arXiv subjects
Publications and source records attributed to Jan Moser.
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.
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 $|\zf|^4$.
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 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.
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.