a proof for Riemann hypothesis (RH) is directly presented and the first ten nontrivial zeros are easily obtained. From now on RH becomes Riemann Theorem (RT) and all its equivalent results and the consequences assuming RH are true.

关于k-Riemann-Liouville分数阶积分的Ostrowski型积分不等式

引入k-Riemann-Liouville分数阶积分和h-凸函数,通过建立k-Riemann-Liouville分数阶积分的Ostrowski型积分不等式,构造了一些新的Hermite-Hadamard型积分不等式.相比较于已有的一些结果,所得结果在积分形式和数据点类型两个方面有一定的优势,使得Hermite-Hadamard型积分不等式适用范围更广.最后给出Ostrowski型k-Riemann-Liouville分数阶积分不等式在概率方面的应用.