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 new one in the theory of the Riemann's zeta-function and simultaneously in the theory of $L_2$-orthogonal systems.
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).
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).
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.
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 introduce the iterations $ϕ^k_1(t)$ of the Jacob's ladder. It is proved, for example, that the mean-value of the product $$Z^2[ϕ^n_1(t)]Z^2[ϕ^{n-1}(t)]... Z^2[ϕ^0_1(t)]$$ over the segment $[T,T+U]$ is asymptotically equal to $\ln^{n+1}T$. Nor the case $n=1$ cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
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 the Jacob's ladder is the asymptotic solution to the new nonlinear integral equations which correspond to the functions $J_ν(x)$ and $|\zf|^4$.
The elementary geometric properties of the Jacob's ladders \cite{7} lead to a class of new formulae for short parts of the Hardy-Littlewood integral. This class of formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.
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.
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.
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 Balasubramanian, Heath-Brown and Ivic.
We use Jacob's ladders to solve the fine problem how to divide of the Hardy-Littlewood integral to equal parts, for example of magnitude $h=6.6\times 10^{-27}$ (the numerical value of elementary Planck quantum). The result of the paper cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
It is proved in this paper that the Jacob's ladders together with the A. Selberg's classical formula (1942) lead to a new kind of formulae for some short trigonometric sums. These formulae cannot be obtained in the classical theory of A. Selberg, and all the less, in the theories of Balasubramanian, Heath-Brown and Ivic.
The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|ζ(1/2+it)|^4$. These formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic.
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 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.
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 represent a kind of synergetic formulae describing cooperative interactions between corresponding sets of values of basic classical functions.
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, by our method of crossbreeding in certain set of $ζ$-factorization formulas, the corresponding complete hybrid formulas. These are playing the role of criterion for selection of new families of $ζ$-kindred real continuous functions.
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.
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 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.
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 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.
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).
It is proved in this paper that there is a fine correlation between the values of $|ζ(1/2+i\vp_2(t))|^4$ and $|ζ(1/2+it)|^4$ where $\vp_2(t)$ stands for the Jacob's ladder of the second order. This new asymptotic formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
It is proved in this paper that there is a nonlocal asymptotic splitting (in the integral sense) of the function $Z^4(t)$ into two factors. The corresponding formula cannot be obtained in the known theories of Balasubramanian, Heath-Brown and Ivic.
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.
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.
In this paper we prove that there is a grafting of complete hybrid formula on meta-functional equation. This synergetic formula gives the result of a $ζ$-chemical reaction between different sets of values of Riemann's zeta-function on the critical strip.
In this paper we obtain new canonical synergetic formula, namely an $ζ$-analogue of next elementary trigonometric formula. This one describes cooperative interactions between corresponding class of elementary functions and the Riemann's zeta-function on a class of disconnected sets on the critical line.
The nonlinear equation which is connected with the main term of the Hardy-Littlewood formula for $ζ^2(1/2+it)$ is studied. In this direction I obtain the fine results which cannot be reached by published methods of Balasubramanian, Heath-Brown and Ivic in the field of the Hardy-Littlewood integral.
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).
In this paper we introduce new class of multiplicative interactions of the $ζ$-oscillating systems generated by a subset of power functions. The main result obtained expresses an analogue of prime decomposition (without the property of uniqueness). Dedicated to recalling of Nicola Tesla's oscillators
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 obtain a set of meta-functional equations as new species of formulas in classical mathematical analysis. Mentioned species are generated by crossbreeding complete hybrid formula as a mother formula. Namely, they are generated by an infinite set of crossbreedings on some subsidiary infinite set of meta-functional equations with one neutral factor.
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 expresses a close binding between some subset of $\{|ζ(s)|\}$ and subset of moduli of certain integral and meromorphic functions.
In this paper we introduce new class of nonlinear interactions of $ζ$-oscillating systems. The main formula is generated by corresponding subset of the set of trigonometric functions. Next, the main formula generates certain set of two-parts forms. For this set the following holds true: the cube of two-part form is asymptotically equal to other two-part form -- short functional algebra.
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 them represents new formula in the theory of the Riemann's zeta-function.
In this paper we obtain new $ζ$-synergetic formula namely an exact secondary complete hybrid formula. This one is generated by some set of trigonometric and power functions together with the square of module of the Riemann's zeta-function on the critical line. By means of this new formula, we define a two-parametric continuum set of a three-parametric invariants.
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.
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.
The oscillations of the function $Z^2(t),\ t\in [0,T]$ around the main part $σ(T)$ 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 by methods of Balasubramanian, Heath-Brown and Ivic.
In this paper we obtain a new-type formula - \emph{a mixed formula} - which connects the functions $|ζ(1/2+it)|$ and $\argζ(1/2+it)$. This formula cannot be obtained in the classical theory of A. Selberg, and, all the less, in the theories of Balasubramanian, Heath-Brown and Ivic.
In this paper a set of 48 crossbreedings on certain cell of meta-functional equations preserving the cell is obtained. In opposite direction we presente an example of very complicated meta-functional equation not obtained in basic cell though it is generated by sequence of internal crossbreedings. Such a formula is called as \emph{cancerous growth} on the basic cell.
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 $ζ$-factorization formulas. This procedure gives the set of elementary meta-functional equations that represents the set of elementary interactions of the Riemann's zeta-function with itself.
In this paper we introduce the notion of $ζ$-crossbreeding in a set of $ζ$-factorization formulas and also the notion of complete hybrid formula as the final result of that crossbreeding. The last formula is used as a criterion for selection of families of $ζ$-kindred elements in class of real continuous functions. Dedicated to recalling of Gregory Mendel's pea-crossbreeding.
It is shown in this paper that there is a fine correlation of the third order between the values of the functions $Z[\vp_1(t)]$ and $\tilde{Z}^2(t)$ which corresponds to two collections of disconnected sets. The corresponding new asymptotic formula cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.
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 new asymptotic formulae cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.
t is proved in this paper that there is a fine correlation between the values of $|ζ(1/2+iφ(t)/2)|^4$ and $|ζ(1/2+it)|^2$ which correspond to two segments with gigantic distance each from other. This new asymptotic formula cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.
It is shown in this paper that there is a fine correlation of the fifth order between the values $Z[ϕ(t)/2+ρ_1]Z[ϕ(t)/2+ρ_2]Z[ϕ(t)/2+ρ_3]$ and $\hat{Z}^2(t)$ which correspond to two collections of disconnected sets. This new asymptotic formula cannot be obtained within known theories of Balasubramanian, Heath-Brown and Ivic.
In this paper we introduce a nonlinear integral equation such that the system of global solution to this equation represents a class of a very narrow beam at $T\to\infty$ (an analogue to the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral. The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown and Ivic.
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 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.
In this paper we obtain new complete hybrid formula for corresponding class of $ζ$-factorization formulas. We demonstrate that this formula is the synergetic one. Namely, this one describes the cooperation between some class of elementary functions and Riemann's zeta-function on a class of disconnected sets on the critical line. Some comparizon between this pure mathematical phenomenon and the Belousov-Zhabotinski chemical reaction is added. \centering DEDICATED TO 110th ANNIVERSARY OF HARDY'S PURE MATHEMATICS
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}.
Jacob's ladders, almost exact decomposition of certain increments of the Hardy-Littlewood integral (1918) by means of the Raabe's integral and the thirteenth equivalent of the Fermat-Wiles theorem
Jacob's ladders, almost linear increments of the Hardy-Littlewood integral (1918) and their relations to the Selberg's formula (1946) and the Fermat-Wiles theorem
