Kuan, Lin and Yu stated more conjectures about zeta-like multizeta values in [7]

POSITIVE CHARACTERISTIC HUEI-JENG CHEN

Abstract. Multizeta values in positive characteristic were first introduced and studied by Thakur. He and Lara Rodr´ıguez [9] discovered and conjectured certain zeta-like fami- lies. Kuan, Lin and Yu stated more conjectures about zeta-like multizeta values in [7]. In the present paper we study and give the transparent formula for certain Anderson-Thakur polynomials. This enables us to confirm the conjectured zeta-like families.

1. Introduction

The study of arithmetic of zeta values begins by Euler’s famous evaluations: for m ∈N, ζ(2m) = −B_2m 2π√

−12m

2(2m)! ,

where B_2m ∈ Q are Bernoulli numbers. Euler’s formula implies that ζ(n)/(2π√

−1)ⁿ is rational if and only if n is even. The multiple zeta values ζ(s₁,· · · , s_r), where s₁, . . . , s_r are positive integers with s1 ≥ 2, are generalizations of zeta values. These numbers were first studied by Euler for the case ofr= 2. Although there exist simple relations between zeta and multiple zeta values, such as ζ(2,1) = ζ(3), sorting out all relations among these multiple zeta values is a much involved problem. Hereris called the depth andw:=Pr

i=1s_i is called the weight ofζ(s1, . . . , sr). We callζ(s1, . . . , sr)Eulerian if the ratioζ(s1, . . . , sr)/ 2π√

−1)^w is rational.

Carlitz introduced and derived an analogue of Euler’s formula for what we now called Carlitz zeta values ζ_A(n) for n ≥1. Let A=F^q[θ] be the polynomial ring in the variable θ over a finite field Fq and K =Fq(θ) be its quotient field. Lett be a variable independent of θ. LetCbe the Carlitz module andeπis a fundamental period ofC. The Carlitz exponential function is defined by exp_C(z) = P

n≥0

z^qn

D_n , where D_n = Qn−1

i=0(θ^qn −θ^qi). We denote by Γ_n+1 ∈ A the Carlitz factorials and BC(n)∈ K by the Bernoulli-Carlitz numbers. Carlitz showed that

ζ_A(n) := X

a∈A+

1

aⁿ = BC(n) Γ_n+1 eπⁿ

if q−1|n. Carlitz’s result implies that ζA(n)/˜πⁿ is rational in K if and only ifq−1|n.

Anderson and Thakur [1] related the interesting valueζA(n) to a special integral point Zn

in C^⊗n(A) via the logarithm map of C^⊗n, where C^⊗n denotes the n-th tensor power of the Carlitz module (viewed as a Carlitz-Tatet-motive). As a consequence, one has thatζ_A(n)/˜πⁿ is rational if and only if Z_n is an Fq[t]-torsion point, and this condition is equivalent to n being divisible byq−1. In [1] a key role is played by a sequence of distinguished polynomials H_n ∈ A[t], now called the Anderson-Thakur polynomials. On the other hand, Yu [13] also showed that the transcendence of ζ_A(n)/˜πⁿ over K is equivalent to Z_n being non-torsion

1

on C^⊗n(A), whence deriving that ζ_A(n)/˜πⁿ is algebraic over K if and only if ζ_A(n)/˜πⁿ is rational in K.

In the last decade, Thakur [10, 11] initiated the study of multizeta values ζ_A(s₁,· · · , s_r), where s₁, . . . , s_r are positive integers. He and his co-workers discovered interesting relations among some multizeta values. Call ζ_A(s₁, . . . , s_r) Eulerian (zeta-like resp.) if the ratio ζA(s1,· · · , sr)/˜π^w (ζA(s1, . . . , sr)/ζA(w) resp.) is rational in K. A basic question in this respect is to find all Eulerian/zeta-like multizeta values. In [9], Lara Rodriguez and Thakur gave particularly precise formulas for certain families of Eulerian/zeta-like multizeta values and conjectured other ones. Their conjectures are supported by numerical data from con- tinued fraction computations. On the other hand, Chang [3] also proved the subtle fact that these ratiosζ_A(s₁,· · · , s_r)/˜π^w,ζ_A(s₁, . . . , s_r)/ζ_A(w) are either rational or transcendental over K.

In an effort to understand relations among multizeta values, Chang, Papanikolas and Yu [4] established an effective criterion for Eulerian/zeta-like multizeta values by constructing an abelian t-module E⁰ defined over A and relating the values ζ_A(s₁,· · · , s_r), ζ_A(w) to specific integral pointsv_s,u_s onE⁰(A). They proved thatζ_A(s₁,· · · , s_r) is Eulerian (zeta-like) if and only ifv_s is an Fq[t]- torsion point (respectively,u_s andv_s have anFq[t]-linear relation inside E⁰(A)). The integral points v_s,u_s are constructed using the Anderson-Thakur polynomials.

Their theory connects possible F^q(θ)-linear relation of ζA(s1,· · · , sr) and ζA(w) explicitly with the possible F^q[t]-linear relation among vs and us insideE⁰(A).

Just recently, Kuan-Lin [7] implemented algorithms basing on the criterion of Chang- Papanikolas-Yu. They have collected more extensive data on zeta-like and Eulerian multizeta values over the polynomial ringsFq[θ]. Particularly in [4, 9], a conjectured rule is spelled out to specify all Eulerian multizeta values. Lists given in [7] suggest more families of zeta-like multizeta values of arbitrary depth. These families are not covered by [9]. It is observed that there should be only a few zeta-like families in higher depth, because of the conjectured

“splicing” condition (cf. [9]). Finding all zeta-like multizeta values is now in sight.

Inspired by this development we study Anderson-Thakur polynomials in more details in this paper, for the purpose of deriving exact rational ratio betweenζ_A(s₁,· · · , s_r) andζ_A(w) whenever such a ratio exists. In particular, we are able to verify : (1) Conjecture 4.6 of [9], (2) Conjecture 5 of [7], (3) the conjectured list of all Eulerian multizeta values given in [4], Section 6.2, are indeed Eulerian. The strategy for proving zeta-like property for given multizeta values is to handle recurrence relations among Anderson-Thakur polynomials H_n. In view of the fact that these H_n are polynomials in both θ and t over F^q, we use Lucas Theorem to establish q-th power recurrence when n has particular q-adic “shape”. Combining with the obvious linear recurrence relating H_n toH_n−qⁱ, we eventually arrive at more transparent formulas for H_n.

The contents of this paper are arranged as follows. In Section 2, we set up preliminaries and introduce the conjectured families of zeta-like multizeta values given in [7] and [9], which we will prove later. In Section 3, we use generalized Lucas Theorem [6, p.75-76] to study Anderson-Thakur polynomials. Then in Section 4 we apply Chang-Papanikolas-Yu’s theorem [4, Theorem 2.5.2] to verify that all previously conjectured families of zeta-like multizeta values are indeed zeta-like with exact formulas given in Theorem 4.4. At the end of this paper we provide ‘recursive’ relations for two very special families of multizeta values and derive that they are Eulerian (Theorem 5.1, 5.2) in Section 5.

2. Preliminaries for Multizeta values 2.1. Notation. We adopt the notation below in the following chapters.

Fq = a finite field with q=p^l elements.

K = Fq(θ), the rational function field in the variable θ.

∞ = zero of 1/θ, the infinite place of K.

| |= the nonarchimedean absolute value on K corresponding to ∞.

K∞= Fq((1/θ)), the completion ofK with respect to the abso- lute value | · |.

A = Fq[θ], the ring of polynomials in the variable θ.

A₊= the set of monic polynomials in A.

Ad= the set of polynomials in A of degree d.

Ad+= Ad∩A+, the set of monic polynomials in A of degreed.

[n] = θ^qn−θ.

D_n=

n−1

Q

i=0

θ^qn−θ^qi = [n][n−1]^q· · ·[1]^qn−1. L_n=

n

Q

i=1

θ^qi −θ= [n][n−1]· · ·[1].

l_n = (−1)ⁿL_n.

t = a variable independent of θ.

2.2. Multizeta values. For s∈N and d∈Z^≥0, put S_d(s) = X

a∈Ad+

1 a^s ∈K.

Fors ∈Nthe Carlitz-Goss zeta values are defined by ζA(s) =

∞

X

d=0

Sd(s) = X

a∈A+

1

a^s ∈K∞.

For a given tuple (s₁,· · · , s_r) ∈ N^r, the Thakur multizeta values of depth r and weight w=P

s_i are defined by

ζ_A(s₁,· · · , s_r) = X

d1>···>dr≥0

S_d1(s₁)· · ·S_dr(s_r) = X

ai∈A+ dega1>···>degar≥0

1 a^s₁¹· · ·a^s_r^r. 2.3. Bernoulli-Carlitz numbers BC(n). For a non-negative integern, we expressn as

n =

∞

X

i=0

niqⁱ (0≤ni ≤q−1, ni = 0 for i0), and we recall the definition of the arithmetic Γ-function Γ_n+1 := Q∞

i=0Dⁿ_iⁱ ∈ A. We de- note by (−θ)^q−11 a fixed (q −1)-th root of −θ. Let C be the Carlitz module and ˜π = (−θ)^q−1q

∞

Y

i=1

(1− θ

θ^qi)⁻¹ be a fundamental period of C. The Carlitz exponential function is defined by exp_C(z) =P

n≥0

z^qn

D_n. The Bernoulli-Carlitz numbers BC(n)∈K are defined by

z

exp_C(z) =X

n≥0

BC(n) Γ_n+1 zⁿ.

When n is ‘even’ i.e.,q−1|n, Carlitz derived an analogue of Euler’s formula as follows:

Lemma 2.4. (Carlitz [2]) (a) For n≥1, ζ_A(n) = BC(n)

Γ_n+1 eπⁿ if q−1|n.

(b) For 0≤i≤n, BC(qⁿ−qⁱ) = (−1)ⁿ⁻ⁱΓ_qⁿ_−qⁱ₊₁ L^q_n−iⁱ . Combining (a), (b) we get ζ_A(qⁿ−1) = (−1)ⁿ

L_n eπ^qn−1.

2.5. Anderson-Thakur polynomials. First we define polynomialsGi ∈F^q[t, θ] fori∈Z^≥0. PutG₀ = 1. For i∈N, let

Gi =

i

Y

j=1

(t^qi −θ^qj).

For n = 0,1,2, . . ., we define the sequence of Anderson-Thakur polynomials H_n ∈ A[t] by the generating function identity

1−

∞

X

i=0

G_i D_i|_θ=tx^qi

!−1

=

∞

X

n=0

H_n Γ_n+1|_θ=txⁿ.

We note that for 0≤n ≤q−1 we have H_n= 1. For any infinite vector a= (a₀, a₁, a₂,· · ·) with integers ai ≥ 0 and aj = 0 for j 0, put m(a):= last index i such that ai 6= 0.

We define C_a := (a0+· · ·+am(a))!

a0!· · ·a_m(a)! . For n ∈ N, a q power weighted partition of n is an infinite vector a satisfying n=P∞

i=0a_iqⁱ. We have the following lemma giving two ways for explicitly writing Anderson–Thakur polynomials:

Lemma 2.6.

(a) For n ∈ Z^≥0, let S_n ={a | n =P

a_iqⁱ, C_a 6≡0 mod p} denote the set of all possible q power weighted partition of n with nonzero C_a mod p. Then

Hn

Γ_n+1|_θ=t = X

a∈Sn

C_a

∞

Y

i=0

( Gi

D_i|_θ=t)^ai. (b) For n∈N,

H_n Γ_n+1|_θ=t =

[logqn]

X

i=0

G_i D_i|_θ=t

H_n−qⁱ Γ_n−qⁱ₊₁|_θ=t.

We will discuss more details about Anderson-Thakur polynomials in Section 3.

2.7. Revisiting Lucas Theorem. To computeC_a modp, a useful tool is a generalization of the Lucas Theorem.

Theorem 2.8. (Dickson[6]) For any infinite vectora, C_a 6≡0 mod p if and only if there is no carrying in computing the suma₀+· · ·+a_m(a) in terms of basepexpansion. Furthermore, if P

a_i =Pm

j=0n_jp^j, a_i =Pm

j=0n_i,jp^j, with 0≤n_j, n_i,j ≤p−1 and n_j =Pm(a)

i=0 n_i,j. Then C_a≡Q

C_nj in Fp, where n_j = (n_0,j,· · · , n_m(a),j,0,0,· · ·).

Proof. See [6, p.75-76].

By Theorem 2.8 we see that C_a mod p can be computed as digits in base p expansion separately. So we try to descend Hn via the maps below. For simplicity we view Ca as elements in F^p.

Definition 2.9. For any infinite vector a with Ca 6= 0, let ˜a = ( ˜a0,a˜1,· · ·), where ai ≡ aei

mod q with 0≤ae_i ≤q−1. We define the following ‘reduction map’ of vectors.

r(a) := (a0−a˜0

q ,a1−a˜1

q ,· · ·).

By Theorem 2.8 we see that C_a=C_˜aC_r(a).

2.10. Binomial series to the Carlitz module. For k ∈ Z^≥0, let Ψ_k(x) be the polynomials inK[x] defined by

exp_C(xlog_C(u)) =X

k≥0

Ψ_k(x)u^qk. Here log_C(z) is the Carlitz logarithm defined by

log_C(z) = X

n≥0

z^qn l_n . Then Ψk(x) can be expressed as follows:

Proposition 2.11. (Anderson-Thakur [1]) Ψ_k(x) =

k

X

i=0

Qi

j=1(θ^qi−θ^qk+j) Di

( x

(−1)^kLk

)^qi. Moreover, Ψ_k(a) = 0 for all a∈Fq[θ] with deg_θa < k and Ψ_k(θ^k) = 1.

This result is another key tool in the proof of Theorem 3.3. For our purpose, we replace θ byt in Anderson-Thakur’s result so that Ψ_k|_θ=t(x)∈Fq(t)[x].

2.12. Conjectures on Eulerian/Zeta-like Multizeta Values. There are families of zeta- like multizeta values of arbitrary depth, for instance, in [9], they showed that for any q, ζ_A(1, q−1,(q−1)q,· · · ,(q−1)qⁿ) is zeta-like by giving the ratio of it to ζ_A(qⁿ⁺¹). There are certainly more families of zeta-like multizeta values of arbitrary depth, the following conjecture is given in [9]:

Conjecture 2.13.

(a) For any q, n≥1 and r≥2,

ζ_A(qⁿ−1,(q−1)qⁿ,· · · ,(q−1)q^n+r−2) = [n+r−2][n+r−3]· · ·[n]

[1]^qn+r−2[2]^qn+r−3· · ·[r−1]^qnζ_A(q^n+r−1−1).

(b) For any q, n≥0,

ζA(1, q²−1,(q−1)q²,· · ·,(q−1)qⁿ⁺¹) = [n+ 2]−1 l₁[n+ 2]

1

l^(q−1)q₁ ⁿl^(q−1)q₂ ⁿ⁻¹· · ·l_n−1^(q−1)q2ln^q2

ζA(qⁿ⁺²).

(c) For q >2, n ≥0 and r≥2,

ζ_A((q−1)qⁿ−1,(q−1)qⁿ⁺¹,· · · ,(q−1)q^n+r−1) = (−1)^r+1[n+r−1][n+r−2]· · ·[n+ 1]

[1]^{(qr−1−1)qn}[2]^{(qr−2−1)qn}· · ·[r−1]^(q−1)qnζ_A(q^n+r−qⁿ−1).

Remark 2.13.1. In [9] Conjecture 2.13 is proved in the depth 2 case. We refer to [9] for more details, in particular [9, Theorem 3.1], where many depth 2 zeta-like multizeta values are given with precise ratio to ζ_A(w).

According to Chang-Papanikolas-Yu criterion for zeta-like multizeta values in [4], Kuan- Lin [7] wrote an algorithm and tested multizeta values with bounded weights and depths by computer. From their output data, they gave another more extensive conjecture about zeta-like families of arbitrary depth and also specific depth 3 zeta-like multizeta values.

Conjecture 2.14. (Kuan-Lin-Yu) Suppose that q >2. Then we have the following families of zeta-like multizeta values:

(a) For q=p^l>2, 1≤p^m ≤q, n >0 and r≥2, consider N_i ∈Z^≥0 for 0≤i≤n−1 such that 1≤P

N_i ≤q−1. If (q−1)(qⁿ−P

N_iqⁱ)≤p^m(q−1)qⁿ⁻¹, then ζ_A(qⁿ−X

N_iqⁱ, p^m(q−1)qⁿ⁻¹,· · · , p^m(q−1)q^n+r−3) is zeta-like. In particular

ζA(1, p^m(q−1), p^mq(q−1),· · · , p^mq^r−2(q−1)) is zeta-like.

(b) In the case of depth r = 3,

ζ_A(1, q(q−1), q³−q²+q−1) = [3]−1

[3][2][1]^q2−q+1ζ_A(q³).

Remark 2.14.1. When n ≥ 1 in Conjecture 2.13 (c), it corresponds to a special case of Conjecture 2.14 (a) by taking p^m = q, N₀ = Nn−1 = 1 and N_i = 0 for 0 < i < n−1 . When n = 0 in Conjecture 2.13 (c), it corresponds to the case when n = 1 and N₀ = 2 in Conjecture 2.14 (a).

Note that when the weight w is ‘even’, the statement that ζ_A(s₁,· · · , s_r) is zeta-like is equivalent to that it is Eulerian. In Section 4 we will prove non-Eulerian part of Conjecture 2.13 and Conjecture 2.14. The Eulerian part of Conjecture 2.13 will be treated in Section 5

3. Investigation into Anderson-Thakur polynomials

In general Anderson-Thakur polynomialsH_nare complicated to investigate. However, for index n having very special q-adic expansion, we can give a nicer and simpler formula for suchH_n. For example, to prove Conjecture 2.14 (b), we need to compute the corresponding Anderson-Thakur polynomialsH₀,H_q²_−q−1,H_q³_−q²_+q−2 and H_q³₋₁. It is known thatH₀ = 1.

On the other hand, it can be directly proved that H_q²_−q−1 = Γ_q²_−q|_θ=t, H_q³₋₁ = Γ_q³|_θ=t and H_q²_−q = Γ_q²_−q+1|_θ=t(t−θ^q)^q−1

L^q−1₁ |_θ=t (These are special cases of Theorem 3.3). Recall that S_m is the set of all q power weighted partition a of m with C_a 6= 0 in Fp. Furthermore, for given b with 0≤b_i ≤q−1, let S_m,b be the subset of S_m collecting a satisfying a_i ≡b_i mod q for alli. By Lemma 2.6 and to use the reduction maps r(a), we can compute H_q³−q²+q−2. Proposition 3.1.

H_q³_−q²_+q−2 =−[2]^q−2|_θ=th

(t−θ^q)^q2−q+1+ [1]^q2−1|_θ=t(t−θ^q)i .

Proof. For any q power weighted partition (a₀, a₁, a₂,0,0,· · ·) of q³ −q² +q −2, we see that a₀ ≡ q−2 = p−2 + (p−1)p+· · ·+ (p−1)p^l−1 mod q. Let a_i ≡ a_i,0 +a_i,1p+

· · ·+ai,l−1p^l−1 modq, where i = 1,2 and 0 ≤ a_i,j ≤ p−1 for j ≥ 0. It follows that if C_a 6= 0, then a_1,0 +a_2,0 ≤ 1 and a_1,j +a_2,j = 0 for j > 0. This implies ˜a = (q − 2,0,0,· · ·) or (q−2,1,0,· · ·) or (q−2,0,1,0,· · ·) and hence S_q³−q²+q−2 is the disjoint union of S_q³−q²+q−2,(q−2,0,0,···), S_q³−q²+q−2,(q−2,1,0,···) and S_q³−q²+q−2,(q−2,0,1,···). The reduction map a7→r(a) induces bijections

r :S_q³_−q²+q−2,(q−2,0,0,···) →S_q²_−q, r :S_q³_−q²+q−2,(q−2,1,0,···) →S_q²_−q−1, r :S_q³_−q²+q−2,(q−2,0,1,···) →S_q²_−2q.

Moveover, we see thatCa =Cr(a)if ˜a= (q−2,0,0,· · ·) andCa =−Cr(a)if ˜a= (q−2,1,0,· · ·) or (q−2,0,1,· · ·). We obtain from Lemma 2.6 that

H_q³−q²+q−2

Γ_q³_−q²_+q−1|_θ=t =X

˜ a

X

a∈S_{q3−q2+q−2,˜a}

C_a( G₀

D₀|_θ=t)^a0Y

i≥1

( G_i D_i|_θ=t)^ai

=X

˜ a

X

a∈S_q3−q2+q−2,˜a

C_˜aC_r(a)( G₀

D₀|_θ=t)^{qa˜˜0+ ˜a0}Y

i≥1

( G_i

D_i|_θ=t)^{qa˜˜i+ ˜ai}

= ( H_q²_−q

Γ_q²−q+1|_θ=t)^q− G1

D₁|_θ=t(H_q²_−q−1

Γ_q²−q|_θ=t)^q− G2

D₂|_θ=t( H_q²_−2q Γ_q²−2q+1|_θ=t)^q On the other hand, by Lemma 2.6 (b) for n=q²−q, we have

H_q²_−2q

Γ_q²−2q+1|_θ=t = D₁|_θ=t G₁

H_q²_−q

Γ_q²−q+1|_θ=t − H_q²_−q−1 Γ_q²−q|_θ=t

.

It follows that H_q³_−q²_+q−2

Γ_q³−q²+q−1|_θ=t = −(t−θ^q)^q2−q+1

[2][1]^q(q−1)|_θ=t + −(t−θ^q)[1]^q−1|θ=t

[2]|_θ=t . By definition of Γ-function, we can easily derive that Γ_q³_−q²_+q−1 =D₂^q−1 and the result follows.

3.2. Formula for Anderson-Thakur polynomials Hm with m = qⁿ−P

Niqⁱ. For n ∈ N, consider a tuple (N₀,· · · , N_n−1) withN_i ∈Z^≥0 satisfying 0≤Pn−1

i=0 N_i ≤q−1. This implies qⁿ−Pn−1

i=0 Niqⁱ−1≥0. We have the following formula for these special polynomials:

Theorem 3.3. Let n∈N and N_i ∈Z^≥0 satisfying 0≤Pn−1

i=0 N_i ≤q−1. Then (3.3.1) H_qⁿ−P

Niqⁱ−1

Γ_qⁿ₋^P_Niqⁱ|_θ=t = (−1)^{Pn−2i=0(n−1−i)Niqi} Qn−2

i=0 L^N_n−1−i^iqi |_θ=t

n−1

Y

v=1

(t−θ^qv)^{Pn−1−vj=0 Njqj}.

The key idea is using Lemma 2.6 and Theorem 2.8 to descend Anderson-Thakur polyno- mials fromH_n to suitable H_m with m < n.

Proof. We will prove by induction on n and N_i. For n = 1 it is clear that Hq−N0−1

Γ_q−N0|_θ=t = 1 satisfying (3.3.1) for any 0 ≤ N₀ ≤ q − 1. For M ≥ 2, suppose that the statement holds for H_qn−Pn−1

i=0 Niqⁱ−1 with 1 ≤ n < M. Our goal is to prove the formula (3.3.1) holds for H_qM−PM−1

i=0 Niqⁱ−1. For the case N₀ = 0, let a = (a₀,· · · , a_M−1,0,0,· · ·) be a q power weighted partition of q^M − PM−1

i=0 N_iqⁱ −1 with C_a 6= 0. Then a₀ ≡ q −1 = (p− 1) +

· · ·+ (p−1)p^l−1 modq. By Theorem 2.8, it forces a_i ≡ 0 mod q for i ≥ 1. Then by considering ˜a = (q −1,0,0,· · ·), we have that r(a) is a q power weighted partition of q^M−1 −PM−2

i=0 N_i⁰qⁱ −1, where N_i⁰ = N_i+1. Conversely, if a⁰ ∈ S_qM−1−PM−2

i=0 N_i⁰qⁱ−1, let a = (qa⁰₀ +q−1, qa⁰₁,· · ·, qa⁰_M−1,0,0,· · ·). We put N₀ = 0 and N_i = N_i−1⁰ for i ≥ 1. Then a ∈ S_qM−PM−1

i=0 Niqⁱ−1 and r(a) = a⁰. Therefore, r : S_qM−PM−1

i=0 Niqⁱ−1 → S_qM−1−PM−2

i=0 N_i⁰qⁱ−1 is bijective. Moreover, Ca =C_r(a). It follows that

H_qM−PM−1 i=0 Niqⁱ−1

Γ_qM−PM−1

i=0 Niqⁱ|_θ=t = X

a∈SqM−PM−1 i=0 Niqi−1

C_a( G₀

D₀|_θ=t)^a0Y

ν≥1

( G_ν D_ν|_θ=t)^aν

= X

a⁰∈S

qM−1−PM−2 i=0 N0

iqi−1

C_a⁰( G0

D₀|_θ=t)^qa00+q−1Y

ν≥1

( Gν

D_ν|_θ=t)^qa0ν

= ( H_qM−1−PM−2 i=0 N_i⁰qⁱ−1

Γ_qM−1−PM−2

i=0 N_i⁰qⁱ|_θ=t)^q

= ((−1)^{PMi=0−3(M−2−i)Ni0qi} QM−3

i=0 L^N_Mⁱ⁰_−2−i^qi |_θ=t

M−2

Y

i=1

(t−θ^qi)^{PMj=0−2−iNj0qj})^q

= (−1)^{PMi=0−3(M−2−i)Ni+1qi+1} QM−3

i=0 L^N_Mⁱ⁺¹_−2−i^qi+1|_θ=t

M−2

Y

i=1

(t−θ^qi)^{PM−2−ij=0 Nj+1qj+1}

= (−1)^{PMi=1−2(M−1−i)Niqi} QM−2

i=1 L^N_Mⁱ_−1−i^qi |_θ=t

M−1

Y

i=1

(t−θ^qi)^{PM−1−ij=0 Njqj}. Note that the last step follows from the induction hypothesis and N₀ = 0.

Now we assume thatN₀ ≥1. By Lemma 2.6 (b) we have (3.3.2) H_qM−PM−1

i=1 Niqⁱ−(N0−1)−1

Γ_qM−PM−1

i=0 Niqⁱ+1|_θ=t = H_qM−PM−1 i=0 Niqⁱ−1

Γ_qM−PM−1

i=0 Niqⁱ|_θ=t +

M−1

X

j=1

G_j D_j|_θ=t

H_qM−PM−1 i=0 Niqⁱ−q^j

Γ_qM−PM−1

i=0 Niqⁱ−q^j+1|_θ=t. Note that q^M −PM−1

i=0 N_jqⁱ−q^j and q^M −PM−1

i=0 N_iqⁱ are congruent to q−N₀ modq. If we consider all possible vectors ˜a= (q−N₀,ae₁,· · · ,a_M˜−1,0,0,· · ·) with 0≤a˜_i ≤q−1 and C_ea6= 0, then P∞

j=0a˜j =PM−1

j=0 a˜j ≤q−1. This impliesq^M−PM−1

i=0 Niqⁱ−q^j−X

i≥0

˜ aiqⁱ ≥0 and q^M −PM−1

i=0 Niqⁱ −X

i≥0

˜

aiqⁱ ≥ 0. Hence for a given ˜a, The reduction map a 7→ r(a) induces bijections

r:S_qM−PM−1

i=0 Niqⁱ−q^j,˜a→S_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−q^j−1−1

r:S_qM−PM−1

i=0 Niqⁱ,˜a →S_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−1. By Lemma 2.6 (a) we have

H_qM−PM−1 i=0 Niqⁱ−q^j

Γ_qM−PM−1

i=0 Niqⁱ−q^j+1|_θ=t =X

˜ a

X

a∈SqM−PM−1

i=0 Niqi−qj ,˜a

Ca( G0

D₀|_θ=t)^a0Y

v≥1

( Gv

D_v|_θ=t)^av

=X

˜ a

X

a⁰∈S

qM−1−PM−1

i=1 (Ni+ ˜ai)qi−1−qj−1−1

C_˜aC_a⁰( G₀

D₀|_θ=t)^{qa00+ ˜a0}Y

v≥1

( G_v

D_v|_θ=t)^{qa0v+ ˜av}

=X

˜ a

C_a˜Y

v≥0

( Gv

D_v|_θ=t)^a˜v(

H_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−q^j−1−1

Γ_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−q^j−1|_θ=t)^q Similarly,

H_qM−PM−1 i=0 Niqⁱ

Γ_qM−PM−1

i=0 Niqⁱ+1|_θ=t =X

˜ a

C˜a

Y

v≥0

( G_v D_v|_θ=t)^a˜v(

H_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−1

Γ_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹|_θ=t)^q. SincePM−1

i=1 N_i+ ˜a_i ≤q−1−N₀+N₀−1 =q−2, by the induction hypothesis onM−1< M, one can show that

H_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−q^j−1−1

Γ_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−q^j−1|_θ=t =

H_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹−1

Γ_qM−1−PM−1

i=1 (Ni+ ˜ai)qⁱ⁻¹|_θ=t

(−1)^{(M−2−(j−1))qj−1} L^q_{M−2−(j−1)}^j−1 |_θ=t

M−2−(j−1)

Y

v=1

(t−θ^qv)^qj−1. It follows that

H_qM−PM−1 i=0 Niqⁱ−q^j

Γ_qM−PM−1

i=0 Niqⁱ−q^j+1|_θ=t = H_qM−PM−1 i=0 Niqⁱ

Γ_qM−PM−1

i=0 Niqⁱ+1|_θ=t

(−1)^M−j−1 L^q_M−j−1^j |_θ=t

M−j−1

Y

v=1

(t−θ^qv)^qj. By (3.3.2), we have

H_qM−PM−1 i=0 Niqⁱ−1

Γ_qM−PM−1

i=0 Niqⁱ|_θ=t = H_qM−PM−1 i=0 Niqⁱ

Γ_qM−PM−1

i=0 Niqⁱ+1|_θ=t(1−

M−1

X

j=1

G_j D_j|_θ=t

(−1)^M−j−1 L^q_M^j_−j−1|_θ=t

M−j−1

Y

v=1

(t−θ^qv)^qj).

Now we begin to prove that the formula (3.3.1) holds for q^M −PM−1

i=0 N_iqⁱ −1. Let f(θ) := 1−

M−1

X

j=0

G_j D_j|_θ=t

(−1)^M−j−1 L^q_M−j−1^j |_θ=t

M−j−1

Y

v=1

(t−θ^qv)^qj.

LetU denotes the collection of all subsets of{1,2,· · ·, M−1}. ForI ={i₁,· · · , i_m} ∈U, we put θ^I =θ^{qi1+···+qim}, and |I|=m, the number of elements in I. Then for i= 0,· · · , M−1,

G_j

M−j−1

Y

v=1

(t−θ^qv)^qj =

M−1

Y

v=1

(t^qj −θ^qv) =X

I∈U

θ^I(−1)^|I|(t^qj)^M−1−|I|. Since for distinct I1, I2,θ^I1 6=θ^I2, we have

f(θ) = 1−X

I∈U

(

M−1

X

j=0

(t^qj)^M−1−|I|

(−1)^M−j−1D_jL^q_M−j−1^j |_θ=t)θ^I(−1)^|I|. Observe that

1

(−1)^M−j−1L^q_M^j_−j−1|_θ=t = Qj

v=1(t^qj −t^qv+M−1) (−1)^M−1L^q_M−1^j |_θ=t . It follows that

f(θ) = 1−X

I∈U

(

M−1

X

j=0

Qi

v=1(t^qj −t^qv+M−1)(t^qj)^M−1−|I|

(−1)^M−1D_jL^q_M−1^j |_θ=t )θ^I(−1)^|I|= 1−X

I∈U

ΨM−1|_θ=t(t^M−1−|I|)θ^I(−1)^|I|. By Proposition 2.11, Ψ_M−1|_θ=t(t^M−1−|I|) = 0 if |I| > 0, and ΨM−1|_θ=t(t^M−1−|I|) = 1 if I is the empty set. In the later condition θ^I(−1)^|I|= 1 and hencef(θ) = 1−1 = 0. This implies 1−

M−1

X

j=1

G_j D_j|_θ=t

(−1)^M−j−1 L^q_M^j_−j−1|_θ=t

M−j−1

Y

v=1

(t−θ^qv)^qj = (−1)^M−1 L_M−1|_θ=t

M−1

Y

v=1

(t−θ^qv) and we deduce that H_qM−PM−1

i=0 Niqⁱ−1

Γ_qM−PM−1

i=0 Niqⁱ|_θ=t = H_qM−PM−1 i=0 Niqⁱ

Γ_qM−PM−1

i=0 Niqⁱ+1|_θ=t

(−1)^M−1 L_M−1|_θ=t

M−1

Y

v=1

(t−θ^qv)

= (−1)^{PMi=0−2(M−1−i)Niqi} QM−2

i=0 L^N_Mⁱ_−1−i^qi |_θ=t

M−1

Y

v=1

(t−θ^qv)^{PM−1−vj=0 Njqj}.

4. Main result on Zeta-like Multizeta values

In this section we will prove Conjecture 2.13 (b) with q >2 and Conjecture 2.14.

4.1. Frobenius twisting. We fix the following automorphism of the field of Laurent series over C^∞, which is referred to asFrobenius twisting:

C^∞((t)) → C^∞((t)), f :=P

iaitⁱ 7→ f⁽⁻¹⁾ :=P

iai

1 qtⁱ.

In [4], the following criterion is proved for deciding zeta-like multizeta values in terms of Anderson-Thakur polynomials.