Is the Infrared Background Excess Explained by the Isotropic Zodiacal Light from the Outer Solar System?

This paper investigates whether an isotropic zodiacal light from the outer Solar system can account for the detected background excess in near-infrared. Assuming that interplanetary dust particles are distributed in a thin spherical shell at the outer Solar system (>200 AU), thermal emission from such cold (<30 K) dust in the shell has a peak at far-infrared (~100 microns). By comparing the calculated thermal emission from the dust shell with the observed background emissions at far-infrared, permissible dust amount in the outer Solar system is obtained. Even if the maximum dust amount is assumed, the isotropic zodiacal light as the reflected sunlight from the dust shell at the outer Solar system cannot explain the detected background excess at near-infrared.Comment: 6 pages, 2 figures, accepted by PAS

Verification of the anecdote about Edwin Hubble and the Nobel Prize

Edwin Powel Hubble is regarded as one of the most important astronomers of 20th century. In despite of his great contributions to the field of astronomy, he never received the Nobel Prize because astronomy was not considered as the field of the Nobel Prize in Physics at that era. There is an anecdote about the relation between Hubble and the Nobel Prize. According to this anecdote, the Nobel Committee decided to award the Nobel Prize in Physics in 1953 to Hubble as the first Nobel laureate as an astronomer (Christianson 1995). However, Hubble was died just before its announcement, and the Nobel prize is not awarded posthumously. Documents of the Nobel selection committee are open after 50 years, thus this anecdote can be verified. I confirmed that the Nobel selection committee endorsed Frederik Zernike as the Nobel laureate in Physics in 1953 on September 15th, 1953, which is 13 days before the Hubble's death in September 28th, 1953. I also confirmed that Hubble and Henry Norris Russell were nominated but they are not endorsed because the Committee concluded their astronomical works were not appropriate for the Nobel Prize in Physics.Comment: 4 pages, 1 figure, Proceedings of the Sixth Symposium on History of Astronomy (March 17 - 18, 2017, Japan

Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems

The shuffle product plays an important role in the study of multiple zeta values. This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of multiple zeta values of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel'd integral expressions for multiple zeta values. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.Comment: 18 page

Infinite series involving hyperbolic functions

In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical results given by Cauchy, Mellin and Kronecker. In the latter part, we give new formulas for some infinite series involving the hyperbolic cosine function.Comment: 18 page

On Witten multiple zeta-functions associated with semisimple Lie algebras V

We study the values of the zeta-function of the root system of type $G_2$ at positive integer points. In our previous work we considered the case when all integers are even, but in the present paper we prove several theorems which include the situation when some of the integers are odd. The underlying reason why we may treat such cases including odd integers is also discussed

An overview and supplements to the theory of functional relations for zeta-functions of root systems

We give an overview of the theory of functional relations for zeta-functions of root systems, and show some new results on functional relations involving zeta-functions of root systems of types $B_r$, $D_r$, $A_3$ and $C_2$. To show those new results, we use two different methods. The first method, for $B_r$, $D_r$, $A_3$, is via generating functions, which is based on the symmetry with respect to Weyl groups, or more generally, on our theory of lattice sums of certain hyperplane arrangements. The second method for $C_2$ is more elementary, using partial fraction decompositions.Comment: 24 page

Zeta-functions of root systems and Poincar\'e polynomials of Weyl groups

We consider a certain linear combination $S(\mathbf{s},\mathbf{y};I;\Delta)$ of zeta-functions of root systems, where $\Delta$ is a root system of rank $r$ and $I\subset\{1,2,\ldots,r\}$. Showing two different expressions of $S(\mathbf{s},\mathbf{y};I;\Delta)$, we find that a certain signed sum of zeta-functions of root systems is equal to a sum involving Bernoulli functions of root systems. This identity gives a non-trivial functional relation among zeta-functions of root systems, if the signed sum does not identically vanish. This is a genralization of the authors' previous result proved in \cite{KMTLondon}, in the case when $I=\emptyset$. We present several explicit examples of such functional relations. A criterion of the non-vanishing of the signed sum, in terms of Poincar{\'e} polynomials of associated Weyl groups, is given. Moreover we prove a certain converse theorem, which implies that the generating function for the case $I=\emptyset$ essentially knows all information on generating functions for general $I$.Comment: 41 page

Desingularization of complex multiple zeta-functions

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.Comment: This paper is the complex part of our original article arXiv:math/1309.3982 which was divided into the complex part and the p-adic par

Sum formulas for double polylogarithms with a shifting parameter and their derivatives

We prove sum formulas for double polylogarithms of Hurwitz type, that is, involving a shifting parameter $b$ in the denominator. These formulas especially imply well-known sum formulas for double zeta values, and sum formulas for double $L$-values. Further, differentiating in $b$, we obtain a kind of weighted sum formula for double polylogarithms and double $L$-values. We also give sum formulas for partial double zeta values with some congruence conditions. Our proofs of those sum formulas are based on certain functional relations for double polylogarithms of Hurwitz type.Comment: 17 page

Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of p-adic multiple L-functions at arbitrary integers

We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as a finite sum of ordinary multiple zeta-functions of the same type. As applications, we explicitly compute special values of desingularized double zeta-functions of Euler-Zagier type. We also extend our previous results concerning a relationship between $p$-adic multiple $L$-functions and $p$-adic multiple star polylogarithms to more general indices with arbitrary (not necessarily all positive) integers.Comment: 36 page