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Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$ where $p$ is a prime number. Suppose that $G=\langle\sigma\rangle \simeq C_p$ acts on $k(x_0,\ldots,x_{p-1})$ by $k$-automorphisms defined as $\sigma:x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0$. Denote by $P$ the set of all prime numbers and define $P_0=\{p\in P:\bm{Q}(\zeta_{p-1})$ is of class number one$\}$ where $\zeta_n$ a primitive $n$-th root of unity in $\bm{C}$ for a positive integer $n$; $P_0$ is a finite set by \cite{MM}. Theorem. Let $k$ be an algebraic number field and $P_k=\{p\in P: p$ is ramified in $k\}$. Then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ for all $p\in P\backslash (P_0\cup P_k)$.

We call a finite set ${\mathcal{D}}\subset {\Bbb Z}^s$ a {\it (self-affine) tile digit set} with respect to an expanding integral matrix ${\bf A}$ if the self-affine set $T({\bf A}, \D)$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given ${\bf A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det {\bf A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case ${\bf A} = p{\bf I}_2$ in ${\Bbb R}^2$. We characterize the tile digit sets by making use of the zeros of the mask polynomial of ${\mathcal{D}}$ associated with a tile criterion of Kenyon [K], together with a recent result of Iosevich {\it et al} on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [IMP].

In two earlier papers with the same title, we studied connections between Voronoi’s formula in the divisor problem and Atkinson’s formula for the mean square of Riemann’s zeta-function. Now we consider this correspondence in terms of segments of sums appearing in these formulae and show that a certain arithmetic conjecture concerning the divisor function implies best possible bounds for the classical error terms Δ($x$) and$E$($T$).

Let$Z$($t$) be the classical Hardy function in the theory of Riemann’s zeta-function. An asymptotic formula with an error term$O$($T$^{1/6}log$T$) is given for the integral of$Z$($t$) over the interval [0,$T$], with special attention paid to the critical cases when the fractional part of $\sqrt{T/2\pi }$ is close to $\frac{1}{4}$ or $\frac{3}{4}$ .