Numerical investigations around a transformation of Landau's formula suggest certain statistical regularities in the distribution of zeros of the Riemann zeta function.
The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these n…
We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ^{'}$, at large distances. We show, that independently of the height, a subset of finite number of successive zeros knows the locations of lower level zeros. The information contained in the subset of zeros is inversely proportional to $ln(γ/(2π))$, where $γ$ is the average zeta of the subset. Because the mean difference of the zeros also decreases as inversely proportional to $ln(γ/(2π))$, each equally long segment of the line $\Re(z)=1/2$ contains equal amount of information. The distributions of differences are skewed…
Gram's Law describes a pattern that frequently occurs in the distribution of the non-trivial zeros of the Riemann zeta function along the critical line. Whenever Gram's Law holds true, it reduces the difficulty of computing the corresponding zeta zeros. In this paper, we provide a model that estimates how often this pattern occurs. The model is based on a conjecture that relates the statistical distribution of the zeta zeros to that of the eigenvalues of random unitary matrices.
We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ'$, at large. We show, that independently of the location of the zeros, their differences have similar statistical properties. The distributions of differences are skewed towards the nearest zeta zero, have local maximum of variance and local minimum of kurtosis at or near each zeta zero. Furthermore, we show that distributions can be fitted with Johnson probability density function, despite the value of skewness or kurtosis of the distribution.
We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical "repulsion" between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarit…
We study distributions of differences of unscaled Riemann zeta zeros, $γ-γ^{'}$, at large. We show, that independently of the location of the zeros, i.e., even for zeros as high as $10^{23}$, their differences have similar statistical properties. The distributions of differences are skewed usually towards the nearest zeta zero. We show, however, that this is not always the case, but depends upon the distance and number of nearby zeros on each side of the corresponding distribution. The skewness, however, always decreases when zeta zero is crossed from left to right, i.e., in increasing directi…
Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from th…
We study the Epstein zeta-function formulated on the $d$-dimensional hypercubic lattice, $ζ^{(d)}(s) = (1/2)\sideset{}{'}\sum_{n_1,\ldots,n_d} (n_1^2+\cdots+n_d^2)^{-s/2}$ where $\Re(s)>d$ and the summation runs over all integers excluding the origin. An analytical continuation of the Epstein function to the whole complex $s$-plane is constructed for spatial dimension $d$ being a continuous variable ranging from $0$ to $\infty$. We are interested in zeros $ρ=ρ_x+{\rm i}ρ_y$ defined by $ζ^{(d)}(ρ) = 0$. Besides the trivial zeros, there exist "critical" zeros (on the critical line) with $ρ_x=\fr…
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical numb…