arXiv subjects
Publications and source records attributed to Kohji Matsumoto.
We consider the value distribution of the difference between logarithms of two symmetric power $L$-functions at $s=σ> 1/2$. We prove that certain averages of those values can be written as integrals involving a density function which is constructed explicitly.
We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the Mellin-Barnes integral formula and the harmonic product formulas. The Mellin-Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence.
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.
We construct $p$-adic multiple $L$-functions in several variables, which are generalizations of the classical Kubota-Leopoldt $p$-adic $L$-functions, by using a specific $p$-adic measure. Our construction is from the $p$-adic analytic side of view, and we establish various fundamental properties of these functions.
This paper deals with a multiple version of zeta- and L-functions both in the complex case and in the p-adic case: [I] Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions (MZFs) at non-positive integer points. (a) We reveal that MZFs turn to be entire on the whole space after taking the desingularization. Further we show that the desingularized function is given by a suitable finite linear combination of MZFs with some arguments shifted. It is also shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones. (b) Twisted MZFs can be continued to entire functions and their special values at non-positive integer points can be explicitly calculated. [II] Our work in the p-adic case is to develop the study on analytic side of the Kubota-Leopoldt p-adic L-functions (pLFs) into the multiple setting. We construct p-adic multiple L-functions (pMLFs), multivariable versions of their pLFs, by using a specific p-adic measure. We establish their various fundamental properties: (a) We establish their intimate connection with the above complex MZFs by showing that the special values of pMLFs at non-positive integers are expressed by the twisted multiple Bernoulli numbers, the special values of the complex MZFs at non-positive integers. (b) We extend Kummer congruence for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers. (c) We extend the vanishing property of the Kubota-Leopoldt pLFs with odd characters to our pMLFs. (d) We establish their close relationship with the p-adic twisted multiple polylogarithms (pTMPLs) by showing that the special values of pMLFs at positive integers are described by those of pTMPLs at roots of unity, which generalizes the previous result of Coleman in the single variable case.
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.
We prove asymptotic formulas for mean square values of the Euler double zeta-function $ζ_2(s_0,s)$, with respect to $\Im s$. Those formulas enable us to propose a double analogue of the Lindel{ö}f hypothesis.
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.
We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also introduce generating functions of special values of those lattice sums, and study their properties by virtue of the theory of convex polytopes. Consequently we evaluate special values of those lattice sums, especially certain special values of zeta-functions of root systems and their affine analogues. In some special cases it is possible to treat sums running over positive integers, which may be regarded as zeta-functions associated with hyperplane arrangements.
We survey the results and the methods in the theory of universality for various zeta and $L$-functions, obtained in these forty years after the first discovery of the universality for the Riemann zeta-function by Voronin.
The distribution of the zeros of the Euler double zeta-function $ζ_2(s_1,s_2)$, in the case when $s_1=s_2$, is studied numerically. Some similarity to the distribution of the zeros of Hurwitz zeta-functions is observed.
We prove two types of functional equations for double series of Euler type with complex coefficients. The first one is a generalization of the functional equation for the Euler double zeta-function, proved in a former work of the second-named author. The second one is more specific, which is proved when the coefficients are Fourier coefficients of cusp forms and the modular relation is essentially used in the course of the proof. As a consequence of functional equation we are able to determine trivial zero divisors.
We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.
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.
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new formulas which especially include the parity results of double and triple zeta values. As another important application, we give certain refinement of restricted sum formulas, which gives restricted sum formulas among MZVs of an arbitrary depth $r$ which were previously known only in the cases of depth $2,3,4$. Furthermore, considering a sub-root system of type $B_r$ analogously, we can give relevant analogues of the Hoffman-Zagier formula, parity results and restricted sum formulas.
We study zeta-functions of weight lattices of compact connected semisimple Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6) and PU(4), and give some functional relations and new classes of evaluation formulas for them.
We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.
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.