Jacob's ladders and the first asymptotic formula for the expression of the fifth order $Z^3[...]{\hat{Z}}^2(t)$ for the collection of disconnected sets

Abstract

It is shown in this paper that there is a fine correlation of the fifth order between the values $Z[\varphi (t)/2+{\rho }_1]Z[\varphi (t)/2+{\rho }_2]Z[\varphi (t)/2+{\rho }_3]$ 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.

References in corpus (3)

• Jacob's ladders and the tangent law for short parts of the Hardy-Littlewood integral
• Jacob's ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral
• Jacob's ladders and the quantization of the Hardy-Littlewood integral

Cited by in corpus (8)

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• Jacob's ladders and new orthogonal systems generated by Jacobi polynomials
• Jacob's ladders, Bessel's functions and the asymptotic solutions of a new class of nonlinear integral equations
• Jacob's ladders and the ${\tilde{Z}}^2$-transformation of the orthogonal system of trigonometric functions
• Jacob's ladders and the ${\tilde{Z}}^2$-transformation of a polynomials in $\ln \vp_1(t)$
• Jacob's ladders and some new consequences from A. Selberg's formula
• Jacob's ladders and some nonlinear integral equations connected with the Poisson-Lobachevsky integral
• Jacob's ladders and the three-points interaction of the Riemann zeta-function with itself