Vol 216, No 4 (2025)

A new integral representation is derived for the square of the modulus of the Riemann zeta function. Estimates of the Laplace transform of the residual term $E_\sigma(T)$ and of the mean value with respect to a Gaussian function of the square of the modulus of the zeta function are obtained.Bibliography: 6 titles.

The problem of the optimal recovery of a solution of a system of linear differential equations from initial information containing a random error is considered. Optimal methods are searched for among all possible (not necessarily linear) recovery methods. Depending on the given variance of random errors, the optimal recovery methods constructed in the paper, which turn out to be linear, use only part of the available information. Bibliography: 17 titles.

Consider a curve $\mathcal C$ defined over an algebraic number field $k$. This work is concerned with the number of generalized Jacobians $J_{\mathfrak{m}}$ of $\mathcal C$ associated with moduli $\mathfrak{m}$ defined over $k$ such that a fixed class of finite order in the Jacobian $J$ of $\mathcal C$ is lifted to a torsion class in the generalized Jacobian $J_{\mathfrak{m}}$. On the one hand it is shown that there are infinitely many generalized Jacobians with the above property, and on the other hand, under some additional constraints on the support of $\mathfrak{m}$ or the structure of $J_{\mathfrak{m}}$, it is shown that the set of generalized Jacobians of this type is finite. In addition, it is proved that there are finitely many generalized Jacobians which have a lift of two given divisors to classes of finite orders in $J_{\mathfrak{m}}$. These results are applied to the problem of the periodicity of continued fractions in the field of formal power series $k((1/x))$ constructed for special elements of the function field $k(\widetilde{\mathcal{C}})$ of a hyperelliptic curve $\widetilde{\mathcal{C}}\colon y^2=f(x)$. In particular, it is shown that for each $n \in \mathbb N$ there is a finite number of monic polynomials $\omega(x) \in k[x]$ of degree at most $n$ such that the element $\omega(x) \sqrt{f(x)}$ has a periodic expansion in a continued fraction. Bibliography: 14 titles.