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.
Let ℓ be a nonnegative integer, and let a and b be two relatively prime integers such that 615 ℓ +3 ⩽ a < b. In this note, assuming the generalized Riemann hypothesis, we prove that there exists a prime p ∈ ( ℓ ab, ( ℓ +1)ab − a − b) that has exactly ( ℓ + 1) different expressions of the form p = ax + by, where x and y are nonnegative integers. This result generalizes, in particular, the recent work of Dai, Ding, and Wang, which confirms the 2020 conjecture of Ramírez Alfonsín and Skałba.
We examine the differential subordination related to the geometric and harmonic means. In particular, the discussed differential subordination generalizes the well-known Briot–Bouquet differential subordination. The main results are applicable to construct nontrivial subclasses of the class of standardly normalized holomorphic functions in the unit disk. In addition, a new type of ordinary differential equation is proposed for study.
Let d denote a fundamental discriminant, and let χ_d be the associated primitive real Dirichlet character. We investigate the discrepancy bounds for the distribution of L(σ ,χ_d) and its corresponding probability model defined by a random Euler product in the critical strip.
We investigate the metric properties of products of consecutive digits in Engel expansions. More precisely, for a nonnegative real number β and positive integer m, we study the set of points in (0, 1] for which the normalized product of m successive Engel digits converges to β. The exact Hausdorff dimension of this set is determined. This result extends the classical metric theory of Engel series and contributes to the dimension theory of digit expansions.
This paper investigates a class of nonlinear equations, specifically the mass-critical inhomogeneous nonlinear fourth-order Schrödinger equation with a nonlocal source term. We establish finite- or infinite-time blowup results for solutions possessing nonpositive energy and nonradial initial data. Our analysis extends the framework developed in [4] to incorporate nonlocal source terms and complements the results of [R. Bai and T. Saanouni, Non global solutions for non-radial inhomogeneous nonlinear Schrödinger equations, Electron. J. Differ. Equ., 2025:55, 2025] in the mass-critical regime.
The generalized convolution is certain operation on the set of probability measures. It is defined by K. Urbanik (see [K. Urbanik, Generalized convolutions, Stud. Math., 23:217–245 1964], [J.K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Classes of measures closed under mixing and convolution. Weak stability, Stud. Math., 167(3):195–213, 2005], and [B.H. Jasiulis, Limit property for regular and weak generalized convolutions, J. Theor. Probab., 23(1):315–327, 2010]). They allow us to define and study the infinite divisibility of probability measures and to construct Lévy processes in the sense of generalized convolutions. This approach was undertaken in [M. Borowiecka-Olszewska, B.H. Jasiulis-Gołdyn, J.K. Misiewicz, and J. Rosiński, Lévy Processes and stochastic integrals in the sense of generalized convolutions, Bernoulli, 21(4):2513–2551, 2015]. It turned out that Lévy processes with respect to generalized convolutions are Markov in the classical sense. Of course, Markov processes are not necessarily Lévy. In this paper, we investigate the converse: when Markov processes are Lévy in the sense of some generalized convolution?
The paper contains a mean square estimate T-HT+H∫|L(λ ,α ,σ +it)|^2dt≪_λ ,α ,σH for the Lerch zeta-function L(λ, α, s) with fixed parameters λ, α ∈ (0, 1], 1/2 < σ ≤ 7/12, and T27/82 ≤ H ≤ Tσ. The estimate is uniform in H. The result extends the mean square estimates for the Hurwitz and Riemann zeta-functions. The obtained bound is applied for universality theorems in short intervals for the function L(λ, α, s).
The Padovan (Pn)n≥0 and Perrin (Rn)n≥0 sequences are third-order linear recurrences, both defined by the relation un = un−2+un−3 for n ≥ 3. They differ in their initial conditions resulting in different sequences. The Padovan sequence begins with P0 = P1 = P2 = 1, whereas the Perrin sequence starts with R0 = 3, R1 = 0, and R2 = 2. Motivated by the work of Gómez and Luca [Tribonacci Diophantine quadruples, Glas. Mat., Ser. III, 50(1):17–24, 2015], we investigate whether there exist quadruples of positive integers a1 < a2 < a3 < a4 such that all pairwise products aiaj + 1 (for i ≠ j) belong to the Padovan or Perrin sequence, and we prove that the answer is negative.
We study the quasilinear Kirchhoff–Schrödinger–Poisson system with logarithmic and critical nonlinearity. Under certain assumptions, we discuss the existence of solutions in two different situations. Firstly, in the homogeneous case, we prove the existence of nontrivial nonnegative and nonpositive solutions, a sequence of high-energy solutions via the perturbation method. Later, in the nonhomogeneous case, we prove that the system admits at least two solutions with different energies by using Ekeland’s variational principle and the mountain pass theorem.
An algebraic structure, once associated with a suitable graph, can be studied through the tools of graph theory. In recent years, algebraic graph theorists have been interested with this issue (especially when the algebraic structure is a group). Bianchi et al. introduced the permutability graph of nonnormal subgroups in 1995. This graphwas then modified to the permutability graph of subgroups, with a vertex set to all proper subgroups of G and the same condition for joining two vertices H and K, HK = KH. Further generalization was done by Muhie et al. by examining the nonpermutability graph of subgroups. Recent research on the spectral properties of the nonpermutability graph of subgroups has focused on F2(G) and sd(G), revealing some new combinatorial formulas involving adjacency and Laplacian matrices. On the other hand, the subgroup commutativity degree sd(G) of G is the probability of finding two commuting subgroups in G at random, and the factorization number F2(G) of a finite group G is the number of all possible factorizations of G = HK as a product of its subgroups H and K. In this work, we present some spectral features of permutability graphs of subgroups, including F2(G) and sd(G). With our new method, we may move forward without assuming that sd(G) ≠ 1.
We prove that for any positive integers d1 < … < dr, a product of shifts ζ(s + idjτ) of the Riemann zeta function has the universality property in the strip 1/2 < σ < 1.
The main purpose of this paper is using analytic methods, an estimate for character sums, and the properties of the third-order characters to study the calculating problems of a certain cubic residues modulo p, an odd prime, and to give an exact computational formula for its counting function. This solves two conjectures proposed by X. Yuan and W. Zhang in [On cubic residues and related problems, Indian J. Pure Appl. Math., 54(3):806–815, 2023].
We consider the randomly weighted sums ∑_i=1^mΘ_iX_i and ∑_j=1^nθ_jY_j for m, n ∈ℕ, where the real-valued random variables Xi; i ∈ ℕ and Yj; j ∈ ℕ, possibly dependent, have heavy-tailed distributions, and the random weights Θi, θj; i, j ∈ ℕ are nonnegative and arbitrarily dependent, but the sequences Xi; i ∈ N, Yj; j ∈ ℕ, and Θi, θj; i, j ∈ℕ are mutually independent. Under some mild conditions on the random weights, we derive the asymptotics of the joint tail probability of the two randomly weighted sums, in which the primary random variables satisfy some dependence assumptions.
The paper is devoted to the study of the Selberg–Steuding class 𝒮. The main result shows that analytic functions are simultaneously approximable by discrete shifts of the L-function from 𝒮, and we prove a similar result on the universality in the density terms. Moreover, in such shifts the imaginary parts γk of the nontrivial zeros of the Riemann zeta-function ζ are involved. In the proof, we use a weaker version of the Montgomery pair correlation conjecture. The results extend a one-dimensional theorem of the first author.
Let f be a normalized primitive cusp form of even integral weight for Γ = SL(2, ℤ), and let g be a normalized Hecke–Maass cusp form. In the present paper, for any prescribed integer ℓ ≥ 2, we intend to investigate the average estimates of the Fourier coefficients λf⊗f⊗…⊗ℓf⊗g(n) of the (ℓ+1)-fold product L-functions L(f ⊗ f ⊗…⊗ℓ f ⊗ g, s) involving f and g, where f ⊗ f ⊗…⊗ℓ f ⊗ g is the (ℓ+1)-fold product associated with f and g. As a direct application, we also derive quantitative results for the sign changes of the sequence λf⊗f⊗…⊗ℓ f⊗g(n)n≥1 in short intervals, with the indices supported at the positive integers and certain binary quadratic forms.
Let λf (n) denote the normalized Fourier coefficients of a primitive holomorphic cusp form f(z) of even integral weight k for the full modular group. In this paper, we investigate the error terms of the summatory function ∑_n≤ xλ_f^2(n^j),j=3,4, and establish the following Ω-results: ∑_n≤ xλ_f^2(n^3)=c_3x+Ω(x^15/32), ∑_n≤ xλ_f^2(n^4)=c_4x+Ω(x^12/25), where c3 and c4 are suitable constants.
Let 𝒜 denote the class of analytic functions f on the unit disk 𝔻 = z ∈ ℂ: |z|, normalized by f(0) = 0 and f′(0) = 1. For –π/2 < α < π/2, let Sα be the subclass of 𝒜 consisting of functions f such that Reeiα(1 + z f″(z)/f′(z)) > 0 for z ∈ 𝔻. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass Sα. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value f″(0).
