This is a survey article on the moments of a class of error terms in analytic number theory. We begin by reviewing the moments of the Dirichlet divisor problem, and then review the hybrid moments of error terms in the divisor problems. We give two new results on the joint distribution of sign changes of Δ_2(x) and Δ_3(x). In the last section, we present some conjectures on the moment results in the well-known Selberg class.
For r=(r1,...,rk),an r-factorization of the complete λ-fold h-uniform n-vertex hypergraph λKhn is a partition of the edges of λKhn into F1,...,Fk such that Fj is rj-regular and spanning for 1 ≤ j ≤ k.This paper shows that for n＞m-1/1-21/(1-h)+h-1,a partial r-factorization of λKhm can be extended to an r-factorization of λKhn if and only if the obvious necessary conditions are satisfied.
In this paper, we study the generalized Dedekind-Rademacher sums considered by Hall, Wilson and Zagier. We establish a formula for the products of two Bernoulli functions. The proof relies on Parseval's formula, Hurwitz's formula, and Lerch's functional equation. The result leads to reciprocity formulas for some generalizations of the classical Dedekind sums. In particular, it is shown that Carlitz's, Berndt's, Hall and Wilson's reciprocity theorems are deduced as special cases.
In this paper, we investigated the problem of describing the form of higher Jordan triple derivations on trivial extension algebras. We show that every higher Jordan triple derivation on a $ 2 $-torsion free $ * $-type trivial extension algebra is a sum of a higher derivation and a higher anti-derivation. As for its applications, higher Jordan triple derivations on triangular algebras are characterized.
We study the Cauchy problem of the Kolmogorov-Fokker-Planck equations and show that the solution enjoys an analytic smoothing effect with L2 initial datum for positive time.
