On Robin’s criterion for the Riemann Hypothesis

Robin’s criterion says that the Riemann Hypothesis is equivalent to \[\forall n\geq 5041, \ \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n,\] where \sigma(n) is the sum of the divisors of n, \gamma represents the Euler–Mascheroni constant, and \log_i denotes the i-fold iterated logarithm. In this note we get the following better effective estimates: \begin{equation*} \forall n\geq3, \ \frac{\sigma(n)}{n}\leq e^{\gamma}\log_2 n+\frac{0.3741}{\log_2^2n}. \end{equation*} The idea employed will lead us to a possible new reformulation of the Riemann Hypothesis in terms of arithmetic functions.

Extensions and Complements

This chapter contains treatments of a range of topics associated with the central limit theorem. These include estimated normalization using methods of heteroscedasticity and autocorrelation consistent variance estimation, the CLT in linear prrocesses, random norming giving rise to a mixed Gaussian limiting distribution, and the Cramér–Wold device and multivariate CLT. The delta method to derive the limit distributions of differentiable functions is described. The law of the iterated logarithm is proved for Gaussian processes.

The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a state-dependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.