We prove that pre-classifiable (see 3.1) simple nuclear tracially $\text{AF}\,\,{{C}^{*}}$-algebras $\left( \text{TAF} \right)$ are classified by their $K$-theory. As a consequence all simple, locally $\text{AH}$ and $\text{TAF}\,\,\,{{C}^{*}}$-algebras are in fact $\text{AH}$ algebras (it is known that there are locally $\text{AH}$ algebras that are not $\text{AH}$). We also prove the following Rationalization Theorem. Let $A$ and $B$ be two unital separable nuclear simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If $A$ and $B$ have the same (ordered and scaled) $K$-theory and ${{K}_{0}}{{\left( A \right)}_{+}}$ is locally finitely generated, then $A\,\otimes \,Q\,\cong \,B\,\otimes \,Q$, where $Q$ is the $\text{UHF}$-algebra with the rational ${{K}_{0}}$. Classification results (with restriction on ${{K}_{0}}$ - theory) for the above ${{C}^{*}}$-algebras are also obtained. For example, we show that, if $A$ and $B$ are unital nuclear separable simple $\text{TAF}\,\,\,{{C}^{*}}$-algebras with the unique normalized trace satisfying the $\text{UCT}$ and with ${{K}_{1}}\left( A \right)\,=\,{{K}_{1}}\left( B \right)$, and $A$ and $B$ have the same rational (scaled ordered) ${{K}_{0}}$, then $A\,\cong \,B$. Similar results are also obtained for some cases in which ${{K}_{0}}$ is non-divisible such as ${{K}_{0}}\left( A \right)\,=\,\mathbf{Z}\left[ 1/2 \right]$.

Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant motivic (Igusa) zeta function, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant.

This paper treats a generalization to singular varieties. Batyrev already considered such a ‘Kontsevich invariant’ for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\mathbb{Q}$-Gorenstein variety $X$ we associate a motivic zeta function and a ‘Kontsevich invariant’ to effective $\mathbb{Q}$-Cartier divisors on $X$ whose support contains the singular locus of $X$.

For a dense ${{G}_{\delta }}$ set of real parameters $\theta $ in [0, 1] (containing the rationals) it is shown that the group ${{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})$ is isomorphic to ${{\mathbb{Z}}^{9}}$, where ${{A}_{\theta }}$ is the rotation ${{\text{C}}^{*}}$-algebra generated by unitaries $U,\,V$ satisfying $VU\,=\,{{e}^{2\pi i\theta }}UV$ and $\sigma $ is the Fourier automorphism of ${{A}_{\theta }}$ defined by $\sigma (U)\,=\,V,\,\sigma (V)\,=\,{{U}^{-1}}$. More precisely, an explicit basis for ${{K}_{0}}$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital ${{\text{C}}^{*}}$-algebras over [0, 1].) The Connes Chern character $\text{ch:}\,{{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})\,\to \,{{H}^{\text{ev}}}{{({{A}_{\theta \,}}{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})}^{*}}$ is shown to be injective for a dense ${{G}_{\delta }}$ set of parameters $\theta $. The main computational tool in this paper is a group homomorphism $\mathbf{T}\,\text{:}\,{{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})\,\to \,{{\mathbb{R}}^{8}}\,\times \,\mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of ${{H}^{\text{ev}}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})$. The range of $\mathbf{T}$ is fully determined for each $\theta $. (We conjecture that this subspace is all of ${{H}^{\text{ev}}}$.)

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\mathbb{C}$. Let $\text{T}{{\text{r}}_{{{\Lambda }_{1}}\,,\ldots ,\,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part ${{G}^{\text{tr}-\text{alg}}}\,\subseteq \,G$. Following Steinberg in the classical case and Brüchert in the affine case, we define the Steinberg map $\chi \,:=\,\left( \text{T}{{\text{r}}_{{{\Lambda }_{1}},\ldots ,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}} \right)$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega :{{\mathbb{C}}^{n}}\times {{\left( {{\mathbb{C}}^{\times }} \right)}^{n-l}}\to C$. We investigate the local behaviour of $\text{ }\!\!\chi\!\!\text{ }$ on $C$ near $\omega \left( \,\left( 0,\ldots 0 \right)\,\times \,\left( 1,\ldots ,1 \right)\, \right)$, and we show that there exists a neighborhood of $\left( 0,...,0 \right)\,\,\times \,\,\left( 1,...,1 \right)$, on which $\text{ }\!\!\chi\!\!\text{ }\,\circ \,\omega $ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on $C$.

For $c\,>\,1$ we denote by ${{\pi }_{c}}\left( x \right)$ the number of integers $n\,\le \,x$ such that $\left\lfloor {{n}^{c}} \right\rfloor $ is prime. In 1953, Piatetski-Shapiro has proved that ${{\pi }_{c}}\left( x \right)\,\sim \,\frac{x}{c\,\log \,x},\,x\to \,+\infty $ holds for $c\,<\,12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c\,<\,1.16117\ldots $.

Let ${{p}_{w}}(n)$ be the weighted partition function defined by the generating function $\Sigma _{n=0}^{\infty }{{p}_{w}}(n){{x}^{n}}=\prod{_{m=1}^{\infty }{{(1-{{x}^{m}})}^{-w(m)}}}$ , where $w\left( m \right)$ is a non-negative arithmetic function. Let ${{P}_{w}}(u)={{\Sigma }_{n\le u}}{{p}_{w}}(n)\,and\,{{N}_{w}}(u)={{\Sigma }_{n\le u}}w(n)$ be the summatory functions for ${{p}_{w}}(n)$ and $w\left( n \right)$, respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions $\Phi \left( u \right)$ and $\text{ }\!\!\lambda\!\!\text{ }\left( u \right)$, an estimate for ${{P}_{w}}\left( u \right)$ of the form log ${{P}_{w}}(u)=\Phi (u)\{1+Ou(1/\lambda (u))\}$$\left( u\to \infty \right)$ implies an estimate for ${{N}_{w}}(u)$ of the form ${{N}_{w}}(u)={{\Phi }^{*}}(u)\{1+O(1/\log \lambda (u))\}$$\left( u\to \infty \right)$ with a suitable function ${{\Phi }^{*}}(u)$ defined in terms of $\Phi \left( u \right)$. We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

A ${{\mathbb{Z}}_{2}}$-action with “maximal number of isolated fixed points” (i.e., with only isolated fixed points such that ${{\dim}_{k}}\left( {{\oplus }_{i}}{{H}^{i}}\left( M;k \right) \right)\,\,=\,\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|,\,k\,=\,\left. {{\mathbb{F}}_{2}} \right)$ on a 3-dimensional, closed manifold determines a binary self-dual code of $\text{length}\,\text{=}\,\left| {{M}^{{{\mathbb{Z}}_{2}}}} \right|$. In turn this code determines the cohomology algebra ${{H}^{*}}\,\left( M;k \right)$ and the equivariant cohomology $H_{{{\mathbb{Z}}_{2}}}^{*}\,\left( M;k \right)$. Hence, from results on binary self-dual codes one gets information about the cohomology type of 3-manifolds which admit involutions with maximal number of isolated fixed points. In particular, “most” cohomology types of closed 3-manifolds do not admit such involutions. Generalizations of the above result are possible in several directions, e.g., one gets that “most” cohomology types (over ${{\mathbb{F}}_{2}}$) of closed 3-manifolds do not admit a non-trivial involution.

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper $\left[ \text{A} \right]$ only the result

$${{\pi }_{K,d}}\left( x \right)\,+\,{{\pi }_{{{K}^{-1}},d}}\left( x \right)\,=\,\frac{1}{h\left( d \right)}\,\frac{x}{\log x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}x} \right)$$

follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result

$${{\pi }_{K,d}}\left( x \right)\,=\,\frac{1}{2h\left( d \right)}\,\frac{x}{\log \,x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}\,x} \right)$$

stated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of $\left[ \text{A} \right]$. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of $\left[ \text{A} \right]$, each and every occurrence of $o\left( s-1 \right)$ should be replaced by $o\left( 1 \right)$.

Sections 5 and 6 of $\left[ \text{A} \right]$ have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of $\left[ \text{A} \right]$.