Mordell–Tornheim Zeta Values, Their Alternating Version,

and Their Finite Analogs

CRYSTAL WANG,JIANQIANG ZHAO∗

Department of Mathematics,

The Bishop’s School,

La Jolla, CA 92037,

UNITED STATES OF AMERICA

∗&RUUHVSRQGLQJ$XWKRU

Abstract: The purpose of this paper is two-fold. First, we consider the classical Mordell–Tornheim zeta values

and their alternating version. It is well-known that these values can be expressed as rational linear combinations

of multiple zeta values (MZVs) and the alternating MZVs, respectively. We show that, however, the spaces

generated by the Mordell–Tornheim zeta values over the rational numbers are in general much smaller than the

MZV space and the alternating MZV space, respectively, which disproves a conjecture of Bachmann, Takeyama

and Tasaka. Second, we study supercongruences of some finite sums of multiple integer variables. This kind of

congruences is a variation of the so called finite multiple zeta values when the moduli are primes instead of prime

powers. In general, these objects can be transformed to finite analogs of the Mordell–Tornheim sums which can

be reduced to multiple harmonic sums. This approach not only simplifies the proof of a few previous results but

also generalizes some of them. At the end of the paper, we provide a general conjecture involving this type of

sums, which is supported by strong numerical evidence.

Key-Words: Mordell–Tornheim zeta values; finite Mordell–Tornheim zeta values; alternating Mordell–Tornheim

zeta values; multiple zeta values; finite multiple zeta values; supercongruence.

Received: April 9, 2023. Revised: November 24, 2023. Accepted: December 12, 2023. Published: December 31, 2023.

1 Introduction

Let Nand N0be the set of positive integers and

nonnegative integers, respectively. The classical

Mordell–Tornheim zeta values (MTZVs) are defined

as follows. Let k≥2be a positive integer. For all

s1, . . . , sk+1 ∈N

T(s1, . . . , sk;sk+1)

∞



m1=1 ···

∞



mk=1

ms1

1···msk

kk

j=1mjsk+1

.(1)

Note that in the literature this function is also de-

noted by ζMT,k(s1, . . . , sk;sk+1). They were first in-

vestigated by Tornheim in the case k= 2, and later

with s1=··· =sk= 1 in [1, 2, 3]. A lot re-

lated works have subsequently appeared, for example,

[4, 5, 6, 7, 8, 9]. On the other hand, by [10, Lemma

3.1] every such value can be expressed as a Q-linear

combination of multiple zeta values (MZVs) which

are defined by

ζ(s1, . . . , sd) := 

0<k1<···<kd

ks1

1. . . ksd

for all s1, . . . , sd−1≥1, sd≥2.

After the seminal works [3, 11] around early

1990s much more results concerning MZVs have

been found The books [12, 13] and the website [14])

are some good references. Consequently, we can de-

rive a lot of relations between MTZVs. A natural

question now arises: can every MZV be expressed as

aQ-linear combination of MTZVs? We will give a

negative answer in section 2.

It is important to consider the alternating version

of MZVs:

ζ(s1, . . . , sd;ϵ1, . . . , ϵd)

:= 

0<k1<···<kd

ϵk1

1. . . , ϵkd

ks1

1. . . ksd

(2)

for for all s1, . . . , sd≥1, ϵ1, . . . , ϵd=±1with

(sd, ϵd)= (1,1). Reference [15] contains a good

summary of their key properties.

Similar to the study of MZVs, we may add some

alternating signs to MTZVs and call them alternating

MTZVs. Or even more generally, we may consider

the multiple variable function

MT s1, . . . , sk;sk+1

z1, . . . , zk;zk+1 

∞



m1=1 ···

∞



mk=1

zm1

1. . . zmk

kzm1+···+mk

k+1

ms1

1···msk

kk

j=1mjsk+1

.(3)

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For instance, Mordell–Tornheim L-functions ([16])

and the second author’s colored Tornheim’s double

series ([17]) are both special cases of (3). When

z1, . . . , zk+1 =±1we call these values alternating

MTZVs and abbreviate them by putting a bar on top

of the arguments whenever the corresponding zj’s are

−1. For example,

T(¯

3,2; 4) =MT 3,2; 4

−1,1; 1

∞



m1,m2=1

(−1)m1

1m2

2(m1+m2)4.

For any n, d ∈Nand s= (s1, . . . , sd)∈Nd, we

define the multiple harmonic sums (MHSs) and their

p-restricted version for primes pby

Hn(s) := 

0<k1<···<kd<n

ks1

1. . . ksd

H(p)

n(s) := 

0<k1<···<kd<n

p∤k1,...,p∤kd

ks1

1. . . ksd

Here, dis called the depth, and |s|:= s1+···+sdthe

weight of the MHS. For example, Hn+1(1) is often

called the nth harmonic number. In general, as n→

∞, we see that Hn(s)→ζ(s)which are the multiple

zeta values (MZVs) when sd>1.

More than fifteen years ago, the author discovered

the following curious congruence (a short proof can

be found in [18])



i+j+k=p

i,j,k>0

ijk ≡ −2Bp−3(mod p)(4)

for all primes p≥3, where B’s are bernoulli numbers

defined by ∞



n=0

n!=x

ex−1.

Since then, several different types of generalizations

have been found in [19, 20, 21, 22]. In this paper, we

will concentrate on congruences of the following type

of sums. Let Ppbe the set of positive integers not di-

visible by p. For any positive integers r,d,s1, . . . , sd

and any prime p, we define

Zpr(s1, . . . , sd) := 

k1+···+kd=pr

k1,...,kd∈Pp

ks1

1···ksd

We will first decompose the above sums into finite

Mordell–Tornheim sums (11)) which in turn can

be studied using the theory of multiple harmonic sum

congruences.

For example, Yang and Cai generalized (4) in [23]

as follows. For α, β, γ ∈N, if w=α+β+γis odd

and prime p > w, then we have

Zpr(α, β, γ)≡pr−1Zp(α, β, γ) (mod pr),(5)

where

Zp(α,β, γ)

≡max{α,β}



j=1

(−1)α+β−j

α+β+γα+β−j−1

α−1

+α+β−j−1

β−1α+β+γ

j

+2(−1)γα+β

αδp−1,α+β+γBp−α−β−γ

modulo p. In section 3 we will extend (5) to the fol-

lowing: if α+β+γis even, then for all r≥1

Zpr(α, β, γ)≡Zp(α, β, γ)p2r−2(mod p2r).(6)

We also determine the value Zp(α, β, γ, λ)when the

weight is odd in Theorem 3.3.

At the end of the paper, we will present a con-

jecture related to some families of finite Mordell–

Tornheim sums.

2 Classical (alternating)

Mordell–Tornheim zeta values

It is well-known [10] that every MTZV can be ex-

pressed as a Q-linear combination of MZVs. How-

ever, it turns out that the space generated by MZVs

is much larger so that MTZVs do not generate whole

MZV space over Q.

Theorem 2.1. Let MZVwand MTZVwbe the Q-

vector spaces generated by MZVs and MTZVs of

weight w≥3, respectively. Then MTZVw=MZVw

for all 3≤w≤14. Further, let Pwbe the Padovan

numbers defined by P2=P3=P4= 1 and Pw=

Pw−2+Pw−3for all w≥5. Then dimQMTZV15 <

P15 = 28 and for all w≥40,

dimQMTZVw< Pw.

Proof. Let p(n)be the partition function of any pos-

itive integer n. A celebrated theorem of Hardy and

Ramanujan gives the asymptotic formula

p(n)∼1

4n√3exp(π2n/3) as n→ ∞,

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found in [24] or [25, p. 70, (5.1.2)]. For any fixed

weight w > 2, let Nwbe the number of MZTVs of

weight n. Then clearly

Nw=

w−1



j=2

(p(j)−1)

and

log(Nw) = O(√w).

On the other hand, it is well-known by Zagier’s con-

jecture that dw=dimQMZVwform the Padovan se-

quence as given in the theorem. Further, we know

that that dw≤Pwby [26, 27, 28]. Thus it is not hard

to see that

log(dw) = O(w).

Consequently the space generated by MTZVs of fixed

weight wshould be much smaller than MZVwfor all

sufficiently large w. It turns out that for all w≥40,

we can use the more accurate bound of Nwby par-

tition functions to obtain that Nw< Pwwith the

computer-aided computation. Hence

dimQMTZVw< Pw

for all w≥40. By straightforward computation with

the aid of MAPLE one can see that for all weight

3≤w≤14, the two Q-spaces are the same. But

for weight w= 15, one already sees that MTZV15 ≤

27 < P15 = 28.

Remark 2.2. We want to remark that our The-

orem 2.1 disproves a conjecture of Bachmann,

Takeyama and Tasaka (29, Conjecture 2.4]).

Similarly, every alternating MTZV can be ex-

pressed as a Q-linear combination of alternating

MZVs. For any w≥3, let AMTwand AMZwbe

the Q-vector spaces generated by alternating MTZVs

and alternating MZVs of weight w, respectively. For

example,

T(1,1; 1) = 2ζ(3), T (¯

1,12; 1) = −ζ(1,¯

3) −π4

72 ,

T(¯

1,¯

1; 1) = 1

4ζ(3), T (¯

1,¯

1,1; 1) = π4

240 ,

T(¯

1,1; 1) = −5

8ζ(3), T (¯

13; 1) = 3ζ(1,¯

3) −π4

240 ,

T(1,2; 1) = π4

72 ,T(¯

1,¯

1; 2) = 2ζ(1,¯

3),

T(1,1; 2) = π4

180 , T (1,¯

2; 1) = ζ(1,¯

3) −7π4

720 ,

T(¯

1,¯

2; 1) = π4

288 , T (¯

1,2; 1) = −ζ(1,¯

3) −π4

240 ,

T(1,1,1; 1) = π4

15 , T (¯

1,1; 2) = −ζ(1,¯

3) −π4

480 ,

where 1nmeans 1 is repeated ntimes. Therefore

AMT3=⟨ζ(3)⟩Qand AMT4=⟨ζ(4), ζ(1,¯

3)⟩Q. As

an alternating analog to Theorem 2.1 we have the fol-

lowing result.

Theorem 2.3. Let Fwbe the Fibonacci numbers de-

fined by F0=F1= 1 and Fw=Fw−1+Fw−2for

all w≥2. Then for all 3≤w≤12 and w≥34, we

have

dimQAMTw< Fw.

Proof. When 3≤w≤12 we computed the set of

generators of AMTw( Remark 2.6). Let Awbe the

number of alternating MTZVs of weight w. For each

MTZV of weight w, we first determine how many

ways to put some alternating signs. Suppose we have

such an MTZV

T({s1}j1, . . . , {sr}jr;w−i), s1<··· < sr,

where for each string swe denote the string obtained

by repeating sexactly jtimes by {s}j. Now for each

{sℓ}jℓ(1≤ℓ≤r) there are jℓ+ 1 ways to put al-

ternating signs because of the symmetry. Thus the

number of ways to put some alternating signs on this

MTZV is



ℓ=1

(jℓ+ 1) subject to the condition



ℓ=1

jℓsℓ=i.

(7)

Then it is not hard to see that for fixed ithe maximal

value of (7) is achieved when i=m(m+1)/2 is a tri-

angular number and the MTZV is T(1,2, . . . , m;w−

i). In this case, the value of (7) is 2mwhere m=

(√8i+ 1 −1)/2. Moreover, the subspace MTZV34

has dimension bounded by the Padovan number P34.

Thus

dimQAMTw≤Aw−Nw+Pw

≤

w−1



i=2 √8i+ 1 −1

2(p(i)−1) −Nw+Pw< Fw

for all w≥34 by computer computation.

In general, we have the following conjecture.

Conjecture 2.4. For every w≥3,AMTwcan be

generated by the following set of elements

Cw:= 









ζ(k; 1, . . . , 1,−1)(2πi)2n:

2n+

λ(k)|k|=w, n ≥0









,(8)

where the product runs through all possible Lyndon

words k= (1) on odd numbers (with 1<3<5<

···) with multiplicity λ(k)so that 2n+kλ(k)|k|=

w. Here, ζ’s are defined by (2).

Proposition 2.5. If Conjecture 2.4 holds then

dimQAMTw≤Fw−2∀w≥3.

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Proof. Deligne proved in [30, Thm. 7.2] that for

all w≥1the Q-vector space AMZwof alternating

MZVs can be generated by

Bw:= 



ζ(k; 1, . . . , 1,−1)(2πi)2n:

2n+

λ(k)|k|=w, n ≥0





(9)

where the product runs through all possible Lyndon

words kon odd numbers (with 1<3<5<···)

with multiplicity λ(k)so that 2n+kλ(k)|k|=w.

Note that the ordering of indices in the definition of

Euler sums is opposite in loc. sit. So the definition

of Lyndon words here has opposite order, too. Fur-

thermore, if a period conjecture of Grothendieck [30,

Conjecture 5.6] holds then Bwis a basis of AMZw. In

particular, ♯Bw=Fwis the Fibonacci number. Note

that the only difference between Bwand Cwis that

k= (1) in Cw. Hence, if Conjecture 2.4 holds then

AMTwis generated by Cw=Bw\ζ(¯

1)Bw−1and

therefore

dimQAMTw≤Fw−Fw−1=Fw−2∀w≥3,

as desired.

Remark 2.6. Using Maple and the table of values for

alternating MZVs provided by [15] we have verified

Conjecture 2.4 for weight w≤12.

It turns out if Grothendieck’s Period Conjecture

[30, Conjecture 5.6] holds then by direct computation

dimQAMTw=Fw−2∀3≤w≤10.

But already in weight w= 11,

dimQAMT11 =F9−1 = 54.

In fact, to find the set of generators for AMT11,

one only needs to modify B11 \ζ(¯

1)B10 by

replacing the two elements ζ(1,1,¯

3)ζ(1,1,1,¯

and ζ(1,1,1,3,1,1,¯

3) by their linear combination

2ζ(1,1,¯

3)ζ(1,1,1,¯

3) + ζ(1,1,1,3,1,1,¯

3).

3 Supercongrence related to finite

Mordell–Tornheim zeta values

Recall that Ppis the set of positive integers not divis-

ible by p. For any prime pand positive integer r, we

call the sum

Tpr(α1, . . . , αm;λ)

:= 

k1,...,km∈Pp

|k|<pr,|k|∈Pp

kα1

1···kαm

m|k|λ

afinite Mordell–Tornheim sum. Here |k|=k1+···+

km. We then define the p-restricted finite Mordell–

Tornheim sums as follows. For any m, n, r ∈Nand

α1, . . . , αm, λ1, . . . , λn∈N0, we set

Tpr(α1, . . . , αm;λ1, . . . , λn)

:= 

|k|=u1<···<un<pr

k1,...,km,u1,un∈Pp

u2−u1,...,un−un−1∈Pp

kα1

1···kαm

muλ1

1···uλn

Here, we call m+n−1the depth and α1+··· +

αm+λ1+··· +λnthe weight of this sum.

By definition we have

Zpr(α1, . . . , αn+1)

=

k1+···+kn+1=pr

k1,...,kn+1∈Pp

kα1

1···kαn+1

n+1

(10)

=

u=k1+···+kn<pr

k1,...,kn,u∈Pp

kα1

1···kαn

n(pr−u)αn+1

= (−1)αn+1 

u=k1+···+kn<pr

k1,...,kn,u∈Pp

1−pr

u−αn+1

kα1

1···kαn

nuαn+1

≡(−1)αn+1 

u=k1+···+kn<pr

k1,...,kn,u∈Pp

1

kα1

1···kαn

nuαn+1

+αn+1pr

kα1

1···kαn

nuαn+1+1 

≡(−1)αn+1 Tpr(α1, . . . , αn;αn+1)

+αn+1prTpr(α1, . . . , αn;αn+1 + 1)(11)

modulo p2r. Therefore, we have decomposed

Zpr(α1, . . . , αn+1)as a sum of finite Mordell–

Tornheim sums.

Define

Hpr(s1, . . . , sd)

:= 

0<u1<···<ud<pr

u1,u2−u1,...,ud−ud−1,ud∈Pp

us1

1···usd

Theorem 3.1. Let pbe a prime, a, b, r ∈Nsuch that

p > w =a+b. If wis odd then

Hpr(a, b)≡pr−1Hp(a, b) (mod pr).

If wis even then

Hpr(a, b)≡p2r−2Hp(a, b) (mod p2r).

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Proof. The case r= 1 is trivial so we may assume

r≥2.

First we assume the weight is even. By Euler’s

theorem, setting m=φ(p2r)−aand n=φ(p2r)−b

we get, modulo p2r,

Hpr(a, b)≡

k<l<pr;k,l,l−k∈Pp

kmln

≡

k<l<pr

k,l∈Pp

kmln−

t<pr−1, k+pt<pr;k∈Pp

km(k+pt)n

≡

k<l<pr

kmln−

t<pr−1

k<pr−pt

km(k+pt)n

since min{m, n}> φ(p2r)−w≥(p2r−1−1)(p−

1) ≥2r. Now



k<l<pr

kmln=



j=0 m+ 1

jBj

m+ 1 

l<pr

lm+1−j+n



j=0 m+ 1

jBj

m+ 1

m+n+1−j



i=0 m+n+ 2 −j

i

×Bipr(m+n+2−j−i)

m+n+ 2 −j

≡pr



j=0 m+ 1

jBjBm+n+1−j

m+ 1

+p2r



j=0 m+ 1

j(m+n+ 1 −j)BjBm+n−j

2(m+ 1)

≡ −prBm+n

2+p2r



j=0 m+ 1

j

×(m+n+ 1 −j)BjBm+n−j

2(m+ 1)

modulo p2r, since m+nis even. Note that in the last

sum above, if (p−1)|jor (p−1)|(m+n−j)then

BjBm+n−jmay not be p-integral, but pBjBm+n−j

must be since (p−1) ∤(m+n). On the other hand,

modulo p2r,



t<pr−1

k<pr−pt

km(k+pt)n



s=0 n

s

t<pr−1, k<pr−pt

(pt)n−skm+s

≡



s=0 n

sm+s



j=0 m+s+ 1

jBj

m+s+ 1

×

t<pr−1

(pr−pt)m+s+1−j(pt)n−s.

Putting ν=m+s+ 1 and µ=m+n, we get



t<pr−1

k<pr−pt

km(k+pt)n

≡



s=0 n

sm+s



j=0 m+s+ 1

jBj

m+s+ 1

×

t<pr−1(ν−j)pr(−pt)m+s−j+ (−pt)ν−j(pt)n−s

≡Bµ

t<pr−1

pr+



s=0

m+s



j=0;

j=µ

µ−j



i=0 n

sν

j

×µ+ 1 −j

i(ν−j)(−1)m+s−jBjBi

ν(µ+ 1 −j)

×pr+µ−j+(r−1)(µ+1−j−i)

−pBµ

t<pr−1



s=0 n

sm+s



j=0;

j=µ

µ+1−j



i=0 ν

j

µ+ 2 −j

i(−1)m+s+1−jBjBi

(m+s+ 1)(µ+ 2 −j)

×pµ+1−j+(r−1)(µ+2−j−i)

≡Bm+n

pr(pr−1−1)

2(mod p2r).

Combining all the above together, we see that

Hpr(a, b)≡ −p2r−1Bm+n

+p2r



j=0 m+ 1

j(m+n+ 1 −j)BjBm+n−j

2(m+ 1)

≡p2r−2Hp(a, b) (mod p2r)

which follows from the proof of the case with r= 1

while keeping m=φ(p2r)−aand n=φ(p2r)−b.

This completes the proof of the theorem when the

weight is even. The proof of the odd weight case

is similar but simpler so we leave it to the interested

reader.

Theorem 3.2. For all r, α, β, γ ∈Nand primes p >

α+β+γ, if α+β+γis odd then we have

Zpr(α, β, γ)≡Zp(α, β, γ)pr−1(mod pr).(12)

Furthermore, if α+β+γis even, then

Zpr(α, β, γ)≡Zp(α, β, γ)p2r−2(mod p2r).

(13)

Proof. By a result of Bradley and Zhou, it can be

shown that all Mordell–Tornheim sums can be re-

duced to the finite multiple zeta values defined in

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the introduction. Indeed, by [10, Lemma 3.1], when

n= 2, we have

Tpr(α, β;γ)

=

u=k1+k2<pr

k1,k2,u∈Pp

α−1



a=0 a+β−1

aka

1kβ

ua+β

β−1



b=0 b+α−1

bkα

1kb

uα+b1

kα

1kβ

2uγ

α−1



a=0 a+β−1

aHpr(α−a, a +β+γ)

β−1



b=0 b+α−1

bHpr(β−b, b +α+γ)



a=1 α+β−a−1

α−aHpr(a, w −a)



b=1 α+β−b−1

β−bHpr(b, w −b),

where w=α+β+γ. Thus the theorem follows from

formula (11) and Theorem 3.1 immediately.

When n= 3, the situation is completely similar

although the formulas are more involved. Let m=

α+β+γand w=α+β+γ+λ. Then

Tpr(α, β, γ;λ)

=

u=k1+k2+k3<pr

k1,k2,k3,u∈Pp

α−1



a=0

β−1



b=0 a+b+γ−1

a, b, γ −1ka

1kb

2kγ

uγ+a+b

α−1



a=0

γ−1



c=0 a+c+β−1

a, c, β −1ka

1kβ

2kc

uβ+a+c

β−1



b=0

γ−1



c=0 b+c+α−1

b, c, α −1kα

1kb

2kc

uα+b+c1

kα

1kβ

2kγ

3uλ



a=1



b=1 m−a−b−1

α−a, β −b, γ −1

×Tpr(a, b, 0; w−a−b)



a=1



c=1 m−a−c−1

α−a, γ −c, β −1

×Tpr(a, c, 0; w−a−c)



b=1



c=1 m−b−c−1

β−b, γ −c, α −1

×Tpr(b, c, 0; w−b−c)

Thus, by [10, Lemma 3.1] or [23, Lemma 2.8]

Tpr(α, β, 0; λ)

α−1



s=0 s+β−1

sHpr(α−s, β +s, λ)

β−1



t=0 t+α−1

tHpr(β−t, α +t, λ).

Then we get, modulo p2r,

(−1)λZpr(α, β, γ, λ)

≡



a=1



b=1 n−a−b−1

α−a, β −b, γ −1a−1



s=0 s+b−1

s

Hpr(a−s, b +s, w −a−b)

+λprHpr(a−s, b +s, w −a−b+ 1)



a=1



b=1 n−a−b−1

α−a, β −b, γ −1b−1



t=0 t+a−1

t

Hpr(b−t, a +t, w −a−b)

+λprHpr(b−t, a +t, w −a−b+ 1)



a=1



c=1 n−a−c−1

α−a, γ −c, β −1a−1



s=0 s+c−1

s

Hpr(a−s, c +s, w −a−c)

+λprHpr(a−s, c +s, w −a−c+ 1)



a=1



c=1 n−a−c−1

α−a, β −b, β −1c−1



t=0 t+a−1

t

Hpr(c−t, a +t, w −a−c)

+λprHpr(c−t, a +t, w −a−c+ 1)



b=1



c=1 n−b−c−1

β−b, γ −c, α −1b−1



s=0 s+c−1

s

Hpr(b−s, c +s, w −b−c)

+λprHpr(b−s, c +s, w −b−c+ 1)



b=1



c=1 n−b−c−1

β−b, γ −c, α −1c−1



t=0 t+b−1

t

Hpr(c−t, b +t, w −b−c)

+λprHpr(c−t, b +t, w −b−c+ 1).

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The above computation quickly yields the following

result.

Theorem 3.3. Let pbe a prime and α, β, γ, λ ∈N

such that w=α+β+γ+λis odd. If p>w+ 2

then we have, modulo p,

Zp(α, β, γ, λ)

≡(−1)λα



a=1



b=1

fn, a, b

α, β, γtp(a, b;w−a−b)



a=1



c=1

fn, a, c

α, γ, βtp(a, c;w−a−c)



b=1



c=1

fn, b, c

β, γ, αtp(b, c;w−b−c)Bp−w

where n=α+β+γ,

tp(α, β;λ)

α−1



s=0 s+β−1

shp(α−s, β +s, λ)

β−1



t=0 t+α−1

thp(β−t, α +t, λ),

fn, a, b

α, β, γ=n−a−b−1

α−a, β −b, γ −1,

and

hp(α, β, γ) = 1

2n(−1)αn

α−(−1)γn

γ.

Proof. Observe that

Hp(α, β, γ) = H(p)

p(α, β, γ) = Hp(α, β, γ).

Taking r= 1 in the above computation, we see that

the theorem follows from [13, Thm. 8.5.13] quickly.

The following conjecture is supported by some ex-

tensive numerical evidence.

Conjecture 3.4. Let r∈N,pbe a prime and s∈Nd

such that p > |s|+ 1.

•If d= 4 and |s|is odd:

Zpr(s)≡p2r−2Zp(s) (mod p2r−1).(14)

•If d= 4 and |s|is even:

Zpr(s)≡pr−1Zp(s) (mod pr).(15)

•If d= 5 and |s|is even:

Zpr(s)≡p2r−2Zp(s) (mod p2r−1).(16)

In general, if r≥2and d+|s|is odd then we have

Zpr(s)≡0 (mod p2r−2).(17)

If r≥2and d+|s|is even then we have

Zpr(s)≡0 (mod pr−1).(18)

In general, the powers of moduli in (14)–(16) can-

not be increased. For example,

Z133(8,1,1,1) ≡134Z13(8,1,1,1) (mod 135),

but

Z133(8,1,1,1) ≡134Z13(8,1,1,1) (mod 136).

We further remark that the patterns in (14)–(16) do

not seems to continue for larger depths even though

(17) and (18) should hold for all d. This is also consis-

tent with the parity phenomenon such that when the

weight and the depth of Zpr(s)have different pari-

ties it can be “reduced further”, similar to the classi-

cal situation for the multiple zeta values. A detailed

description of a conjectural link between the classical

version of these values and their “finite” analogs can

be found in Chapter 8 of [13].

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Volume 22, 2023

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Contribution of individual authors.

Jianqiang Zhao wrote the paper after several discus-

sions with Crystal Wang.

Sources of funding.

Jianqiang Zhao was supported partially by the Jacobs

Prize from The Bishop’s School.

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Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.107

Crystal Wang, Jianqiang Zhao

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Volume 22, 2023