Large gaps between the zeros of the Riemann zeta function

On the Riemann Zeta Function and the fractional part of rational powers

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta Function.

On the zeros of the second derivative of the Riemann zeta function under the Riemann hypothesis

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, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we extend the results of Akatsuka to the second derivative of the Riemann zeta function.

A discrete mean value of the derivative of the Riemann zeta function

In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $ζ'(ρ)$ where $ζ(s)$ is the Riemann zeta function and $ρ$ is a non-trivial zero of the Riemann zeta function.

Distinct zeros of the Riemann zeta-functions

In this paper, we prove that there are more than 66.036% of zeros of the Riemann zeta-function are distinct.

On Maslanka's representation for the Riemann zeta-function

A rigorous proof is given of the hypergeometric-like representation of the Riemann-zeta function discovered by K. Maslanka.

松本 zeta 函数的离散普适性与黎曼 zeta 函数的非平凡零点

Discrete universality for Matsumoto zeta-functions and the nontrivial zeros of the Riemann zeta-function

本文将 Garunkštis 等人关于黎曼 zeta 函数经非平凡零点筛选的离散普适性定理推广到松本 zeta 函数。

On gaps between zeros of the Riemann zeta function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.

Zeros of the Riemann zeta function on the critical line

it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line

The eighth moment of the Riemann zeta function

In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth moment of the Riemann zeta function and work of Conrey-Gonek and Ivić. A key input is a sharp bound for a certain shifted moment of the Riemann zeta function, assuming the Riemann hypothesis.

Gaps between zeros of Dedekind zeta-functions of quadratic number fields. II

Let $K$ be a quadratic number field and $ζ_K(s)$ be the associated Dedekind zeta-function. We show that there are infinitely many normalized gaps between consecutive zeros of $ζ_K(s)$ on the critical line which are greater than $2.866$ times the average spacing.

黎曼 zeta 函数零点上三角乘积的 Zeta 正则化

Zeta Regularized Trigonometric Products Over Zeros Of The Riemann Zeta Function

本文证明黎曼 zeta 函数非平凡零点上三角乘积的 zeta 正则化公式，计算其偏差，并讨论其与弱黎曼假设的猜想性联系。

Exponential moments of the logarithm of the Riemann zeta-function twisted by arguments

We discuss moments of the Riemann zeta-function in this paper. The purpose of this paper is to give an upper bound of exponential moments of the logarithm of the Riemann zeta-function twisted by arguments. Our results contain an improvement of Najnudel result for exponential moments of the argument of the Riemann zeta-function and an unconditional upper bound of the moments.

通过黎曼ζ函数检测狄利克雷L函数的零点

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。

Some identities for the Riemann zeta-function II

Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and Müntz's identity.

BesselK Series for the Riemann Zeta function

This paper provides some expansions of Riemann xi function, $ξ$, as a series of Bessel K functions.

A note on the fourth power moment of the Riemann zeta-function

We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.

黎曼zi函数在调和点处的联合极值

Joint extreme values of the Riemann zeta function at harmonic points

本文通过共振方法改进了黎曼zi函数在调和点处联合极值的估计，提供了新的行为见解，并恢复了部分已知极值结果。

黎曼zeta函数矩的上界中隐式常数的优化

Optimization of the Implicit Constant for Upper Bounds for Moments of the Riemann zeta Function

本文优化了Harper证明的黎曼zeta函数矩的精细上界中的隐式常数，并在特定条件下计算了Soundararajan证明的上界中的隐式常数。

Trigonometric inequalities and the Riemann zeta-function

We very slightly improve the leading constant of the (currently best) proven asymptotic zero-free region of the Riemann zeta-function, by using an easy improvement to a trigonometric polynomial.

Universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function

In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on some line parallel to the real axis.

On the zeros of the $k$-th derivative of the Riemann zeta function under the Riemann hypothesis

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, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates for the first derivative of the Riemann zeta function under the truth of the Riemann hypothesis. In this paper, we generalize the results of Akatsuka to the $k$-th derivative (for positive integer $k$) of the Riemann zeta function.

黎曼ζ函数论证的显式极值

Explicit extreme values of the argument of the Riemann zeta-function

研究黎曼ζ函数在短区间内论证的显式极值，改进Conrey和Turnage-Butterbaugh关于ζ函数零点r-gap的结果。

Strategies To Evaluate The Riemann Zeta Function

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta function. The results originate from attempts to extend the zeta function by classical means on the complex plane. This is particularly of interest for representations which converge rapidly in a given area of the complex plane, or for the purpose to make error bounds available.

Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros

Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.

The inverses of tails of the Riemann zeta function

We present some bounds of the inverses of tails of the Riemann zeta function on $0 < s < 1$ and compute the integer parts of the inverses of tails of the Riemann zeta function for $s=\frac{1}{2}, \frac{1}{3}$ and $\frac{1}{4}$.

Critical zeros of the Riemann zeta-function

In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on the critical line. This uses the recent result of Bettin, Bui, Li and Radziwill [BBLR] on the twisted fourth moment of the Riemann zeta-function.

关于黎曼ζ函数的解析延拓

On the analytic continuation of the Riemann zeta function

通过详细分析黎曼ζ函数的解析延拓，推导出多个可能在物理和数学中有用的新恒等式。

New Formulas for the Riemann Zeta Function

A new method for continuing the usual Dirichlet series that defines the Riemann zeta function $ζ(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.

General infinite series evaluations involving Fibonacci numbers and the Riemann Zeta function

The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.

A conjecture for the sixth power moment of the Riemann zeta-function

The authors conjecture an asymptotic expression for the sixth power moment of the Riemann zeta function. They establish related results on the asymptotics of the zeta function that support the conjecture.

Lower bounds for moments of the derivative of the Riemann zeta function

We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k \geq 0$.

Mixed moments of the Riemann zeta function

We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$ conjecture.

The reciprocals of tails of the alternating Riemann zeta function

In this paper, we give the integer parts of reciprocals of tails of the alternating Riemann zeta function at $s=1,2,3,4$ by using several new inequalities and elementary method.

Discrete Moments of the Riemann Zeta-Function near $σ=1$

We employ mean value estimates of Weyl sums in order to obtain discrete second moments of the Riemann Zeta-function with respect to polynomials near the vertical line $1+i\mathbb{R}$.

On Criteria about Estimation of the Riemann Zeta Function on the Line $σ=1$

In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $σ=1$.

A note on the gaps between consecutive zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and infinitely often they differ by at least 2.69 times the average spacing.

Une region explicite sans zero pour la fonction Zeta de Riemann

The Riemann Zeta function $ζ(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$

Twin Primes and the Zeros of the Riemann Zeta Function

The Legendre type relation for the counting function of ordinary twin primes is reworked in terms of the inverse of the Riemann zeta function. Its analysis sheds light on the distribution of the zeros of the Riemann zeta function in the critical strip and their links to primes and the twin prime problem.

黎曼ζ函数连续零点间的小间隔

Small gaps between consecutive zeros of the Riemann zeta-function

本文引入共振相关方法研究黎曼ζ函数连续零点间的小间隔，结合Montgomery对称方法和Montgomery-Odlyzko方法，突破0.515障碍，证明在黎曼假设下μ<0.50895。

Riemann's Zeta Function. Numerical Evaluation via its Alternating Relative η(s)

The paper describes a method for calculating values of Riemann's Zeta function within the critical strip 0< σ <1 and on its boundary. The approach is based on the "Alternating Zeta function" η(s). The actual Riemann Zeta function ζ(s) is easily obtained from η(s). The obtained accuracy, within certain limits of the described method is discussed. A decent scientific calculator suffices to carry out the involved computations.

A reciprocal sum related to the Riemann zeta function at s=6

We introduce an explicit formula for a reciprocal sum related to the Riemann zeta function at s=6, and pose one question related to a computational formula for larger values of s.

Multiple finite Riemann zeta functions

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite counterparts in connection with symmetric polynomials and some arithmetic quantities called powerful numbers.

黎曼zeta函数的平移矩

Shifted moments of the Riemann zeta function

本文证明黎曼假设蕴含Chandee关于黎曼zeta函数平移矩的猜想，证明基于Harper关于临界线上$2k$次矩精确上界的思想。

Numerical study of the derivative of the Riemann zeta function at zeros

The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.

The twisted fourth moment of the Riemann zeta function

We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - ε}$

On the Distribution of the non-trivial Roots of Riemann's Zeta Function. An alternative Approach

Expressing Weierstrass type infinite products in terms of Stieltjes integrals is discussed. The asymptotic behavior of particular types of infinite products is compared against the asymptotic behavior of the entire function Xi(s), well-known in Riemann zeta function theory. An approximate formula for the distribution of the non-trivial roots of Riemann's zeta function is obtained.

Pair Correlation of the zeros of the Riemann zeta function in longer ranges

In this paper, we study a more general pair correlation function, $F_h(x,T)$, of the zeros of the Riemann zeta function. It provides information on the distribution of larger differences between the zeros.

Linear statistics for zeros of Riemann's zeta function

We consider a smooth counting function of the scaled zeros of the Riemann zeta function, around height T. We show that the first few moments tend to the Gaussian moments, with the exact number depending on the statistic considered.

On the large values of the Riemann zeta-function on the critical line - II

We prove some new bounds for the maximum of Riemann zeta-function on very short segments of the critical line. All the theorems are based on the Riemann hypothesis.

Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.

Moments of the Riemann zeta-function

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in families of $L$-functions.

On the $a$-points of the derivatives of the Riemann zeta function

We prove three results on the $a$-points of the derivatives of the Riemann zeta function. The first result is a formula of the Riemann-von Mangoldt type; we estimate the number of the $a$-points of the derivatives of the Riemann zeta function. The second result is on certain exponential sum involving $a$-points. The third result is an analogue of the zero density theorem. We count the $a$-points of the derivatives of the Riemann zeta function in $1/2-(\log\log T)^2/\log T<\Re s<1/2+(\log\log T)^2/\log T$.

Both real and imaginary parts of the function F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s)

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function, have infinitely many zeroes for any value of Re(s).

Elliptic curves of high rank and the Riemann zeta function on the one line

We describe some experiments that show a connection between elliptic curves of high rank and the Riemann zeta function on the one line. We also discuss a couple of statistics involving $L$-functions where the zeta function on the one line plays a prominent role.