An Elementary Problem Equivalent to the Riemann Hypothesis
Jeffrey C. Lagarias
本文证明了黎曼假设等价于一个包含调和数H_n（1到n的整数倒数之和）的初等不等式序列。这是对盖伊·罗宾的一个标准的修改。
This paper shows the equivalence of the Riemann hypothesis to an sequence of elementary inequalities involving the harmonic numbers H_n, the sum of the reciprocals of the integers from 1 to n. It is a modification of a criterion due to Guy Robin.
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In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L-functions.
Identities involving Mobius function values (u(j),u(k)) are used to generate a Riemann Hypothesis equivalent.
A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis
我们证明了Riemann的xi函数是严格增加（分别，严格减少）的模沿任何零自由，开放的右（分别，左）半平面中的每一个水平半线。一个推论是对黎曼假设的重新表述。
A proof of Riemann's hypothesis via Hadamard-Weierstrass factorization
利用黎曼ξ函数的哈达玛 - 魏尔斯特拉斯因式分解定理，对黎曼假设进行讨论与证明。
A new necessary and sufficient condition for the Riemann hypothesis
本文给出了黎曼假设的一个新的等价条件，该条件是黎曼ζ函数的值在偶数正整数处的有限有理组合的序条件.
In this note, we give an equivalent condition for a self-dual weight enumerator of genus three to satisfy the Riemann hypothesis. We also observe the truth and falsehood of the Riemann hypothesis for some families of invariant polynomials.
Character Sums and the Riemann Hypothesis
本文证明一个看似简单的不等式蕴含黎曼猜想，并提出通过Legendre符号和来研究该不等式的方法。
We show that the $θ=\infty$ conjecture implies the Riemann hypothesis.
On the Riemann Hypothesis for Drinfeld Modules
本文通过仅使用全球函数域和Drinfeld模理论中的基本概念，给出了Drinfeld模黎曼猜想的简短证明。
On the Balazard-Saias criterion for the Riemann Hypothesis
通过改变狄利克雷级数的截断值，对黎曼假设的Balazard-Saias准则进行了修正.他们建立了一些与修正准则有关的渐近结果。
On the zeros of the second derivative of the Riemann zeta function under the Riemann hypothesis
Berndt、Levinson、Montgomery和Akatsuka研究了黎曼ζ函数的导数的零点个数和非实零点实部的分布。伯恩特，莱文森和蒙哥马利调查的一般情况下，同时赤冢给尖锐的估计一阶导数的黎曼ζ函数的真理下的黎曼假设。本文将Akatsuka的结果推广到黎曼ζ函数的二阶导数。
In this short note I present a tauberian conjecture that I consider to be the simplest and the best tauberian reformulation of RH using good variation theory. The method applies also to the Grand Riemann Hypothesis.
We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an equivalence between such asymptotics and the Riemann Hypothesis.
We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3\cdot10^{12}$. That is, all zeroes $β+ iγ$ of the Riemann zeta-function with $0<γ\leq 3\cdot 10^{12}$ have $β= 1/2$.
Some Sufficient Conditions for the Riemann hypothesis
黎曼假设（RH）是众所周知的。本文给出了RH的几个充分条件。第一个条件与因子和函数有关，另一个条件与切比雪夫函数有关。
Modulus representation of the Riemann $ξ$ function and polynomial inequalities equivalent to the Riemann hypothesis
本文利用 Jacobi theta 函数给出 Riemann ξ 函数的模表示，并证明 Riemann 假设等价于一族多项式不等式，同时给出初步结果。
We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.
We prove an explicit error term for the $ψ(x,χ)$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.
In this mainly expository article, we revisit some formal aspects of B{á}ez-Duarte's criterion for the Riemann hypothesis. In particular, starting from Weingartner's formulation of the criterion, we define an arithmetical function $ν$, which is equal to the M{ö}bius function if, and only if the Riemann hypothesis is true. We record the basic properties of the Dirichlet series of $ν$, and state a few questions. KEYWORDS: Riemann hypothesis, arithmetical functions, Dirichlet series, Hilbert space
Riemann Hypothesis for Goss $t$-adic Zeta Function
本文证明了高斯$v$ -adic zeta函数$ζ_{v}(s)$的黎曼假设，当$v$是一阶$\mathbb{F}_{q}[t]$的素数时.
Sufficient and equivalent criteria for the Riemann Hypothesis
本文给出了经典黎曼假设的几个新的充分条件和新的等价判据。值得注意的是，还有其他关于$ζ$的陈述和评论，可以在整篇论文中找到。
Zeta functions for linear codes were defined by Iwan Duursma in 1999. They were generalized to the case of some invariant polynomials by the preset author. One of the most important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis.
Yet Another Riemann Hypothesis
这个简短的说明提出了一个特殊的推广黎曼假设，作为行动的置换群的元素的连分数。这个问题很难通过传统的分析技术来解决，因此本说明侧重于提供一个数值调查。这些结果表明，一个广泛的类以前未检查的功能可能服从黎曼假设在一般情况下，甚至共享非平凡的零点特别。
Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a derivative function. It is proved that these are equivalent, and that, if the Riemann hypothesis holds, then all zeros of the zeta function on the critical line are simple. The Riemann hypothesis is then shown to imply the third proposition holds, this being a new necessary condition for the Riemann hypothesis. The third proposition is shown to be equivalent to the…
The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's $τ$-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.
The asymptotic representation of some series and the Riemann hypothesis
给出了一类级数的渐近表示的一个猜想。这个猜想包含了黎曼假设，并且也表明了zeta函数的非平凡零点的简单性。
Control theory and the Riemann hypothesis: A roadmap
从数学控制理论的观点来看待黎曼假设的另一种方式被认为是。控制理论的传递函数是通过反转黎曼ζ函数的值来构造的，其中s=1处的不稳定极点已经从该函数中剥离。利用拉氏逆变换，对控制系统的冲激响应进行了级数展开。如果级数收敛，并且冲激响应的未证明的增长猜想为真，则黎曼假设也为真。
In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function $Ξ(t)=ξ(\tfrac{1}{2}+it)$. First, we prove that positivity of all local maxima and negativity of all local minima of $Ξ(t)$ form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of $Ξ(t)$ is a saddle point of the function $\Re\{ξ(s)\}$, we prove that the above properties of local extrema of $Ξ(t)$ are also a sufficient condition for the Riemann hypothesis to hold at $t\gg 1$. We pres…
Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of $S(t)$, $S_1(t)$, and $ζ\left(1/2+\mathrm{i}t\right)$ while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional explicit bound on gaps between consecutive zeros of the Riemann zeta-function.
On the zeros of the $k$-th derivative of the Riemann zeta function under the Riemann hypothesis
Berndt、Levinson、Montgomery和Akatsuka研究了黎曼ζ函数的导数的零点个数和非实零点实部的分布。伯恩特，莱文森和蒙哥马利调查的一般情况下，同时赤冢给尖锐的估计一阶导数的黎曼ζ函数的真理下的黎曼假设。本文将Akatsuka的结果推广到黎曼ζ函数的$k$阶导数（对正整数$k$）.
Detecting zeros of Dirichlet $L$-functions via the Riemann zeta-function
在假设黎曼猜想成立的基础上，证明黎曼ζ函数非平凡零点的某种垂直分布等价于狄利克雷L函数的广义黎曼猜想，并在两者均成立时，实现通过黎曼ζ函数零点检测狄利克雷L函数零点。
Assuming the Riemann Hypothesis, we provide effective upper and lower estimates for $\left|ζ(s)\right|$ right to the critical line. As an application we make explicit Titchmarsh's conditional bound for the Mertens function and Montgomery--Vaughan's conditional bound for the number of $k$-free numbers.
An abundance of invariant polynomials satisfying the Riemann hypothesis
1999年，Iwan Duursma定义了线性码的zeta函数作为其汉明重量计数器的生成函数。它也可以定义为其他齐次多项式不对应于现有的代码。如果齐次多项式在MacWilliams变换下是不变的，那么它的zeta函数满足一个函数方程，我们可以用公式表示类似的黎曼假设。就现有的代码而言，黎曼假设被认为是密切相关的极值性质。本文证明了存在大量的满足黎曼假设的MacWilliams变换不变量多项式。证明进行了明确的建设，这样的多项式。为了证明某类不变多项式的黎曼假设，我们建立了类似于Enestrom Kakeya定理的一个定理。
Small values of the Euler function and the Riemann hypothesis
设$\vfi$为欧拉函数，$\ga$为欧拉常数，$N_k$为第一个$k$素数的乘积。在本文中，我们考虑函数$c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$。在Riemann假设下，证明了$c(N_k)$是有界的，并给出了明确的界，当Riemann假设不成立时，$c(N_k)$上下无界.
Interpolation and Extrapolation Statements equivalent to the Riemann Hypothesis
本文通过插值理论提出与黎曼猜想真假等价的命题，并利用Jawerth-Milman外推的关键结果改进Beurling-Nyman准则，给出更精确的充分条件。
We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert ζ\vert$. The main ingredient in our argument is an identity which links the zeros of a function $f$ defined on the interval $[-a,a]$ and the values of its derivatives of odd order at $\pm a$.
On some reasons for doubting the Riemann hypothesis
对黎曼假设的真实性的几个论点进行了广泛的讨论。其中包括Lehmer现象、Davenport-Heilbronn zeta函数、临界线上$|ζ(1/2+it)|$的大值和平均值以及一类卷积函数的零点。前两个主题是经典的，其余的是与作者的研究。
We use subsequence and moving average ergodic theorems applied to Boole's transformation and its variants and their invariant measures on the real line to give new characterisations of the Lindelh{ö}f Hypothesis and the Riemann hypothesis. These ideas are then used to study the value distribution of Dirichlet L series, and the zeta functions of Dedekind, Hurwitz and Riemann and their derivatives. This builds on earlier work of R. L. using Birkhoff's ergodic theorem and probability theory.
On Balazard, Saias, and Yor's equivalence to the Riemann Hypothesis
巴拉扎德、塞亚斯和约尔证明了黎曼假设等价于黎曼ζ函数的对数沿着等于零的临界线的某个加权积分。假设的黎曼假设，我们调查的速度，截断版本的这个积分趋于零，回答问题的Borwein，布拉德利和克兰德尔和证伪一个猜想的同一作者。我们的技术的一个简单的修改给出了一个新的证明的经典欧米茄定理的函数S（t）的理论中的黎曼ζ函数。
Towards the Deep Riemann Hypothesis for $\mathrm{GL}_{n}$
该研究阐释一般线性群$\u2006\mathrm{GL}_{n}\u2006$的深度黎曼假设，其可条件性改进素数定理误差项并超出大黎曼假设预测，还从该假设视角讨论了$\u2006\mathrm{GL}_{n}\u2006$上Satake参数的切比雪夫偏差。
The purpose of the present paper is to provide a general overview of a variety of results related to a category of cotangent sums which have been proven to be associated to the so-called Nyman-Beurling criterion for the Riemann Hypothesis. These sums are also related to the Estermann Zeta function.
A square-free integer is a positive integer that is not divisible by the square of any prime. Merten's function, $M(x)$ is defined as the difference between the number of square free integers with an even number of prime factors and the number of those with an odd number of prime factors up to $x$. It's well known that $M(x)=O(x^{1/2+ε})$ is equivalent to Riemann Hypothesis. In this paper, we derive a new equation involving Merten's function that leads to a promising technique for Riemann Hypothesis.
The Generalized Riemann Hypothesis from zeros of a single L-function
本文针对每个本原 Dirichlet 特征，基于其关联 L-函数的零点提出假设，并证明该假设与广义黎曼假设等价。
For any real $β_0\in[\tfrac12,1)$, let ${\rm GRH}[β_0]$ be the assertion that for every Dirichlet character $χ$ and all zeros $ρ=β+iγ$ of $L(s,χ)$, one has $β\leβ_0$ (in particular, ${\rm GRH}[\frac12]$ is the Generalized Riemann Hypothesis). In this paper, we show that the validity of ${\rm GRH}[\frac{9}{10}]$ depends only on certain distributional properties of the zeros of the Riemann zeta function $ζ(s)$. No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.
Bounding zeta on the 1-line under the partial Riemann hypothesis
本文假设Riemann假设直至高度$T$成立，给出了Riemann zeta-函数在$\Re{s}=1$直线上的显式界，并改进了对数导数及其倒数在有限区域内的界。
We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace $\mathcal{H}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to $\mathcal{H}_0^\bot$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_\mathbb{N}=\{T_k:\, k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypot…
In 1916, Riesz proved that the Riemann hypothesis is equivalent to the bound $\sum_{n=1}^\infty \frac{μ(n)}{n^2} \exp\left( - \frac{x}{n^2} \right) = O_ε \left( x^{-\frac{3}{4} + ε} \right)$, as $x \rightarrow\infty$, for any $ε>0$. Around the same time, Hardy and Littlewood gave another equivalent criteria for the Riemann hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann hypothesis. In particular, we obtain the bound…
An equivalent inequality for the Riemann hypothesis
本文提出一个与黎曼假设等价的纯分析不等式，其证明基于黎曼ξ函数模的表示，并进一步将黎曼假设等价于布朗运动构造的波函数不会演化成完全可区分状态这一物理陈述。
Zeros of the Riemann zeta function and its derivatives have been studied by many mathematicians. Among, the number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, Akatsuka, and the author. Berndt, Levinson, and Montgomery investigated the unconditional case, while Akatsuka and the author gave sharper estimates under the truth of the Riemann hypothesis. In this paper, we prove similar results related to the first derivative of the Dirichlet $L$-functions associated with primi…
A few equalities involving integrals of the logarithm of the Riemann zeta-function and equivalent to the Riemann hypothesis II
本文是我们最近的论文相同的标题，arXiv：0806.1596v1 [math.NT]，其中一些积分等式涉及积分的对数的黎曼ζ函数的介绍，它表明，其中一些是等价的黎曼假设。建立了一些新的这种类型的等式;与前面的论文相反，现在的重点是积分涉及的参数的黎曼ζ函数（虚部的对数），而不是对数的模块（实部的对数）。使用这些等式进行数值研究，以测试黎曼假设的初步结果。我们的积分等式，连同在以前的文件中给出的等式，包括所有早期已知的这种标准，即王，Volchkov和Balazard-Saias-Yor标准，这是某些特殊情况下的一般方法提出。
We establish the Langlands-Shahidi method over a global field of characteristic p. We then focus on the unitary groups and prove global and local Langlands functoriality to general linear groups for generic representations. Main applications are to the Ramanujan Conjecture and Riemann Hypothesis.
The Riemann Hypothesis for the Goss zeta function for F_q[T]
设q是素数p的幂次，证明了Carlitz的一个以q为参数的断言。Diaz-Vargas对F_p[T]上Goss zeta函数的黎曼假设的证明依赖于他对特定情形q = p下Carlitz断言的验证。我们对一般情形的证明允许我们将Diaz-Vargas的证明推广到F_q[T]上。
Some series equivalent to the extended Riemann hypothesis for Dedekind zeta functions
本文研究 Dedekind zeta 函数的 ERH，证明若干基于其非平凡零点的无穷级数闭式与 ERH 逻辑等价，为该猜想提供新的级数刻画。
A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis
令$ρ(x)=x-[x]$，$χ=χ_{(0,1)}$。在$L_2(0,\infty)$中，考虑由$\{ρ_a | a \geq 1\}$生成的子空间$\B$，其中$ρ_a(x):=ρ(\frac{1}{ax})$。根据Nyman-Beurling准则，黎曼假设等价于陈述$χ\in\bar{\B}$。一段时间以来，人们一直在讨论并证明黎曼假设等价于$χ\in\bar{\Bnat}$的更强陈述，其中$\Bnat$是由$\{ρ_a | a\in\Nat\}$生成的小得多的子空间。
Robin's criterion states that the Riemann hypothesis is equivalent to $σ(n) < e^γn \log\log n$ for all integers $n \geq 5041$, where $σ(n)$ is the sum of divisors of $n$ and $γ$ is the Euler-Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that $σ(n) < \frac{e^γ}{2} n \log\log n$ for all odd numbers $n \geq 3^4 \cdot 5^3 \cdot 7^2 \cdot 11 \cdots 67$. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to $σ(n) < H_n + \exp{H_n}\log{H_n}$ for all integers $n \geq 1$, where $H_n$ is the $n$th harmonic number. W…
The Landau function $g(n)$ is the maximal order of an element of the symmetric group of degree $n$; it is also the largest product of powers of primes whose sum is $\le n$. The main result of this article is that the property " For all $n > 0$ , $log g(n) < li^{-1} (n))$ " (where $li^{-1}(n)$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.
The explicit upper bound of the multiple integral of $S(t)$ on the Riemann Hypothesis
我们证明明确的上界的功能$S_m(T)$，定义的反复集成的参数的黎曼ζ函数。$S(T)$和$S_1(T)$的显式上界已由A.藤井我们的结果是Fujii结果的推广。
Analysis of Voros criterion: what derivatives involving the logarithm of the Riemann xi-function at z=1/2 should be non-negative for the Riemann hypothesis holds true
最近，Voros发现的总和涉及某些权力的z-1/2，其中，当采取了黎曼xi-功能零/ρ，必须是积极的黎曼假设成立，反之亦然。在这里，我们分析这些和，把它们写成只涉及/rho-1/2的非负偶次幂的表达式，并证明黎曼假设对于导数（1/（2n-1）！）* 的非负性是等价的。其中F_（2n）=4*Sum_（k=0）^（n-1）（（n-k）*A_（k，n）*（z-1/2）^（2k），系数A_（k，n）=a^（2k-2n）*Sum_（l=k）^（n）（C_（2n）^（2l）*C_（l）^（k）），C_（j）^（m）是二项式系数，对于任何n=1，2，3...任何实数a>1/14。
Determinantal approach to a proof of the Riemann hypothesis
本文讨论行列式方法在证明黎曼假设中的应用。我们从这样一个事实出发：如果某个双无限集的行列式都是正的，那么这个假设是真的。这种方法扩展了Csordas，Norfolk和Varga在1986年的工作，并广泛使用了Karlin所描述的结果。我们已经发现并证明或证实了问题的五个方面的关系，如引言中所总结的。进一步的努力很可能导致更大的进展。
The Generalized Riemann Hypothesis from zeros of the zeta function
本文证明了所有 Dirichlet L-函数的广义 Riemann 假设可由 Riemann zeta 函数零点的特定猜想性质推导得出，反之在 GRH 下该零点亦满足这些性质。
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