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Register a new userProbability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions
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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.
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We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.
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 $Phi(u,v,beta)$.
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.
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.
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.
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