arXiv subjects

Jan Moser

Publications and source records attributed to Jan Moser.

At least 73 records · Page 4Linked to original sources

arXiv2014

Microscopic interpolation formula for the Riemann $Z(t)$-function and new algorithm for asymptotic solution of some Diophantine equation with Leibnitz coefficients

In this paper we use the values of the Riemann $Z(t)$-function in order to construct certain quasi-orthonormal system of vectors. On this basis we prove a formula for microscopic interpolation of the function $Z(t)$. Simultaneously we have obtained a new algorithm how to construct asymptotic solutions of Diophantine equation with Leibnitz coefficients. This paper is English version of the paper \cite{3} except the Appendix.

math.CA↗

arXiv2014

Jacob's ladders, heterogeneous quadrature formulae, big asymmetry and related formulae for the Riemann zeta-function

In this paper we obtain as our main result new class of formulae expressing correlation integrals of the third-order in $Z$ on disconnected sets $\mathring{G}_1(x),\mathring{G}_2(y)$ by means of an autocorrelative sum of the second order in $Z$. Moreover, the distance of the sets $\mathring{G}_1(x),\mathring{G}_2(y)$ from the set of arguments of autocorrelative sum is extremely big, namely $\sim Aπ(T),\ T\to\infty$, where $π(T)$ is the prime-counting function.

math.CA↗

arXiv2013

New consequences of the Riemann-Siegel formula and a law of asymptotic equality of signum-areas of $Z(t)$ function

In this paper we obtain the first mean-value theorems for the function $Z(t)$ on some disconnected sets. Next, we obtain a geometric law that controls chaotic behavior of the graph of the function $Z(t)$. This paper is the English version of the papers \cite{8} and \cite{9}, except of the Appendix that connects our results with the theory of Jacob's ladders, namely new third-order formulae have been obtained.

math.CA↗

arXiv2013

Jacob's ladders and $|ζ|^{-2}$-representation of some functionals generated by a new class of transcendental integrals

In this paper we introduce a new infinite set of transcendental integrals. Each of them is expressed by corresponding value of the function $|\zf|^{-2}$. Such a property is another argument about universality of the Riemann zeta-function $\zf$ in the field of pure mathematics.

math.CA↗

arXiv2013

Jacob's ladders and some generalizations of certain Ramachandra's inequality

In this paper we obtain some essential generalizations of certain Ramachandra's inequality, i. e. we obtain new lower estimates for the energies of some complicated signals generated by the Riemann zeta-function on the critical line.

math.CA↗

arXiv2013

Riemann's hypothesis and some infinite set of microscopic universes of the Einstein's type in the early period of the evolution of the Universe

We obtain in this paper, as a consequence of the Riemann hypothesis, certain class of topological deformations of the graph of the function $|\zf|$. These are used to construct an infinite set of microscopic universes (on the Planck's scale) of the Einstein type. Dedicated to the 90th anniversary of the A.S. Edington's book \emph{The mathematical theory of relativity}.

physics.gen-ph↗

arXiv2013

Distribution of the roots of the equations $Z(t)=0$, $Z'(t)=0$ in the theory of the Riemann zeta-function

Let the symbols $\{γ\},\ \{t_0\};\ t_0\not=γ$ denote the sequences of the roots of the equations $$Z(t)=0,\quad \text{and}\qquad Z'(t)=0,$$ respectively, and $$m(t_0)=\min\{γ"-t_0,t_0-γ'\},\quad Q(t_0)=\max\{γ"-t_0,t_0-γ'\},\quad γ'<t_0<γ",$$ where $γ',γ"$ are the neighboring zeroes. We have proved the following in this paper: on the Riemann hypothesis we have $$\frac{Q(t_0)}{m(t_0)}<t_0\ln^2t_0\ln_2t_0\ln_3t_0,\quad t_0\to\infty.$$ This paper is the English version of the paper of ref. \cite{5}.

math.CA↗

arXiv2013

On the roots of the equation $Z'(t)=0$

We have proved in this paper that the Lindel\" of hypothesis generates essential contraction of distances between consecutive odd-order zeros of the function $Z'(t)$. This paper is the translation of the paper \cite{11} into the English except part 8 that we added in order to point out the I. M. Vinogradov' scepticism on possibilities of the method of trigonometric sums.

math.CA↗

arXiv2013

Jacob's ladders, conjugate integrals, external mean-values and other properties of a multiply $π(T)$-autocorrelation of the function $|\zf|^2$

In this paper we obtain a new class of transformation formulae (without an explicit presence of a derivative) for the integrals containing products of factors $|\zf|^2$ with respect to two components of a disconnected set on the critical line.

math.CA↗

arXiv2012

Jacob's ladders, their iterations and the new class of integrals connected with parts of the Hardy-Littlewood integral of the function $|ζ(1/2+it)|^2$

In this paper we introduce the iterations of the Jacob's ladder and the new type of integral containing certain product of the factors $|ζ|^2$ corresponding to the components of some disconnected set of the critical line. Next, we obtain an asymptotic formula for this integral, its factorization and, for example, the essential generalization of two Selberg's formulae (1946).

math.CA↗

arXiv2012

Jacob's ladders and new class of integrals containing product of factors $ζ^2$

In this paper we obtain new properties of a signal generated by the Riemann zeta-function on the critical line. At the same time we obtain an asymptotic formula for a new class of transcendental integrals connected with the Riemann zeta-function.

math.CA↗

arXiv2012

Jacob's ladders and certain asymptotic multiplicative formula for the function $|\zf|^2$

In this paper it is proved that a mean-value of the product of some factors $|ζ|^2$ is asymptotically equal to the product of the mean-values of $ζ|^2$, and this holds true for every fixed number of the factors.

math.CA↗

arXiv2011

On the order of the Titchmarsh's sum in the theory of the Riemann zeta-function and on the biquadratic effect in the information theory

We obtain in this paper the solution of the classical problem on the order of the Titchmarsh's sum (1934). Simultaneously, we obtain a connection of this problem and the Kotelnikoff-Whittaker-Nyquist's theorem from the information theory.

math.CA↗

arXiv2011

The validity of the analog of the Riemann hypothesis for some parts of $ζ(s)$ and the new formula for $π(x)$

An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $π(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the solutions of the similar integral equations. \noindent This paper is the English version of the paper of reference \cite{1}.

math.CA↗

arXiv2011

A minimal integral of the Riemann $Ξ$-function

In this paper we obtain an equilibrium sequence $\{ω_n\}$ for which the following holds true: the areas (measures) of the figures corresponding to the positive and negative parts, respectively, of the graph of the function $Ξ(t),\ t\in[ω_n,ω_{n+1}]$ are equal. Dedicated to the 500th anniversary of rabbi Löw.

math.CA↗

arXiv2011

Jacob's ladders and the three-points interaction of the Riemann zeta-function with itself

It is proved that some set of the values of $|ζ(σ_0+i\vp_1(t))|^2$ on every fixed line $σ=σ_0>1$ generates a corresponding set of the values of $|ζ(\frac 12+it)|^2$ on the critical line $σ=\frac 12$ (i.e. we have an analogue of the Faraday law).

math.CA↗

arXiv2011

Riemann hypothesis and some new integrals connected with the integral negativity of the remainder in the formula for the prime-counting function $π(x)$

In this paper a new integral for the remainder of $π(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.

math.CA↗

arXiv2011

Jacob's ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations

In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth order expression $|ζ(\frac{1}{2}+i\vp_1(t))|^4|\zf|^2$. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivic. Dedicated to the 75th aniversary of Anatolii Alekseevich Karatsuba.

math.CA↗