We study natural families of \bar {\partial } $\bar {\partial }$-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

In this paper we investigate the zeros of the Estermann zeta function E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s}$ as a function of a complex variable s$s$, where k$k$ and \ell $\ell$ are coprime integers and {\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha }$ is the generalized divisor function with a fixed complex number \alpha $\alpha$. In particular, we study the question on how the zeros of E(s; k/ \ell , \alpha )$E(s; k/ \ell , \alpha )$ depend on the parameters k/ \ell $k/ \ell$ and \alpha $\alpha$. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

For any positive integer n$n$, let f(n)$f(n)$ denote the number of solutions to the Diophantine equation

\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*} $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$

x, y, z$x, y, z$

f(n)\gt 0$f(n)\gt 0$

n\geq 2$n\geq 2$

f(n)$f(n)$

f(p)$f(p)$

n$n$

p$p$

\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*} $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$

Let p$p$ be a prime. In this paper, we present a detailed p$p$-adic analysis on factorials and double factorials and their congruences. We give good bounds for the p$p$-adic sizes of the coefficients of the divided universal Bernoulli number {B}_{n} / n${B}_{n} / n$ when n$n$ is divisible by p- 1$p- 1$. Using these, we then establish the universal Kummer congruences modulo powers of a prime p$p$ for the divided universal Bernoulli numbers {B}_{n} / n${B}_{n} / n$ when n$n$ is divisible by p- 1$p- 1$.

Let \Lambda $\Lambda$ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If \Lambda $\Lambda$ admits a trivial maximal 1-orthogonal subcategory of \text{mod } \Lambda $\text{mod } \Lambda$, then, for any indecomposable module M\in \text{mod } \Lambda $M\in \text{mod } \Lambda$, the projective dimension of M$M$ is equal to one if and only if its injective dimension is also equal to one, and M$M$ is injective if the projective dimension of M$M$ is equal to two. In this case, we further get that \Lambda $\Lambda$ is a tilted algebra.