Luis Baez-Duarte
A formal description of a functional analysis approach to the Riemann zeta-functional equation that provides in principle an infinity of different proofs based on work by the author on the existence of dilation-invariant unitary operators in L2 of the positive real line.
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Assuming the Riemann hypothesis (RH) and using Littlewood's conditional estimates for the Riemann zeta-function, we provide an estimate related to an approach of Y. Motohashi to the zero-free region.
It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $ϕ$ with Nevanlinna characteristic satisfying $T(r, ϕ)=o(r)$ as $r\to \infty$
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $ζ(s)$, $s=σ+i t$, $0\leq σ\leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Φ(u,v,β)$, $…
Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived here provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large $α$ behaviour of the modified Hurwitz zeta function $ζ_1(s,α)$, $s\in \mathbf{C}$, $α\in \mathbf{R}$, which is also presented here.
We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).
We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the Keiper--Li coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic …
In the paper it is discussed the relations of the Riemann $ζ-$function to classes of generating functions of multiply positive sequences according to Schoenberg (also called Pólya frequency sequences). Key words: multiply positive sequences, totally positive sequences, Pólya frequency sequences, Laguerre-Pólya class, Riemann $ζ-$function.
We use symmetric Poisson-Schwarz formulas for analytic functions $f$ in the half-plane ${Re}(s)>\frac12$ with $\bar{f(\bar{s})}=f(s)$ in order to derive factorisation theorems for the Riemann zeta function. We prove a variant of the Balazard-Saias-Yor theorem and obtain explicit formulas for functions which are important for the distribution of prime numbers. In contrast to Riemann's classical explicit formula, these representations use integrals along the critical line ${Re}(s)=\frac12$ and Blaschke zeta zeroes.
Series expansions for the Riemann zeta function
该研究证明了含临界线的复垂直带形区域内黎曼ζ函数的收敛无穷级数表示结论，据此重推已知展开式并发现了基于不完全伽马函数等的新级数表示。
In this paper, we give a connection between the Riemann hypothesis and uniqueness of the Riemann zeta function and an analogue for L-functions.
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.
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.
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.
In this paper, we prove that there are more than 66.036% of zeros of the Riemann zeta-function are distinct.
A rigorous proof is given of the hypergeometric-like representation of the Riemann-zeta function discovered by K. Maslanka.
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。
We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the Riemann zeta function whose spacing is 2.9125 times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.
it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line
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.
Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.
Assuming the Generalized Riemann Hypothesis(GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.
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。
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.
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 are proved, using properties of the Mellin transform and Müntz's identity.
This paper provides some expansions of Riemann xi function, $ξ$, as a series of Bessel K functions.
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.
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。
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。
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.
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.
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.
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。
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.
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.
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 $\frac{1}{4}$.
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.
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。
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.
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.
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.
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 $k \geq 0$.
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.
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.
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 $1+i\mathbb{R}$.
In this paper we give criteria about estimation of derivatives of the Riemann Zeta Function on the line $σ=1$.
The Riemann Zeta function $ζ(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$
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.
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.
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.
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.
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。
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.
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.
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - ε}$
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.
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.
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.
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.
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.
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