Related papers: Jacob's ladders and some new consequences from A. …
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 a new equivalent of the Fermat-Wiles theorem based on a kind of cross-bred of Ingham and D. R. Heath-Brown formula. Further, we prove the existence of infinite set of finite chains of a kind of equivalent expressions…
In this paper we obtain two new points of contact between Jacob's ladders and Fermat-Wiles theorem. They are generated by a logarithmic modification of the Hardy-Littlewood integral. Furthermore, we present a kind of asymptotic laws of…
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…
In this paper we give some new consequences that follow from our formula for increments of the Hardy-Littlewood integral. The main of these ones are $\mathcal{T}_1$ and $\mathcal{T}_2$ equivalents of the Fermat-Wiles theorem.
The main result of this paper is new formula connecting certain $zeta$-integral on the critical line with a $\zeta$-integral in the critical strip. Further, a kind of cross-breeding of the Hardy-Littlewood-Ingham formula and Ingham formula…
In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal…
We generalise the work of Sarnak-Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik-Selberg Conjecture.
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…
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 prove that there is a continuum set of increments with some minimal structure for the Hardy - Littlewood integral. The result implies a number of new properties of the Hardy - Littlewood integral.
The nonlinear equation which is connected with the main term of the Hardy-Littlewood formula for $\zeta^2(1/2+it)$ is studied. In this direction I obtain the fine results which cannot be reached by published methods of Balasubramanian,…
In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the…
In this paper we introduce the iterations $\phi^k_1(t)$ of the Jacob's ladder. It is proved, for example, that the mean-value of the product $$Z^2[\phi^n_1(t)]Z^2[\phi^{n-1}(t)]... Z^2[\phi^0_1(t)]$$ over the segment $[T,T+U]$ is…
In connection of our proof (1980) of the Titchmarsh's hypothesis (1934), we have obtained two asymptotic formulae (1991). In this paper we obtain three new $\zeta$-equivalents of the Fermat-Wiles theorem based on the mentioned asymptotic…
We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…
Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and…
Selberg sums are the analogues over finite fields of certain integrals studied by Selberg in in 1940s. The original versions of these sums were introduced by R.J.Evans in 1981 and, following an elegant idea of G.W.Anderson in 1991 they were…
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…
In this note, we provide evidence for a certain twisted version of the parity conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green, Konstantinou and the author. To do this, we use arithmetic duality theorems for…