The aim of the present article is to render the spectral theory of mean values of automorphic $L$-functions -- in a unified fashion. This is an outcome of the investigation commenced with the parts XII and XIV, where a framework was laid on the basis of the theory of automorphic representations and a general approach to the mean values was envisaged. We restrict ourselves to the situation offered by the full modular group, solely for the sake of simplicity. Details and extensions are to be published elsewhere.

This article concerns the value distribution of the third Painlevé transcendents in sectorial domains around fixed singular points. We show that the cardinality of the zeros of a third Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimation.

The paper is concerned with the asymptotic analysis of a minimizer of an $n$-Ginzburg-Landau type functional. The convergence rate of the module of minimizers is presented when the parameter $\varepsilon$ goes to zero. This conclusion shows that the functional converges to $\frac{1}{n}\int|\nabla u_n|^n$ locally when $\varepsilon \to 0$, where $u_n$ is an $n$-harmonic map.

Let $p$ be a prime number. A number field $F$ satisfies the condition $(H_p)$ when any tame cyclic extention $N/F$ of degree $p$ has a normal integral basis. For the case $p=2$, it is shown by Mann that $F$ satisfies $(H_2)$ only when $h_F=1$ where $h_F$ is the class number of $F$. We prove that if an imaginary quadratic field $F$ satisfies $(H_p)$ for some $p$, then $h_F=1$.