Related papers: Zeta functions and Blow-Nash equivalence

To a given real polynomial function f $\in$ R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z $\pm$ top,0 (f\,; s) $\in$ Q(s), analogous to the topological zeta function of Denef and Loeser in the…

This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that…

We collect some properties of the motivic zeta functions and the motivic nearby fiber defined by Denef and Loeser. In particular, we calculate the relative dual of the motivic nearby fiber. We give a candidate for a nearby cycle morphism on…

We introduce a notion of integration on the category of proper birational maps to a given variety $X$, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers…

We offer an equivariant analogue of the monodromy zeta function of a germ invariant with respect to an action of finite group G as an element of the Grothendieck ring of finite (Z x G)-sets. We formulate equivariant analogues of the…

We elaborate notions of integration over the space of arcs factorized by the natural $C^*$-action and over the space of non-parametrized arcs (branches). There are offered two motivic versions of the zeta function of the classical monodromy…

Given a germ of an analytic variety $X$ and a germ of a holomorphic function $f$ with a stratified isolated singularity with respect to the logarithmic stratification of $X$, we show that under certain conditions on the singularity type of…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

We introduce the etale framework to study Igusa zeta functions in several variables, generalizing the machinery of vanishing cycles in the univariate case. We define the etale Alexander modules, associated to a morphism of varieties F from…

A notion of Milnor fibration for meromorphic functions and the corresponding concepts of monodromy and monodromy zeta function have been introduced in [GZLM1]. In this article we define the topological zeta function for meromorphic germs…

This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced…

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the…

In this paper, we compare the two definitions of Bloch group, and survey the elements in Bloch group. We confirm the Lichtenbaum conjecture on the field $Q(\zeta_p)$ under the assumption the truth of the base of the Bloch group of…

It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special…

We define the doubling zeta integral for smooth families of representations of classical groups. Following this we prove a rationality result for these zeta integrals and show that they satisfy a functional equation. Moreover, we show that…

We prove new relations on zeta function at even arguments and Dirichlet $L$ function at odd. The key idea is to make use of the Taylor series and partial fraction decomposition of cotangent and secant functions as we discuss in calculus and…

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…

In arXiv:1408.4708, Xu defines the dlt motivic zeta function associated to a regular function $f$ on a smooth variety $X$ over a field of characteristic zero. This is an adaptation of the classical motivic zeta function that was introduced…

The \emph{Barnes $\zeta$-function} is \[ \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to $\C$.…

We revisit congruence zeta functions of smooth projective varieties over finite fields in the framework of Scholze's Berkovich motives. Via this formalism and categorical traces, we construct a new zeta function, and show that it agree with…