Convergence of Euler products in the critical strip is directly related to a proof of the generalized Riemann hypothesis. Moreover its behavior on the critical line is called the deep Riemann hypothesis (DRH). Kimura-Koyama-Kurokawa recently proved DRH over function fields in case the $L$-function is regular at $s=1$ [3]. In this paper we generalize their results to Selberg zeta functions. Our results imply the DRH for principal congruence groups over function fields.

In this paper, we prove the validity of the cohomological Hasse principle for $\mathrm{H}^{1}$ of semisimple simply connected algebraic groups defined over infinite algebraic extensions of global fields and also some local–global principles for (skew-)hermitian forms defined over such fields.

A numerical Godeaux surface $X$ is a minimal surface of general type with $\chi(\mathcal{O}_{X})=K_{X}^{2}=1$. Over $\mathbf{C}$ such surfaces have $p_{g}(X)=h^{1}(\mathcal{O}_{X})=0$, but $p_{g}=h^{1}(\mathcal{O}_{X})=1$ also occurs in characteristic $p>0$. Keum and Lee~[9] studied Godeaux surfaces over $\mathbf{C}$ with an involution, and these were classified by Calabri, Ciliberto, and Mendes Lopes~[4]. In characteristic $p\ge 5$, we obtain the same bound $|\mathrm{Tors}\,X|\le 5$ as in characteristic 0, and we show that the quotient $X/\sigma$ of $X$ by its involution is rational, or is birational to an Enriques surface. Moreover, we give explicit examples in characteristic 5 of quintic hypersurfaces $Y$ with an action of each of the group schemes $G$ of order 5, and having extra symmetry by $\mathrm{Aut}\,G\cong\mathbf{Z}/4\mathbf{Z}$, hence by the \textit{holomorph} $H_{20}=\mathrm{Hol}\,G=G\rtimes\mathbf{Z}/4\mathbf{Z}$ of $G$.

Let $f$ be a nonconstant meromorphic functions, $n, k$ be two positive integers. Suppose that $f^{n}$ and $(f^{n})^{(k)}$ share the value $a( \ne 0,\infty )$ CM. If either (1) $n>k+2$, or (2) $n>k+1$ and $\bar{N}(r,f)=\lambda T(r,f)(\lambda \in [0,\frac{1}{2}))$, then $f^{n}=(f^{n})^{(k)}$ and $f$ assumes the form \begin{equation*} f(\mathrm{z}) = \mathrm{ce}^{\frac{λ}{n}z} \end{equation*} where $c$ is a nonzero constant and ${\lambda}^{k}=1$.

In this paper, we elucidate the well-known Wilton’s formula for the product of two Riemann zeta functions. A proof of Wilton’s expression for product of two zeta functions was given by M. Nakajima in~[5] using the Atkinson dissection. On the similar line we derive Wilton’s formula using the Riesz sum of the order $\kappa=1$.