arXiv subjects
Publications and source records attributed to Jan Moser.
In our previous papers, we have introduced within the theory of the Riemann zeta function the following notions: Jacob's ladders, oscillating systems, $ζ$-factorization, metamorphoses, \dots In this paper we obtain $ζ$-analogue of an elementary trigonometric identity and other interactions between oscillating systems.
In this paper we show that the $\mathcal{Z}_{ζ,Q^2}$-transformation of every unbounded signal based on increasing power function is a telegraphic signal, i.e. the unit rectangular signal.
In this paper we obtain a new set of metamorphoses of the oscillating Q-system by using the Euler's integral. We split the final state of mentioned metamorphoses into three distinct parts: the signal, the noise and finally appropriate error term. We have also proved that the set of distinct metamorphoses of that class is infinite one.
The notion of the Jacob's ladders, reversely iterated integrals and the $ζ$-factorization is used in this paper in order to obtain new results in study of the function $\arg\zf$. Namely, we obtain new formulae for non-local and non-linear interaction of the functions $|\zf|$ and $\arg\zf$, and also a set of metamorphoses of the oscillating Q-system.
In this paper we study certain stochastic process that is generated by the Riemann-Siegel formula. Further, we construct corresponding statistical model by a way similar to those used in telecommunication. We define statistical arc length of the Riemann $Z(t)$-curve in this model and obtain an asymptotic formula for that length. This paper is English remake of our work of reference \cite{4}.
In this paper we introduce complicated oscillating system, namely quotient of two multiforms based on Riemann-Siegel formula. We prove that there is an infinite set of metamorphoses of this system (=chrysalis) on critical line $σ=\frac 12$ into a butterfly (=infinite series of M\" obius functions in the region of absolute convergence $σ>1$).
In this paper we study properties of some sums of members of the sequence $\{Z[t_ν(τ)]\}$. Our results are expressed in statements proving essential influence of the Lindel\" of hypothesis on corresponding formulae. In this paper: the parts 1 -- 6 are English version of our paper \cite{6}, and the part 7 of this work contains current results, namely new set of metamorphosis of the main multiform from our paper \cite{7}.
In this paper we use Jacob's ladders together with fundamental Hardy-Littlewood formula (1921) to prove the so-called $ζ$-factorization formula on the critical line. Simultaneously, we obtain a set of control parameters of metamorphosis of a multiform connected with the Riemann-Siegel formula.
The main subject of this paper is the mean-value of the function $|ζ(s)|^{2k-1}$ in the critical strip. On Lindel\" of hypothesis we give a solution to this question for some class of disconnected sets. This paper is English version of our paper \cite{5}.
In this paper we obtain an extension of the set of non-local equalities by adding to it new set of local equalities. Namely, we obtain an invariant set of equalities on the set of reversely iterated integrals (energies). In other words, we obtain a new continuum set of constraints on behaviour of the function $\zf,\ t\to\infty$.
In this paper we obtain new theorems about classes of exceptional sets for the Selberg's theorem C (1942). Our theorems, as based on discrete method, are not accessible for Karatsuba's theory (1984) since this theory is a continuous theory. This paper is English version of our paper \cite{8}, the results of our paper \cite{9} are added too.
In this paper we improve classical Hardy-Littlewood exponent $1/2$ by about $16.6\%$ 62 years after the original result. This result is the first step to prove the Selberg's hypothesis (1942). In order to reach our purpose we use discrete method. This paper is the English version of our paper \cite{4}.
New class of integral identities concerning constraints on behavior of the Riemann's zeta function on the critical line is introduced in this paper. Namely, we have obtained new kind of $σ$-additivity and $σ$-multiplicativity in the class of reverse iterated integrals (energies).
We obtain in this paper new application of the classical E.C. Titchmarsh' discrete method (1934) in the theory of the Riemann $\zf$ - function. Namely, we shall prove the first localized $Ω$-theorem for short trigonometric sum. This paper is the English version of the work of reference \cite{4}.
Certain completely logarithmic formula for a set of reversely iterated integrals (energies) is proved in this paper. Namely, in this case we have that integral powers of $\ln T$ are contained on input as well as on output of corresponding integrals (energies).
The main subject to study in this paper are properties of the sequence of reversely iterated segments. Especially, we will examine properties of chaotic behavior of the sequence of measures of corresponding segments. Our results are not accessible within current methods in the theory of Riemann zeta-function.
On Riemann hypothesis it is proved in this paper that the arc length of the Riemann $Z$-curve is asymptotically equal to the double sum of local maxima of the function $Z(t)$ on corresponding segment. This paper is English remake of our paper \cite{9}, with short appendix concerning new integral generated by Jacob's ladders added.
It is proved in this paper that continuum set of $L_2$-orthogonal systems generated by the Riemann zeta-function on the critical line corresponds to every fixed $L_2$-orthogonal system on a fixed segment. This theorem serves as a resource for new set of integrals not accessible by the current methods in the theory of the Riemann zeta-function. \noindent Dedicated to the 100th anniversary of G.H. Hardy's fundamental theorem: the function $\zf$ has an infinite set of zeros, \cite{1}.