In this paper, we study the distribution of the index of Farey fractions whose denominators are products of k distinct primes. We obtain asymptotic formulas for the first and second moments after twisting by an arbitrary Dirichlet character.
We examine the maximal cardinality of sets {a_1, a_2,… ,a_m} consisting of nonzero elements in the ring 𝒪_K of algebraic integers in an imaginary quadratic field K having the property that a_ia_j +ε = x_ij^2 for some fixed ε∈{ -1,1} , where x_ij∈𝒪_K for all i, j such that 1 ≤ i < j ≤ m . The lower bound 4 is known since long, while the upper bound 24 was proved in our recent work [Bull. London Math. Soc. 56 (2024), 274–287]. Here we improve on this result by showing that the cardinality is at most 22.
This paper investigates the Cauchy problem for a weakly dissipative generalized Camassa-Holm equation. It is proven that if the initial data and their corresponding derivatives of the system decay exponentially at infinity, then the solution to the system exhibits the same exponential decay property at infinity.
We study surgery descriptions of orientation-reversing periodic diffeomorphisms of compact orientable 3-manifolds in terms of standard symmetries of model manifolds. We develop a method for reducing the singular set by equivariant Dehn surgery, eliminating singular points with orientation-preserving local symmetries and connecting surface components. For periodic diffeomorphisms with reduced singular sets, we prove that such surgery descriptions exist in S^3 whenever the singular set contains no surface or contains a separating surface. The remaining cases are addressed through generalized surgery descriptions on other model manifolds. Furthermore, we show that the surgery descriptions can be chosen so that every surgery coefficient is an integer or a half-integer.
In the original publication of this article [Monatsh. Math. 209 (2026), 185–213.], oversight occurred in the typographical notation of the extremal functions, where the presence of the absolute value in both the upper limit of integration and the differential inadvertently compromised their analyticity. Additionally, the parameter α was omitted from its absolute value notation in several subsequent upper bounds and norm estimates throughout Section 3. This corrigendum rectifies these expressions to their rigorous analytic forms and ensures the mathematical precision of the sharpness computations. These corrections do not alter the validity of the main results or the underlying analytical proofs.
