Let K3 be a nonnormal cubic extension over the rational field ℚ. Let a_K_3(n) be the number of integral ideals of the field K3 of norm n, also regarded as the nth Dirichlet coefficient of the Dedekind zeta function ζ_K_3(s) . In this paper, we focus on the asymptotic formula for the lth power of the Dirichlet coefficients a_K_3(n) . We first improve earlier results on the second moment. We then generalize them to all l ≥ 4. Moreover, we refine the results over the sparse sequence {n_1^2+n_2^2} and establish asymptotic formulas for the variance of a_K_3^l(n) .
In this paper, we continue the research of [S. Jokubauskienė and V. Čekanavičius, Multivariate Hipp-type compound Poisson approximations for lattice distributions I, Lith. Math. J., 65(3):337–356, 2025]. We approximate the distribution of the sum of independent lattice vectors by the accompanying compound Poisson distribution and Hipptype signed compound Poisson measures. We establish upper bounds for nonuniform local estimates, generalizing and extending known one-dimensional estimates. We obtain general lower bounds for local and ℓr metrics. Particularly, we prove that in the scheme of sequences, upper and lower bounds have the same order of accuracy.
We study Volterra–Lévy processeswith kernels regularly varying at infinity. We prove the weak convergence of rescaled processes to Gaussian limits with explicit covariance structures and establish large and moderate deviation principles. The analysis combines regular variation techniques, cumulant methods, and the Gärtner–Ellis theorem, extending classical asymptotic results for Lévy processes to the Volterra framework. The results are obtained for finite-dimensional distributions.
Let S denote the class of univalent functions in the open unit disk 𝔻 = z ∈ ℂ: |z| < 1 of the form f(z)=z+∑_n=2^∞a_nz^n . We establish sharp upper bounds for the second Toeplitz determinant of logarithmic coefficients and the logarithmic coefficients of the inverse functions for the classes of starlike and convex functions. Moreover, we examine the invariance property of our main results.
In the ring of arithmetic functions, several characterizations of multiplicative /completely multiplicative and additive /completely additive functions are established. They are derived by employing concepts related to unitary convolution, including generalized unitary Möbius functions and logarithmic and exponential operators. The distributivity of arithmetic functions through discriminative products is also investigated, leading to unitary analogues of classical results obtained via the Dirichlet convolution. These results provide necessary and/or sufficient conditions for multiplicative/ completely multiplicative, and additive /completely additive behavior, extending earlier studies.
