Related papers: Jacob's ladders, new properties of the function $\…
In our previous papers, we have introduced within the theory of the Riemann zeta function the following notions: Jacob's ladders, oscillating systems, $\zeta$-factorization, metamorphoses, \dots In this paper we obtain $\zeta$-analogue of…
In this paper we introduce new class of nonlinear interactions of $\zeta$-oscillating systems. The main formula is generated by corresponding subset of the set of trigonometric functions. Next, the main formula generates certain set of…
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…
It is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the Bessel's functions. In this direction, a new class of the nonlinear integral equations is introduced.
In this paper we introduce the iterations of the Jacob's ladder and the new type of integral containing certain product of the factors $|\zeta|^2$ corresponding to the components of some disconnected set of the critical line. Next, we…
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…
In this paper we obtain a new-type formula - \emph{a mixed formula} - which connects the functions $|\zeta(1/2+it)|$ and $\arg\zeta(1/2+it)$. This formula cannot be obtained in the classical theory of A. Selberg, and, all the less, in the…
In this paper new classes of $L_2$-orthogonal functions are constructed as iterated $L_2$-orthogonal systems. In order to do this we use the theory of the Riemann's zeta-function as well as our theory of Jacob's ladders. The main result is…
We obtain some new properties of the signal generated by the Riemann zeta-function in this paper. Namely, we show the connection between the function $\zf$ and a nonlinear integral equation related to the Poisson-Lobachevsky integral.
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…
In this paper we introduce new class of multiplicative interactions of the $\zeta$-oscillating systems generated by a subset of power functions. The main result obtained expresses an analogue of prime decomposition (without the property of…
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
In this paper we obtain a set of five new transmutations of the mother formula. Further, we obtain the second set of ten exact metafunctional equations by crossbreeding on every two elements of the previous set. Elements of the last set…
In this paper we obtain the set of grafts from some set of 12 elementary functions on 12 critical strips. We make use this directly for grafting of elements of the corresponding set of $\zeta$-factorization formulas. This procedure gives…
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…
In this paper we use Jacob's ladders together with fundamental Hardy-Littlewood formula (1921) to prove the so-called $\zeta$-factorization formula on the critical line. Simultaneously, we obtain a set of control parameters of metamorphosis…
In this paper we obtain new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem based on the elementary Fourier orthogonal system, Riemann's zeta-function and Jacob's ladders.
In this paper we obtain new canonical synergetic formula, namely an $\zeta$-analogue of next elementary trigonometric formula. This one describes cooperative interactions between corresponding class of elementary functions and the Riemann's…
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…
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…