Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1508.06920

Desingularization of complex multiple zeta-functions

Hidekazu Furusho·Yasushi Komori·Kohji Matsumoto·Hirofumi Tsumura
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with infinitely many singular hyperplanes) turn to be entire on the whole space after taking the desingularization. The desingularized function is given by a suitable finite `linear' combination of multiple zeta-functions with some arguments shifted. It is shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. We also discuss twisted multiple zeta-functions, which can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated.

Explore related subjects

math.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hidekazu Furusho, Yasushi Komori, Kohji Matsumoto, Hirofumi Tsumura. 2015-08-28. Desingularization of complex multiple zeta-functions. https://arxiv.org/abs/1508.06920

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Documentation for the ratpoints program

This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.

math.NT↗
arXiv2026

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

math.NT↗
arXiv2026

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Let $\mathbb{WP}^n_{\mathbf{q}}$ be a weighted projective space with weights $\mathbf{q} = (q_0, \dots, q_n)$, $q = \operatorname{lcm}(q_i)$, and let $ϕ\colon \mathbb{WP}^n_{\mathbf{q}} \to \mathbb{P}^n$, $[x_i] \mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of $\mathbb{P}^n(\mathbb{Q})$ is the image of a rational point of $\mathbb{WP}^n_{\mathbf{q}}$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ of bounded weighted height $\mathfrak{h} = H(ϕ(\,\cdot\,))^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$ Z^{\circ}_{\mathfrak{h}}\big( \mathbb{WP}^n_{\mathbf{q}}(\mathbb{Q}), X \big) = X^{q\,a(\mathbf{q})} P_{\mathbf{q}}(\log X) + O\big( X^{q\,a(\mathbf{q}) - θ} \big), \qquad θ> 0, $$ with $P_{\mathbf{q}}$ of exact degree $β(\mathbf{q})$, where $a(\mathbf{q})$ and $β(\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q \leq q\,a(\mathbf{q}) \leq q(n+1)$, $Q = \sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1) - q\,a(\mathbf{q})$ measures the sparsity of the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights $(2,2,3,3)$. The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

math.NT↗