Xiaolong Wu
This paper shows that, in the critical strip, the Riemann zeta function $ζ(s)$ have the same set of zeros as $F(s):=\int_0^\infty t^{s-1}(e^t+1)^{-1}dt$, and then discusses the behavior of $F(s)$.
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An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no unique solution for any given non-zero complex numbers which means Riemann zeta function is entirely divergent. Infinitely many zeros of Riemann zeta function produced unfortunately those zeros also gives us non-zero values of Riemann zeta function. Among those zeros some of them are zeros of Riemann hypothesis. The present paper also discussed on eta function(…
In this article, with a new approach, which is not discussed in the literature yet, the limit of the Riemann zeta function or Euler-Riemann zeta function is approximately explored by applying Dirichlet's rearrangement theorem for absolutely convergent series to the Riemann zeta function by rearranging its terms as geometric series for sufficiently large $n$. The limit of the Riemann zeta function or Euler-Riemann zeta functions, $\lim_{n\to\infty} ζ(z)$, is first time explored. The limit obtained here is a very promising for the nontrivial or complex zeros of the Rieman zeta function.
On the complex zeros of the Riemann Zeta-function
本文证明Zeta函数的拟黎曼假设可推出黎曼假设，从而为验证黎曼假设提供新途径。
A proof of the Riemann hypothesis using the reflection principle is presented.
The main result of the paper is a definition of possible ways of the confirmation of the Riemann hypothesis based on the properties of the vector system of the second approximate equation of the Riemann Zeta function. The paper uses a feature of calculating the Riemann Zeta function in the critical strip, where its approximate value is determined by partial sums of the Dirichlet series, which it is given. These expressions are called the first and second approximate equation of the Riemann Zeta function. The representation of the terms of the Dirichlet series by vectors allows us when analyzin…
In this paper, we introduce a geometrical summation method that makes the original Riemann series converge over the critical strip. This method gives an analytical function, that coincides with zêta. This point of view allows us to introduce a quantity of interest that seems to give a characterization of the non-trivial zeros of the Riemann zêta function.
Applying Littlewood's lemma in connection to Riemann's Hypothesis and exploiting the symmetry of Riemann's $xi$ function we show that almost all nontrivial Riemann's Zeta zeros are on the critical line.
In this paper, we present a proof of the Riemann hypothesis. We show that zeros of the Riemann zeta function should be on the line with the real value 1/2, in the region where the real part of complex variable is between 0 and 1.
Some Series Related to Extended Riemann Hypothesis for Dedekind Zeta Functions
本文在广义黎曼假设下，推导了与狄利克雷ζ函数相关的级数闭式，并证明闭式成立则所有高零点位于临界线，从而获得黎曼假设的等价陈述。
A Qualitative Property of the Riemann zeta function
论文被作者撤回。它包含各种错误。
Proof of generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions
在Hecke证明$ζ_{k}$的函数方程和[ $M_{A}$ ]（黎曼假设的代数证明）中证明黎曼假设的方法的基础上，给出了代数数域$k$ -的zeta函数$ζ_{k}$的广义黎曼假设（简称gRH）的一个简短证明. Dirichlet L-函数的广义黎曼假设是$ζ_{k}$的（gRH）和连接Dedekind zeta与L-函数的适当乘积公式的直接结果。
We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix has two dimensions, and the second member is the vector (1, $-ζ(2)$). The elements of this vector are linearly independent on the Rationals; and from this independence, we determined that $ζ(j)$ is irrational for each j=3,4,5,6,7,8,9,....
The cycle problem: an intriguing periodicity to the zeros of the Riemann zeta function
对n^rho的对数积分的实部值求和，其中rho是黎曼ζ函数的连续零点系列之一，揭示了总和的意外周期性。这就是周期问题。
An equation satisfied by all non-trivial zeros $ρ$ of the Riemann zeta function $ζ$
证明了如果$ρ$是黎曼ζ函数$ζ$的非平凡零点，则$$2^ρ+ \frac{1}{ρ- 1} + 1/2 = ρ\int_{1}^{\infty} {t + 1/2} t^{-ρ-1} dt$$，其中，${x}$是$x$的分数部分.
Recursion Relations and Functional Equations for the Riemann Zeta Function
引入了黎曼ζ函数的新的递推关系。它们的推导是从标准的函数方程开始的。新的函数方程具有实增量和虚增量两种形式，可应用于整个复平面。我们已经开发了各种版本的递归关系消除每个系数函数，留下平原zeta函数
The functional equation for Riemann's Zeta function is studied, from which it is shown why all of the non-trivial, full-zeros of the Zeta function $ζ(s)$ will only occur on the critical line {$σ=1/2$} where {$s=σ+I ρ$}, thereby establishing the truth of Riemann's hypothesis. Further, two relatively simple transcendental equations are obtained; the numerical solution of these equations locates all of the zeros of {$ζ(s)$} on the critical line.
On Elementary Methods To Evaluate Values of The Riemann Zeta Function and Another Closely Related Infinite Series At Natural Numbers
本文给出了求黎曼ζ函数在自然数为偶数时的值以及求一个密切相关的级数在自然数为奇数时的值的一种初等方法。还介绍了另一种方法，特别是用于评价$ζ(2),$。
The zeros of the Riemann-zeta function and the transition from pseudo-random to harmonic behavior
本文基于黎曼ζ函数的非平凡零点，提出了一种新的函数。这样的函数表现出一个有趣的行为：当函数的自变量增加时，它从伪随机行为变为频率降低的谐波行为。
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