In this paper, we present some new inequalities for the gamma function. The main tools are the multiple-correction method developed in our previous works, and a generalized Morticis lemma.
Let $ba=(a_1,a_2,ldots,a_k)$, where $a_j (j=1,ldots,k)$ are positive integers such that $a_1 leq a_2 leq cdots leq a_k$. Let $d(ba;n)=sum_{n_1^{a_1}cdots n_k^{a_k}=n}1$ and $Delta(ba;x)$ be the error term of the summatory function of $d(ba;n)$. In t
his paper we show an asymptotic formula of the mean square of $Delta(ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.
In 1956, Tong established an asymptotic formula for the mean square of the error term in the summatory function of the Piltz divisor function $d_3(n).$ The aim of this paper is to generalize Tongs method to a class of Dirichlet series that satisfy a
functional equation. As an application, we can establish the asymptotic formulas for the mean square of the error terms for a class of functions in the well-known Selberg class. The Tong-type identity and formula established in this paper can be viewed as an analogue of the well-known Voronois formula.
The main aim of this paper is to further develop the multiple-correction method that formulated in our previous works~cite{CXY, Cao}. As its applications, we establish a kind of hybrid-type finite continued fraction approximations related to BBP-type
series of the constant $pi$ and other classical constants, such as Catalan constant, $pi^2$, etc.
