Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHigh points of a random model of the Riemann-zeta function and Gaussian multiplicative chaos
83
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to Gaussian multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.
rate research
Read More
This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, 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 $Phi(u,v,beta)$, $uin mathbb{C}$, $vin mathbb{C}$, $beta in mathbb{R}$. Generalizing the methodology used in the study of $zeta(s)$, we derive asymptotic formulae for $Phi(u,v,beta)$.
Voronins theorem on the `Universality of Riemann zeta function is shown to imply that Riemann zeta function is a fractal (in the sense that Mandelbrot set is a fractal) and a concrete ``representation of the ``giant book of theorems that Paul Halmos referred to.
This paper reviews known results which connect Riemanns integral representations of his zeta function, involving Jacobis theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemanns zeta function which are related to these laws.
In 2008 I thought I found a proof of the Riemann Hypothesis, but there was an error. In the Spring 2020 I believed to have fixed the error, but it cannot be fixed. I describe here where the error was. It took me several days to find the error in a careful checking before a possible submission to a payable review offered by one leading journal. There were three simple lemmas and one simple theorem, all were correct, yet there was an error: what Lemma 2 proved was not exactly what Lemma 3 needed. So, it was the connection of the lemmas. This paper came out empty, but I have found a different proof of the Riemann Hypothesis and it seems so far correct. In the discussion at the end of this paper I raise a matter that I think is of importance to the review process in mathematics.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
