Related papers: Jacob's ladders, nonlinear interactions between $\…
The oscillations of the function $Z^2(t),\ t\in [0,T]$ around the main part $\sigma(T)$ of its mean-value are studied in this paper. It is proved that an almost equality of the corresponding areas holds true. This result cannot be obtained…
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,...
It is shown in this paper that there is a fine correlation of the fourth order between the functions $Z^2[\vp_1(t)]$ and $\tilde{Z}^2(t)$, respectively. This correlation is with respect to two collections of disconnected sets. Corresponding…
It is shown in this paper that there is a continuum set of orthogonal systems relative to the weight function $\tilde{Z}^2(t)$. The corresponding integrals cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
In this paper we obtain new functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theorem. These are generated by our asymptotic formulae (1981) for the function $Z'(t)$ which we are now using to prove essential influence of…
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 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…
In this paper we use operation of crossbreeding on the set of six transmutations of corresponding asymptotic complete hybrid formulas from our previous paper. We obtain in result the set of fifteen exact meta-functional equations. Every of…
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 $|\zeta(\frac{1}{2}+i\vp_1(t))|^4|\zf|^2$. These formulae cannot be…
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…
In this paper we introduce a generating vector-operator acting on the class of functions $L_2([a,a+2l])$. This operator produces (for arbitrarily fixed $[a,a+2l]$) infinite number of new generation $L_2$-systems. Every element of the…
It is proved that some set of the values of $|\zeta(\sigma_0+i\vp_1(t))|^2$ on every fixed line $\sigma=\sigma_0>1$ generates a corresponding set of the values of $|\zeta(\frac 12+it)|^2$ on the critical line $\sigma=\frac 12$ (i.e. we have…
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…
We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice…
The elementary geometric properties of Jacob's ladders lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral. This class of asymptotic formulae cannot be obtained by methods of…
In this paper we have obtained new phenomenon lying in the following: every fixed asymptotic complete hybrid formula (we call it as mother formula) generates infinite set of new formulas (transmutations) such that every new formula…
It is shown in this paper that there is a fine correlation of the fifth order between the values $Z[\phi(t)/2+\rho_1]Z[\phi(t)/2+\rho_2]Z[\phi(t)/2+\rho_3]$ and $\hat{Z}^2(t)$ which correspond to two collections of disconnected sets. This…
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 it is proved that a mean-value of the product of some factors $|\zeta|^2$ is asymptotically equal to the product of the mean-values of $\zeta|^2$, and this holds true for every fixed number of the factors.
We revisit the proposed theoretical model for a small but nonzero cosmological constant which seems supported increasingly better by recent observations. The model features two scalar fields which interact with each other through a…