Desingularization of complex multiple zeta-functions, fundamentals of $p$-adic multiple $L$-functions, and evaluation of their special values
This paper deals with a multiple version of zeta- and -functions
both in the complex case and in the -adic case:
(I). Our motivation in the complex case is to find suitable rigorous meaning of the values of multivariable multiple zeta-functions at non-positive integer points.
(a). A desingularization of multiple zeta-functions (of the generalized Euler-Zagier type): We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with whose singularities lying on infinitely many hyperplanes) 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 multiple zeta-functions 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 multiple zeta-functions: Those can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated.
(II). Our work in the -adic case is to develop the study on analytic side of the Kubota-Leopoldt -adic -functions into the multiple setting. We construct -adic multiple -functions, multivariable versions of their -adic -functions, by using a specific -adic measure. We establish their various fundamental properties:
(a). Evaluation at non-positive integers: We establish their intimate connection with the above complex multiple zeta-functions by showing that the special values of the -adic multiple -functions at non-positive integers are expressed by the twisted multiple Bernoulli numbers, the special values of the complex multiple zeta-functions at non-positive integers.
(b). Multiple Kummer congruence: We extend Kummer congruence for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers.
(c). Functional relations with a parity condition: We extend the vanishing property of the Kubota-Leopoldt -adic -functions with odd characters to our -adic multiple -functions.
(d). Evaluation at positive integers: We establish their close relationship with the -adic twisted multiple polylogarithms by showing that the special values of the -adic multiple -functions at positive integers are described by those of the -adic twisted multiple polylogarithms at roots of unity, which generalizes the previous result of Coleman in the single variable case.
Key words and phrases:Complex multiple zeta-function, desingularization, -adic multiple -function, -adic multiple polylogarithm, multiple Bernoulli numbers, multiple Kummer congruence, Coleman’s -adic integration theory
2010 Mathematics Subject Classification:Primary 11M32, 11S40, 11G55; Secondary 11M41, 11S80
- 0 Introduction
- 1 Complex multiple zeta-functions
- 2 -adic multiple -functions
- 3 Special values of -adic multiple -functions at non-positive integers
- 4 Special values of -adic multiple -functions at positive integers
The aim of the present paper is to consider multiple zeta- and -functions both in the complex case and in the -adic case, and especially study their special values at integer points.
Let our story begin with the multiple zeta-function of the generalized Euler-Zagier type defined by
for complex variables , where are complex parameters whose real parts are all positive. Series (0.1) converges absolutely in the region
The first work which established the meromorphic continuation of (0.1) is Essouabri’s thesis . The third-named author [38, Theorem 1] showed that (0.1) can be continued meromorphically to the whole complex space with infinitely many (possible) singular hyperplanes.
A special case of (0.1) is the multiple zeta-function of Euler-Zagier type defined by
which is absolutely convergent in .
Its special value when are positive integers
makes sense when .
It is called the multiple zeta value (abbreviated as MZV),
history of whose study goes back to the work of Euler  published in 1776
On the other hand, after the work of meromorphic continuation of (0.1) mentioned above, it is natural to ask how is the behavior of when are positive (or non-negative) integers. However, as we will mention in Section 1.1, in most cases these points are on singular loci, and hence they are points of indeterminacy. Therefore we can raise the following fundamental problem.
Are there any ‘rigorous’ ways to give a meaning of for ?
Several approaches to this problem have been done so far. Guo and Zhang , Manchon and Paycha  and also Guo, Paycha and Zhang  discussed a kind of renormalization method. In the present paper we will develop yet another approach, called the desingularization, in Subsection 1.4. The Riemann zeta-function is a meromorphic function on the complex plane with a simple and unique pole at . Hence is an entire function. This simple fact may be regarded as a technique to resolve a singularity of and yield an entire function. Our desingularization method is motivated by this simple observation. For , multiple zeta-functions have infinitely many singular loci (see Figure 1 and (1.45) for the case ). We will show that a suitable finite sum of multiple zeta-functions will cause cancellations of all of those singularities to produce an entire function whose special values at non-positive integers are described explicitly in terms of Bernoulli numbers.
Another possible approach to the above Problem 0.1 is to consider the twisted multiple series. Let be roots of unity. For with (), define the multiple zeta-function of the generalized Euler-Zagier-Lerch type by
Because of the existence of the twisting factor , we can see (in Theorem 1.13 below) that, if all is not equal to 1, series (0.4) can be continued to an entire function, hence its values at non-positive integer points have a rigorous meaning. Moreover we will show that those values can be written explicitly in terms of twisted multiple Bernoulli numbers.
The above second approach naturally leads us to the theory of -adic multiple -functions. Our next main theme in the present paper is to search for a multiple analogue of Kubota-Leopoldt -adic -functions.
In the 1960s, Kubota and Leopoldt  gave the first construction of the -adic analogue of the Dirichlet -function associated with the Dirichlet character , which is called the -adic -function denoted by . This can be regarded as a -adic interpolation of values of the Dirichlet -function at non-positive integers, based on Kummer’s congruences for Bernoulli numbers.
Iwasawa  proposed a different way of constructing -adic -functions, based on the study of the arithmetic of Galois modules associated with certain towers of algebraic number fields called -extensions (see also ). In particular, his method shows that the -adic -function can be expressed by use of the power series. His study, called the Iwasawa theory, developed spectacularly with recognition of the importance of -adic -functions.
In the 1970s, other constructions of -adic -functions were given, for example, by Amice and Fresnel , Coates , Koblitz  and Serre . More generally -adic -functions over totally real number fields were constructed by Barsky , Cassou-Noguès , Deligne and Ribet . These works are based on the -adic properties of values of abelian -functions over totally real number fields at non-positive integers. In fact, the approach of Deligne and Ribet were built on the theory of -adic Hilbert modular forms, while those of Barsky and of Cassou-Noguès were built on the pioneering work of Shintani . Shintani’s work also inspired the theory of multivariable -adic -functions. In fact, motivated by , Imai  constructed certain multivariable -adic -functions and Hida  constructed -adic analogues of Shintani’s -functions.
Recently, the second, the third and the fourth named author  introduced a certain double analogue of the Kubota-Leopoldt -adic -function, which could also be seen as a -adic analogue of the double () zeta function of a specific case. Its evaluation at non-positive integers and a certain functional relation with the Kubota-Leopoldt -adic -function were shown.
In the present paper we generalize the method in  to construct the multivariable -adic multiple -function,
It is a -adic valued function for -adic variables , which is attached to each (: the Teichmüller character, ), and with .
It can be seen as a multiple analogue of the Kubota-Leopoldt -adic -function and the above mentioned -adic double -function and also regarded as a -adic analogues of the complex multivariable multiple zeta-functions (0.4) in a sense. Actually it can be constructed by a multiple analogue of the -adic gamma transform of a -adic measure of Koblitz . We investigate -adic properties of (0.5), particularly its -adic analyticity in the parameter and its -adic continuity in the parameter .
Evaluation of (0.5) at integral points is one of our main themes of this paper: By explicitly describing its special values at all non-positive integer points in terms on twisted Bernoulli numbers, we show that our -adic multiple -function (0.5) is closely connected to the complex multiple zeta function (0.4) of the generalized Euler-Zagier-Lerch type via their special values at non-positive integers. Our evaluation yields two particular results; multiple Kummer congruence and functional relations. The multiple Kummer congruence is a certain -adic congruence among the special values, which includes an ordinal Kummer congruence for Bernoulli numbers as a very special case and where we also discover a newly-looked (or not?) type of congruence for Bernoulli numbers. The functional relations are linear relations among -adic multiple -functions as a function, which reduce (0.5) as a linear sum of (0.5) with lower . The relations is attached to a parity of . It extends the known fact in the single variable case that the Kubota-Leopoldt -adic -function with odd character is identically zero function. As the double variable case, we recover the result of  that a certain -adic double -function is equal to the Kubota-Leopoldt -adic -function up to a minor factor.
Whilst, as for evaluation of (0.5) at all positive integers, we show that the special value of -adic multiple -function (0.5) with at any positive integer points is given by the special values of -adic twisted multiple polylogarithms (-adic TMPL’s, in short)
at unity. The above -adic TMPL (0.6) is a -adic valued function for -adic variable , which is attached to each positive integers and certain -adic parameters (which are occasionally roots of unity.) We construct the function by using Coleman’s -adic iterated integration theory . The construction generalizes that of -adic polylogarithm by Coleman  and that of -adic multiple polylogarithm and -adic multiple zeta (-)values by the first-named author [22, 23] and Yamashita . Our result here shows that there is a close connection between the theory of -adic multiple -functions of the Kubota-Leopoldt type initiated in , and the theory of -adic multiple polylogarithms developed by the first-named author [22, 23] which were introduced under a very different motivation. It also generalizes the previous result of Coleman  which connects the Kubota-Leopoldt -adic -function with his -adic polylogarithm. In Remark 4.4 of  it is written that it is unclear whether there is some connection between these two theories or not. Our results in the present paper give an answer to this question.
We remark that our multivariable -adic multiple -functions are different from -adic multiple zeta functions by Tangedal and Young , which are one variable -adic functions stemming from the works of Shintani, Cassou-Noguès, Yoshida and Kashio, though pursuing relationship between our works and theirs might be a significant research. A relationship to non-commutative Iwasawa theory (Coates, Fukaya, Kato, Sujatha and Venjakob , etc) might be another direction. The theory is a generalization of Iwasawa theory but it looks lacking its analytic side. It is not clear whether our work in this paper lead any direction related to this or not, which might be worthy to discuss further.
Here is the plan of this paper: In Section 1, we will introduce the twisted -ple Bernoulli numbers associated with integers such that , roots of unity and complex numbers (Definition 1.6) as generalizations of ordinary Bernoulli numbers. We will show that is analytically continued to the whole space as an entire function and interpolates at non-positive integers when all are not (Theorem 1.13). On the other hand, when are all equal to , then is meromorphically continued to the whole space with whose singularities lying on infinitely many hyperplanes. We will introduce and develop the method of desingularization, which is to resolve all singularities of . We will construct a function , called the desingularized multiple zeta-function (Definition 1.16), which will be shown to be entire (Theorem 1.19). Furthermore we will show that can be expressed as a finite ‘linear’ combination of , the multiple zeta-functions of the generalized Euler-Zagier type whose arguments are appropriately shifted by integers (Theorem 1.23). It is where we stress that summing up suitable finite combinations of causes that a marvelous cancellation of all of their infinitely many singular hyperplanes occurs and consequently turns to be entire. We will also describe the special values of at non-positive integers in terms of ordinary Bernoulli numbers (Theorem 1.22).
In Section 2, we will construct the -adic -ple -function associated with , and a positive integer with (Definition 2.9). The case essentially coincides with the Kubota-Leopoldt -adic -function (Example 2.12). Also the case coincides with the -adic double -function introduced in  as mentioned above (Example 2.13). Our main technique of construction is due to Koblitz . By using a specific -adic measure, we will define as a multiple version of the -adic -transform that provides a -adic analyticity in (Theorem 2.14). We will further show a non-trivial fact that the map can be continuously extended to any -adic integer as a -adic continuous function (Theorem 2.17).
In Section 3, we will describe the values of at non-positive integers as a sum of (Theorem 3.1). We will see that our is a -adic interpolation of a certain sum (3.2) of the complex multiple zeta-functions (Remark 3.2). As an application of the theorem, we will obtain a multiple version of the Kummer congruences (Theorem 3.10). In fact, the case coincides with the ordinary Kummer congruences for Bernoulli numbers. Also we will give some functional relations with a parity condition among -adic multiple -functions (Theorem 3.15). The functional relations can be regarded as multiple versions of the well-known fact that is the zero-function (see Example 3.19). We will see that they also recover the functional relations shown in  as a special case (Example 3.20).
In Section 4, we will describe the special values of at positive integers. In Theorem 4.41 we will establish their close relation to those of -adic TMPL’s (cf. Definition 4.29) at roots of unity, which is an extension of the previous result of Coleman . For this aim, we will introduce -adic rigid TMPL’s and -adic partial TMPL’s (Definition 4.4 and 4.16 respectively) as in-between functions. In Subsection 4.1 the above special values at positive integers will be shown to be connected with the special values of -adic rigid TMPL’s at roots unity (Theorem 4.9). Basic properties of these in-between functions will be presented in Subsection 4.2. We will show an explicit relationship between -adic rigid TMPL’s and -adic TMPL’s (Theorem 4.35) by transmitting through their connections with -adic partial TMPL’s to obtain Theorem 4.41 in Subsection 4.3.
1. Complex multiple zeta-functions
In this section, we will first recall the analytic properties of complex multiple zeta-functions of Euler-Zagier type (0.3) and of the zeta-function of Lerch type (1.10) which interpolates the twisted Bernoulli numbers (1.4). Next we will introduce multiple twisted Bernoulli numbers (Definition 1.6) which are connected with multiple zeta-functions of the generalized Euler-Zagier-Lerch type (0.4). It will be shown that the functions in the non-unity case (1.20) are analytically continued to the whole space as entire functions and interpolate these numbers at non-positive integers (Theorem 1.13). In the final subsection, we will develop our method of desingularization (Definition 1.16) of multiple zeta-functions of the generalized Euler-Zagier type (0.1). They are meromorphically continued to the whole space with whose singularities lying on infinitely many hyperplanes. Our desingularization is a method to reduce them into entire functions (Theorem 1.19). We will further show that the desingularized functions are given by a suitable finite ‘linear’ combination of multiple zeta-functions (0.1) with some arguments shifted (Theorem 1.23). It is where we see a miraculous cancellation of all of their infinitely many singular hyperplanes occurring there by taking a suitable finite combination of these functions. We will prove that certain combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized functions (Theorem 1.22). Our method might be said as a multiple series analogue of the procedure reducing the Riemann zeta function into the entire function (Example 1.18). These observations lead to the construction of -adic multiple -functions which will be discussed in the next section.
1.1. Basic facts
Let , , , , and be the set of natural numbers, non-negative integers, rational integers, rational numbers, real numbers and complex numbers, respectively. Let be the algebraic closure of . For , denote by and the real and the imaginary parts of , respectively.
Let be a primitive Dirichlet character and denote the conductor of by . The Dirichlet -function associated with is defined by
In the case , namely the trivial character with , is equal to .
It is well-known that is an entire function when , and is a meromorphic function on with a simple pole at , and satisfies
where and are the Bernoulli numbers
respectively (see [47, Theorem 4.2]). Note that () except for .
The multiple zeta-function of Euler-Zagier type is defined by (0.3). As was mentioned in the Introduction, the research of (0.3) goes back to a paper of Euler. In the late 1990s, several authors investigated its analytic properties, though their results have not been published (for the details, see the survey article ). In the early 2000s, Zhao  and Akiyama, Egami and Tanigawa  independently showed that the multiple zeta-function (0.3) can be meromorphically continued to . Furthermore, the exact locations of singularities of (0.3) were explicitly determined as follows.
Theorem 1.1 ([1, Theorem 1]).
The multiple zeta-function (0.3) can be meromorphically continued to with infinitely many singular hyperplanes
Multiple zeta values, namely the special values of (0.3) at positive integers, are equal to the special values of multiple polylogarithm
(which is an -variable complex analytic function converging on polyunit disk) at unity, namely
for with .
In the last section of this paper a -adic analogue of the equality (1.3) will be attained.
As stated above, the multiple zeta-function (0.3) but for the single variable case () has infinitely many singular hyperplanes and almost all non-positive integer points lie there. It causes an indeterminacy of its special values at non-positive integers. For example, according to [1, 2],
1.2. Twisted multiple Bernoulli numbers
We will review Koblitz’ definition of twisted Bernoulli numbers. Then we will introduce twisted multiple Bernoulli numbers, their multiple analogue, in Definition 1.6 and investigate their expression as combinations of twisted Bernoulli numbers in Proposition 1.7.
Definition 1.2 ([32, p. 456]).
For any root of unity , we define the twisted Bernoulli numbers by
where we formally let .
Koblitz  generally defined the twisted Bernoulli numbers associated with primitive Dirichlet characters. The above correspond to .
In the case , we have
In the case , we have and , where are what is called the Frobenius-Euler numbers associated with defined by
Let be the group of th roots of unity. Using the relation
for an indeterminate , we obtain the following.
Let . For ,
We see that (1.8) can also be given from the relation
Now we define certain multiple analogues of twisted Bernoulli numbers.
Let , and let be roots of unity. Set
twisted multiple Bernoulli numbers
where we note as mentioned before. In the case , we have . Note that if then is holomorphic around the origin with respect to the parameters , hence the singular part does not appear on the right-hand side of (1.13).
Let and be roots of unity. Then can be expressed as a polynomial in and with -coefficients, that is, a rational function in and with -coefficients.
The following series will be treated in our desingularization method in subsection 1.4.
For and with , define
In particular when , by use of (1.7), we have
We note that is holomorphic around the origin with respect to the parameters , and tends to as . We also note that the Bernoulli numbers appear in the Maclaurin expansion of the limit
These are important points in our arguments on desingularization methods developed in Subsection 1.4.
1.3. Multiple zeta-functions
Corresponding to the twisted multiple Bernoulli numbers is the multiple zeta-function of the generalized Euler-Zagier-Lerch type (0.4) defined in the Introduction, which is a multiple analogue of . This function can be continued analytically to the whole space and interpolates at non-positive integers (Theorem 1.13).
Assume . Using the well-known relation