Multiple zeta functions at regular integer points

We show the recurrence relations of the Euler-Zagier multiple zeta-function which describes the $r$-fold function with one variable specialized to a non-positive integer as a rational linear combination of $(r-1)$-fold functions, which extends the previous results of Akiyama-Egami-Tanigawa and Matsumoto. As an application, we obtain an explicit method to calculate the special values of the multiple zeta-function at any integer point (the arguments could be neither all-positive nor all-non-positive) as a rational linear summation of the multiple zeta values.


Introduction
The Euler-Zagier multiple zeta-function (MZF for short) is the complex analytic function defined by the following series; where r is a positive integer and s 1 , s 2 , . . ., s r are complex variables.When r = 1, it is the so-called Riemann zeta function.It converges absolutely in the region Re(s r ) > 1, Re(s r−1 + s r ) > 2, . . ., Re(s The special value of the MZF at positive integer points is called the multiple zeta value (MZV for short).The MZVs appear in calculations of a certain invariant in knot theory, in calculations of a certain integration in mathematical physics and in various areas of mathematics.See [Z2] for more details.In the early 2000s, Zhao ([Z1]) and Akiyama-Egami-Tanigawa ( [AET]) independently showed that the Date: September 12, 2022.
MZF can be meromorphically continued to C r .In particular, in [AET], the set of singularities of the MZF is determined as follows; s r = 1, s r−1 + s r = 2, 1, 0, −2, −4, −6, . . ., k i=1 s r−i+1 ∈ Z ≤k , (k = 3, 4, . . ., r). (0.1) This shows that all-non-positive integer points 1 mostly lie on the above singularities.In addition, it is known that the special values at all-non-positive integer points of the MZF are indeterminate and depend on a direction of the limit to choose.Akiyama-Egami-Tanikawa defined the special values of the MZF at allnon-positive integer points with a certain limit by using the following recurrence relation of the MZF and presented certain linear relations among those limit values.
Theorem 0.3.(Theorem 2.16).The special value of the MZF at any regular integer point can be presented explicitly as a rational linear combination of MZVs.
Explicit formulas of the special values at all regular points up to depth 3 are presented in § 2.2.
The plan of our paper goes as follows.In §1, we review the harmonic algebra and show certain relations among words.In §2, we translate the formulae of the words obtained in § 1 into the formulae of MZFs and prove the recurrence relations of MZFs.And then we prove the above main results and give some examples.

Algebraic framework
We prepare an algebra to deal formally with the computation of the MZF and show some relations on that algebra.
1.1.The harmonic algebra.In this subsection, we define an algebra with the structure of the harmonic product.This construction is based on [FK].
e. S N is the commutative semigroup generated by S).Let S • N be the non-commutative free monoid generated by S N .We denote the empty word by 1 (as the unit) and denote each elements of S • N by (u 1 , u 2 , . . ., u k ) ∈ S • N , (u j ∈ S N ) as a sequence.We set H := Q S N to be the non-commutative polynomial ring generated by S N .
Notice that H is the Q-vector space generated by S • N .We sometimes call the element of S N word.
We next review the harmonic product.
Then the pair (H, * ) forms the commutative, associative, and unital Q-algebra.We prepare some useful notations.
The map l : S • N → N 0 is defined as follows; for any ω The map Rev : and Rev(1) := 1.
to be the Q-linear space generated by the multiple zeta functions.We assume that ζ(∅) := 1.
Note that Z r is the subspace of meromorphic functions on C r .We regard Z r as the subset of Z r+1 naturally for r ∈ N 0 .We look at the simplest example.
Example 1.6.When r = 1, Z 1 is Note that by construction ζ * is a Q-algebraic homomorphism from (H, * ) to Z.
1.2.Calculation of the harmonic product.In this subsection, we prepare some algebraic relations among words which are required to prove our main results in §2.
By using this corollary, we can obtain the following algebraic relation among the words.
Proposition 1.10.Let n ∈ N and u, v 1 , v 2 , . . ., v n ∈ S N Then we have the following equation for any word ω ∈ S Here the term of i = 1 in the summation is Proof.We show the claim by induction on n.The claim for n = 1 immediately follows from the harmonic product.Indeed we have Let n ≥ 2. We calculate the left-hand side of the claim by using the harmonic product; By the induction assumption on the case of n − 1 in the first term, we have By applying the harmonic product to the summand of the third term, we have The sum of the fourth and sixth terms where By using Corollary 1.9 with Hence we have In other words, we get Similarly the sum of the fifth and seventh terms is calculated to be where By using Corollary 1.9 with In other words, we get By (1.3), (1.4), (1.5), and (1.6), we obtain the equation (1.1).

Main results
We extend the recurrence relation of Theorem (0.1) and give several examples of the special values of the MZFs at regular integers.
2.1.The recurrence relation of multiple zeta functions.In this subsection, we give a proof of the extended recurrence relation of the MZF in Theorem 2.12.
We rewrite (1.1) as the relation among MZFs and compute both sides by using the original recurrence relation (0.1).
Here the term of i = 1 in the summation means Proof.We obtain the claim by mapping both sides of (1.1) using ζ * in Definition 1.7.

Note that ζ
(2.2) Here the term of i = 1 in the summation is Proof.It is obtained as a special case of Proposition 2.1 with ω = (ω, u), u = s j , n = r − j and v i = s j+i (i = 1, . . ., r − j).
We prepare some notations which will be employed in a proof of the extended recurrence relation of the MZF.
where we put We note that the second equality holds by (2.2).
The equation (2.2) is nothing but (2.3) Our strategy is to transform this equation (2.3) to an equivalent form (2.16), from which we will deduce the extended recurrence relation (Theorem 2.12).
To start with, we review the original recurrence relation of the MZF.
Theorem 2.4.(cf.[M, §4, (4.4) and §6]).Let ǫ be a small positive real number and M , n ∈ N. We have (2.4) The symbol (M − ǫ) stands for the path of integration along the vertical line from Let ǫ be a positive real number, M ∈ N and v ∈ S N .We set Then (2.4) is represented as (2.5) The following lemma is the calculation of the left-hand side of (2.3).
To simplify calculation of the right-hand side of (2.3), we prepare Lemma 2.6 and Notation 2.7 below.
where the empty summation is interpreted as 0.
Proof.It is an immediate consequence from The following notations are for a preparation of calculation of the equation (2.9) in Lemma 2.8.
The following is on the last term of right-hand side of (2.3).
Lemma 2.8.Let r ∈ N, j ∈ {2, . . ., r − 1}, ω ∈ S • N and u ∈ S N .The following holds; (2.9) Proof.To calculate Q ′ , we use the following set-theoritical decomposition; for i = 2, . . ., r − j, we have (2.10) Here the term of l = 2 in the above decomposition is ( for k = 3, . . ., i. Thanks to the decomposition, we have By using the harmonic product, we have In the above equation, we set By applying (2.5) to each term in the above equation, we obtain (2.11) with S 1 , S 2 , S 3 , S 4 in Notation 2.7 and We note that the each summand of the first and third terms in the right-hand side of above equation (2.11) are the product of two multiple zeta functions given by (−1) i−1 Q(ω, u, s j+i + • • • + s j − 1; s j+i+1 , . . ., s r ) and (−1) i−1 Q(ω, u + s j+i + • • • + s j − 1; s j+i+1 , . . ., s r ) respectively.We also note that the summation of the summand of the second and fifth terms is equal to Q ′ (ω, u + s j − 1; s j+1 , . . ., s r ) by definition.By putting m = i + 1 − l in the fourth and the sixth terms, we obtain Since the summand of the above equation with respect to each l is described in terms of Q ′ , we obtain Since Lemma 2.6 says that the summation of the first and fourth terms in the above equation is equal to 1 sj −1 ζ(ω, u, s j+1 +s j −1, s j+2 , . . ., s r ) and the summation of the third and fifth terms is equal to 1 sj −1 ζ(ω, u + s j+1 + s j − 1, s j+2 , . . ., s r ), we get the equation (2.9).By using Lemma 2.5 and Lemma 2.8, one can reformulate (2.3) as follows.
The following calculates the third and fourth terms of right-hand side of (2.12) Lemma 2.10.Let r, M ∈ N, j ∈ {2, . . ., r − 1}, ω ∈ S • N and u ∈ S N .The following holds; (2.14) Proof.The proof goes similarly to that of Lemma 2.8.We first calculate To save the space, we set By the definition of S 1 (k) and of S 2 (k), we have In the above equation, we set We note that the each summand of the first and third terms in the above equation are the product of two multiple zeta functions given by (−1) i−1 Q(ω, u, s j+i + • • • + s j + k; s j+i+1 , . . ., s r ) and (−1) i−1 Q(ω, u + s j+i + • • •+ s j + k; s j+i+1 , . . ., s r ) respectively.We also note that the summation of the summand of the second and fifth terms is equal to Q ′ (ω, u+s j +k; s j+1 , . . ., s r ) by definition.By putting m = i+1−l in the fourth and the sixth terms, we obtain Since the summand of the above equation with respect to each l is described in terms of Q ′ , we obtain Since Lemma 2.6 says that the summation of the first and fourth terms in the righthand side above equation is equal to ) and the summation of the third and fifth terms is equal to (2.15) By (2.15), we have Hence, by (2.8), we obtain (2.14).
The extended recurrence relation of the MZF is obtained by specializing the equation (2.16).
Theorem 2.12.Let r ∈ N ≥2 , j ∈ {2, . . ., r − 1} and n j ∈ N 0 .We have (2.17) Proof.It is obtained as a special case of Proposition 2.11 with ω = (s 1 , . . ., s j−2 ), u = s j−1 , M − 1 = n j and s j = −n j , where we set ω = 1 (the empty word) when j = 2.We note that the integral terms appearing in S 3 and S 4 are 0 because turns to be 0 when The case j = 1 is treated below.
Proposition 2.13.Let r ∈ N ≥2 and n 1 ∈ N 0 .We have (2.18) Proof.By (2.10), we have By (2.4), we have where By the definition of Q in Notation 2.3 and by putting m = i + 1 − l in the third and fourth terms, we have Since the summand of the above equation with respect to l is described in terms of Q ′ , we have By applying Lemma 2.6 to the summation of the first and third terms and that of the second and fourth terms in the right-hand side of the above equation, we have By (2.3), we get Therefore, setting s 1 = −n 1 for n 1 ∈ N 0 and M − 1 = n 1 in (2.1), we have S 5 = 0 by the same arguments as the proof of Theorem 2.12 and we obtain the equation (2.18).
Proof.By combining Theorem 0.1, Theorem 2.12, and Proposition 2.13, we obtain the claim.

Special values of multiple zeta functions at regular integer points.
In this subsection, we explain that special values of MZF at any regular integer points are calculated explicitly by (2.19).And we present explicit formulas of those special values up to depth 3.
To begin with, we mention that Theorem 2.14 gives the recurrence relation among special values of MZF at regular integer points.
To state our main theorem precisely, we prepare some notations.For an index k We also define the following Q linear subspace of R generated by MZVs.For a ∈ N 0 and b ∈ N ≥2 , we define Our main theorem is stated as follows.
Theorem 2.16.The equation ( 2.20) allows us to express the special value of the MZF at a regular integer point as a rational linear combination of MZVs.In more detail, when an index n = (n 1 , . . ., n r ) ∈ Z r with d = dp n + and w = wt n + is not on the set of singularities given by (0.1), we have ζ r (n 1 , . . ., n r ) ∈ Z ≤d ≤w .Proof.We show the claim by induction on r.Let r = 2.If the variables s 1 and s 2 are both positive integers, ζ 2 (s 1 , s 2 ) is a double zeta value.Hence we may assume that at least one of s 1 and s 2 is non-positive.By [FKMT1, §4, Lemma 4.1], we have k+1 (see Example 2.17).Hence, we obtain the claim for r = 2.
Let r > 2. If the all variables s 1 ,. . .,s r are positive integers, ζ r (s 1 , . . ., s r ) is nothing but a multiple zeta value.Hence we may consider the special value for an index (n 1 , . . ., −n j , . . ., n r ) ∈ Z r (j ∈ {1, . . ., r}, n j ∈ N 0 ) which are not on the set of singularities of MZF.By using the equation (2.20) and by the induction assumption, one can easily check that ζ r (n 1 , . . ., −n j , . . ., n r ) is in Z ≤d ≤w with d = dp n + and w = wt n + .Thus we obtain the claim.
In precise, the equation (2.20) allows us to calculate special values of ζ r (s 1 , . . ., s r ) at regular integer points explicitly.The following is an example in the case of r = 1.
Example 2.17.It is well known that ζ(n) is the Riemann zeta value when n ∈ Z ≥1 and for n ∈ Z ≤0 , we have where B n is the Bernoulli number which is a rational number defined by The following is an example in the case of r = 2.