In the number fields case, we prove a version of Lerch’s formula forOKHil, the ring of integers of the Hilbert class field of a number fieldK

A NOTE ON LERCH’S FORMULAE FOR EULER QUOTIENTS

Alex Samuel Bamunoba¹

Department of Mathematics, University of Stellenbosch, South Africa.

bamunoba@aims.ac.za, bamunoba@gmail.com

Received: 4/7/14, Revised: 9/4/15, Accepted: 10/8/15, Published: 11/13/15

Abstract

Lerch’s formulae for Euler quotients in the ringsZandFq[t] have already been studied.

In this paper, we extend the study of these quotients to number fields and the Carlitz module. In the number fields case, we prove a version of Lerch’s formula forOK^Hil, the ring of integers of the Hilbert class field of a number fieldK. In theF^q[t] case, we replace the usual multiplication inFq[t] with the Carlitz module action⇢and prove two new versions of this formula. In addition, we relate these congruences to Carlitz Wieferich primes inFq[t]. All our proofs use properties of Carlitz polynomials.

1. Introduction

Letpbe an odd prime,aandnbe integers witha6= 0,±1, andn >1. TheFermat- Euler Theorem (also known as the Euler Totient Theorem) asserts that, if aand n are coprime, then a^'(n) ⌘1(mod n), where '(n) := #(Z/nZ)^⇤. Ifn = pand a is coprime to p, then '(p) = p 1 and soa^{p 1} ⌘ 1(mod p). This is the well known

“little theorem” of Fermat. These two congruences motivate the following definitions.

• Ifais coprime top, then theFermat quotient foraandpis defined to be q(a, p) := a^{p 1} 1

p .

• Ifais coprime ton, then theEuler quotient foraandnis defined to be q(a, n) := a^'(n) 1

n .

1The author was supported by the AIMS - DAAD In Country Scholarship Award (A/13/90157), the Post Graduate Merit Bursary Scheme at the University of Stellenbosch, South Africa and the government of Canada’s International Development Research Centre (IDRC) and within the frame- work of the AIMS Research for Africa Project.

There is a lot of literature that discusses the history and properties of these quotients, e.g., [8] and [10]. The questions addressed in these studies can be summarized into two categories: the first category deals with questions on the divisibility of the Fermat quotients whereas in the second category, one fixes a prime p, interpretsq(·, p) as a function and then estimates #{q(a, p) : 0ap 1}and the multiplicity of points in the image ofq(·, p), all considered modulop. For an in-depth study of these two categories, see [1] and [13]. For details on the multiplicativeFq[t]-analogues, see [6].

Amongst the many properties of the Fermat-Euler quotients, the one of interest to us is the congruence due to M. Lerch, see [8] and [7]. This congruence (or formula) relates the quotientq(a, n) to the sum over all representatives of elements of (Z/nZ)^⇤. The statement and proof of Lerch’s result in Theorem 1 are from [1, Theorem 9.3].

Theorem 1 ([1, Theorem 9.3]). Ifa andnare coprime, then

q(a, n) = a^'(n) 1

n ⌘

Xn (r,n)=1r=1

1 ar

har n

i

(modn),

where[x] is the greatest integer less than or equal tox.

Proof. Letr 1 be an integer less than and coprime ton. We writear⌘c( mod n), where c is a generator of a residue class in (Z/nZ)^⇤. Then ar = bn+c for some b = [^ar_n] 2 Z. As c goes through all residue classes in (Z/nZ)^⇤, so does r. Let S denote the product of all such representatives of the residue classes of (Z/nZ)^⇤. So

S= Yn

r=1

(r,n)=1

c= Yn

r=1

(r,n)=1

⇣ar nhar n

i⌘=a^'(n)S Yn

r=1

(r,n)=1

⇣1 n ar

har n

i⌘. (1)

Divide both sides of equation (1) byS to get

1 =a^'(n) Yn

r=1

(r,n)=1

⇣1 n ar

⌘⌘a^'(n) 0 B@1 n

Xn

r=1

(r,n)=1

1 ar

har n

i 1

CA(modn²).

Equivalently,

a^'(n) 1⌘a^'(n)n 0 B@

Xn

r=1

(r,n)=1

1 ar

har n

i 1

CA(mod n²). (2)

Divide both sides of congruence (2) bynand usea^'(n)⌘1( modn) on the right.

Upon interpretingq(·, p) as an operator, we obtain Theorem 2.

Theorem 2. Let a, bbe integers coprime to p. The quotientq(·, p)satisfies

q(ab, p)⌘q(a, p) +q(b, p)(mod p), q(a+bp, p)⌘q(a, p) b

a(modp)and 2q(2, p)⌘

p 1

X2

i=1

1

i (modp).

Remark 3. The congruences in Theorem 2 were discovered by G. Eisenstein [10].

Remark 4. The last congruence relation in Theorem 2 was first proved by J.

Sylvester.

Since the Fermat-Euler quotients are somehow hard to compute, it is natural to relate their sums over residue classes, and other quantities all defined modulo p.

It was Johnson [10] who gave a practical method for determiningq(a, p) using this philosophy. A sketch of the proof of his result follows. Lets2Zbe the least positive integer witha^s=±1 +tp, for somet2Z+. Thenp= 1 +sufor someu2Z+ and

q(a, p) = a^{p 1} 1

p = a^su 1

p = (±1 +pt)^u 1

p = (1±pt)^u 1 p

⌘ (1±upt) 1

p ⌘±ut⌘ ⌥t

s(modp).

There is also a link between Fermat quotients and Wieferich primes to base a. A Wieferich prime to base ais a primep(coprime to a) satisfyinga^{p 1} ⌘1(modp²).

This happens precisely when q(a, p) ⌘0(mod p). We shall briefly comment on the Carlitzian analogue of this result in Section 4, but for details, see [2] and [14].

The remainder of the paper is structured as follows. In Section 2, we shall state, and prove the OK^Hil-analogue of Lerch’s formula, state and prove properties of the OK^Hil-analogue of Fermat quotient (interpreted as an operator) and Johnson’s result.

In Section 3, we shall state (without proof) Y. Meemark and S. Chinwarakorn’sFq[t]- analogue of Lerch’s formula together with the properties of the associated Fermat quotient operator. In Section 4, we shall describe the Carlitz module and Carlitz cy- clotomic polynomials. Lastly, we shall prove two Carlitz module analogues of Lerch’s formula and some properties of the Carlitz-Fermat quotient operator.

2. Number Fields Analogue of Lerch’s Formula for Euler Quotients LetKbe a number field,OK be the ring of integers inK,nbe a nonzero ideal, andp be a nonzero prime ideal ofOK. SinceKis a number field, the ringOK is a finitely generated Z-module. SinceZis a principal ideal domain (PID), the quotientOK/n is finite. The norm ofnis defined as|n|:= #(OK/n) and'(n) := #(OK/n)^⇤. This 'is the number field extension of the Euler totient function in Section 1.

The property ofZ utilized a lot in Section 1 without explicit mention is the fact thatZ is a PID. In general, OK is not a PID, and so the proof of Lerch’s formula can not be adapted for general number fields. However, thePrincipal Ideal Theorem guarantees that we can recover unique factorization by considering the ring of integers in the Hilbert class field ofK. Of course, ifOK is a unique factorization domain then Kis its own Hilbert class field. In the ringOK^Hil,Fermat’s Theoremstates that, ifpis a prime ideal ofOK^Hil anda2OK^Hil witha /2p, thena^'(p)⌘1(modp). TheEuler Totient Theorem states that ifa2OK^Hil is such thata /2n, thena^'(n)⌘1(modn).

Let⇡,⇡^⇤2OK^Hil be the generators ofpandn, respectively, as (nonzero) ideals of OK^Hil. This naturally gives rise to the definition of Fermat and Euler quotients as

q(a,p) :=a^{|p| 1} 1

⇡ , and q(a,n) :=a^'(n) 1

⇡^⇤ , respectively. Theorem 5 is the ringOK^Hil-analogue to Lerch’s formula.

Theorem 5. Let a2OK^Hil andnbe a nonzeroOK^Hil-ideal. Ifa /2n, then

q(a,n) =a^'(n) 1

⇡^⇤ ⌘ X

r2OKHil, r /2n

|rO_KHil|<|n|

1 ar

har

⇡^⇤ i

(modn),

wheren=⇡^⇤OK^Hil and[_⇡^ar_⇤] is the quotient when aris divided by⇡^⇤.

Proof. Letr2OK^Hil nand |rOK^Hil|<|n|. Sincer /2n, we writear⌘c(modn), where cis a generator of a residue class in (OK^Hil/n)^⇤. Then ar= ⇡^⇤+c for some 2OK^Hil and so = [_⇡^ar_⇤]. Asc goes through residue classes in (OK^Hil/n)^⇤, so does r. LetS denote the product of representatives of elements of (OK^Hil/n)^⇤. Then

S= Y

r2OKHil n

|rO_KHil|<|n|

c= Y

r2OKHil n

|rO_KHil|<|n|

⇣ar ⇡^⇤har

⇡^⇤ i⌘

=a^'(n)S Y

r2OKHil n

|rO_KHil|<|n|

✓ 1 ⇡^⇤

ar har

⇡^⇤ i◆.

Divide through byS to get

1 =a^'(n) Y

r2OKHil n

|rO_KHil|<|n|

✓ 1 ⇡^⇤

ar

◆

⌘a^'(n) 0 BB

B@1 ⇡^⇤ X

r2OKHil n

|rO_KHil|<|n|

1 ar

har

⇡^⇤ i

1 CC

CA(mod n²).

Equivalently,

a^'(n) 1⌘a^'(n)⇡^⇤ 0 BB B@

X

r2OKHil n

|rO_KHil|<|n|

1 ar

har

⇡^⇤ i

1 CC

CA(mod n²). (3)

Divide both sides of congruence (3) by⇡^⇤and usea^'(n)⌘1( mod n) on the right.

Theorem 6. Let a, b 2 OK^Hil, let p be a prime ideal of OK^Hil, and let ⇡ be the uniformizer of p. The Fermat quotient operator q(·,p)satisfies

q(ab,p)⌘q(a,p) +q(b,p)(modp) and q(a+b⇡,p)⌘q(a,p) b

a(mod p).

Proof. Take⇡to be the uniformizer of the prime idealp. Then q(ab,p) = (ab)^{|p| 1} 1

⇡ = (ab)^{|p| 1} b^{|p| 1}+b^{|p| 1} 1

⇡

= b^{|p| 1}(a^{|p| 1} 1) +b^{|p| 1} 1

⇡ ⌘q(a,p) +q(b,p)(modp).

To prove the second congruence, we proceed as follows. We have q(a+b⇡,p) =(a+b⇡)^{|p| 1} 1

⇡ = a^{|p| 1}+ (|p| 1)a^{|p| 2}b⇡+· · ·+ (b⇡)^{|p| 1} 1

⇡

⌘ (a^{|p| 1} 1)

⇡ a^{|p| 2}b⌘q(a,p) b

a(modp).

It is not hard to show that ifpis coprime to 2, then q(2,p)⌘ 1

2 X

r2OKHil p

|rO_KHil|<|p|

1

r(modp).

Letsbe the least positive integer such that a^s⌘↵(modp), where↵2(OK^Hil)^⇤. Then a^s=↵+t⇡, wheret2OK^Hil andp=⇡OK^Hil. So|⇡|= 1 +su, u2(OK^Hil)^⇤ and

q(a,p) = a^{|p| 1} 1

⇡ =a^su 1

⇡ = (↵+t⇡)^u 1

⇡ = (1 + t⇡)^u 1

⇡

⌘ (1 + ut⇡) 1

⇡ ⌘ t

s(modp).

This is theOK^Hil-analogue of Johnson’s result.

Remark 7. The units±1 in Johnson’s result [10] are now replaced by↵, 2O_K^⇤Hil. Recently, J. Sauerberg, L. Shu [12] and other several authors have studied the multiplicativeFq[t]-analogues of these results. In Section 3, we state their findings.

3. Multiplicative Fq[t]-analogue of Lerch’s Formula for Euler Quotients LetA:=Fq[t] be the ring of polynomials in the variabletdefined over the finite field Fq and let P be a monic irreducible in A. For each a, m 2 A {0}, the absolute value of ais defined as |a| := #(A/aA) = q^deg(a). For this ring, the Euler Totient Theorem states that, if m 2 A is coprime to a, then a^'(m) ⌘ 1(mod m), where '(m) := #(A/mA)^⇤. If m = P, then we get an Fq[t]-analogue of Fermat’s Little Theorem [11, Chapters 1, 3]. TheFermat andEuler quotients are then defined as

q(a, P) := a^{|P| 1} 1

P andq(a, m) := a^'(m) 1

m ,

respectively. In Theorems 8 and 9, we give the twoFq[t]-analogues of Lerch’s formula as proved by Y. Meemark and S. Chinwarakorn. For their proofs, refer to [9].

Theorem 8 ([9], Theorem 2). If a, m2Aare coprime, then

q(a, m) = a^'(m) 1

m ⌘ X

deg(R)<deg(m)

(R,m)=1

1 aR

aR

m (modm),

where[^aR_m]is the quotient when aR is divided bym.

Fixd|q 1. For the primeP, thedth power residue symbol (_P^·)_d is defined as

⇣a P

⌘

d⌘

(a^{|P|d 1}(modP), (a, P) = 1,

0, otherwise.

Meemark and Chinwarakorn [9] defined theFermat quotient of degree dto baseaas q_d(a, P) :=a^{|P|d 1} (_P^a)_d

P .

Furthermore, Meemark and Chinwarakorn proved the following result.

Theorem 9 ([9], Theorem 3). If a2A is coprime toP, then

q_d(a, P)⌘

Ca

R0 (_P^a)d

P

! +⇣a

P

⌘

d

X

deg(R)<deg(P)

(R,P)=1 (_P^a)d=1

1 aR

aR

P (mod P),

where

R₀= Y

deg(R)<deg(P)

(R,P)=1 (_P^a)d=1

R and C_a= Y

deg(R)<deg(P)

(R,P)=1 (_P^a)d=1

✓ aR P

aR P

◆ .

Moreover, if there exists an ↵2F^⇤q such that(_P^a)d= (_P^↵)d, then qd(a, P)⌘↵^qd1deg(P) X

deg(R)<deg(P)

(R,P)=1 (_P^a)d=1

1 aR

aR

P (modP).

The properties of the quotient operatorq(·, P) are summarized in Theorem 10.

Theorem 10. Leta, b2A. For any primeP, we have thatq(·, P)satisfies q(ab, P)⌘q(a, P) +q(b, P)(modP), q(a+bP, P)⌘q(a, P) b

a(modP)and q(↵, P)⌘0(modP)for any↵2F^⇤q.

The proofs of the congruences in Theorem 10 are straightforward calculations from the definition ofq(·, P) and are therefore left for the reader.

Let s be the least positive integer for which a^s = ↵+bP for some b 2 A and

↵2F^⇤q. So|P|= 1 +sufor some positive integeru, (take ↵ ⌘1(modP)) and q(a, P) =a^{|P| 1} 1

P = (↵+bP)^u 1

P =(1 + bP)^u 1

P ⌘ b

s(modP).

This is the version of Johnson’s result associated with the Fermat quotient q(a, P).

Remark 11. Here, the units±1 in Johnson’s result are replaced by↵, 2F^⇤q. The above analogues are built out of the multiplicative parallels of Z inA. We obtain the additive versions by using the analogy coming from the Carlitz module.

4. Carlitz Fq[t]-module Analogues of Lerch’s Formula

We shall maintain A := Fq[t], let A₊ be the set of monic polynomials in A, let k be the fraction field of A and let F be an algebraically closed field containing k.

Furthermore, let⌧ be theqth power Frobenius map onF and letF{⌧}be the ring of twisted polynomials overFwith commutation relation⌧w=w^q⌧for allw2F. The ringF{⌧}is isomorphic to the non-commutative ring ofFq- linear polynomials inx with coefficients inFand multiplication defined by composition of polynomials. The map⇢:A!F{⌧}satisfyingt7!⇢t=⌧+t⌧⁰is called theCarlitz homomorphism.

With eachm2A {0},⇢associates the separable polynomial⇢m(x) :=⇢m(⌧)(x) called theCarlitz m-polynomial. Proposition 12 shows one way to compute⇢_m(x).

Proposition 12 ([5], Proposition 3.3.10). Let m2A {0}. Then

⇢m(x) =a_m,deg(m)x^|m|+· · ·+am,0x,

wherea_m,0=m, and[i]a_m,i = (a_{m,i 1})^q a_{m,i 1},i= 1, . . . ,deg(m).

As an example, we compute⇢_t²₊₁(x) using Proposition 12. Givenm=t²+ 12A, we havea_m,0=t²+ 1 anda_m,2= 1, sincemis a degree 2 monic polynomial. Lastly,

a_m,1= a^q_m,0 a_m,0

t^q t = (t²+ 1)^q (t²+ 1)

t^q t =t^2q t²

t^q t = (t^q t)(t^q+t)

t^q t =t^q+t.

So⇢_t2(x) =a_m,2x^q2+a_m,1x^q+a_m,0x=x^q2+ (t^q+t)x^q+ (t²+ 1)x.

Remark 13. We have a few consequences of Proposition 12, 1. Ifm is a monic polynomial inA, thena_m,deg(m)= 1.

2. Ifm=P^s whereP is a prime, thenam,i ⌘0(modP) for alli6= deg(m).

Remark 13 (2) implies that for anya, P 2A, we have that⇢_P(a)⌘a^|P|(modP).

As a consequence, we get an analogue for Fermat’s Little Theorem for the Carlitz module, i.e., for any a 2 A, ⇢P 1(a) ⌘ 0(mod P). This version of the theorem does not require aandP to be coprime. The requirement (a, P) = 1 gives rise to a definition of a Fermat quotient (we divide byato exclude the casea⌘0(modP)):

q_C(a, P) := ⇢_{P 1}(a)

aP = a^|P| a aP + 1

P

1+deg(P)X

i=0

a_P,ia^{qi 1}. We shall later refer to this as the Carlitz-Fermat quotient of type I.

For simplicity, we shall often refer to “Carlitz-something” as “c-something”. For example, Carlitz-Fermat quotient will become c-Fermat quotient.

Theorem 14. Leta, P 2A. IfaandP are coprime, then q_C(a, P) =⇢_{P 1}(a)

aP ⌘ X

deg(R)<deg(P)

(R,P)=1

1 aR

✓aR

P + (R)a^qdeg(R)

◆

(mod P),

where (R) = 1 ifR is monic and (R) = 0otherwise.

Before we prove Theorem 14, let us first recall the fundamental numbers used in the arithmetic of the ring A. We shall use the following notation: for each positive integeri, [i] :=t^qi t,L_i:= [i][i 1]· · ·[1] andD_i:= [i][i 1]^q· · ·[1]^{qi 1}. The symbol [i] is the product of monic irreducible polynomials of degree dividingi,L_iis the least common multiple of monic polynomials of degreeiandD_iis the product of all monic polynomials of degreei[5, Proposition 3.1.6] and [15, page 44]. To define [0], we use the philosophy that the empty product is equal to 1. So,L0=D0= [0] = 1.

For each i2Z 0, set

Si:= ( 1)ⁱ

Li = X

R2Ai+

1

R, (4)

where the sum runs over monic polynomials of degreei, [15].

Proof of Theorem 14. Suppose that deg(P) =n. By Proposition 12, we getaP,0=P and [i]a_P,i = a^q_{P,i 1} a_{P,i 1} for i = 1, . . . , n 1. Taking coefficients modulo P² givesa_P,0=P and [i]a_P,i=a^q_{P,i 1} a_{P,i 1}⌘ a_{P,i 1}(modP²) fori= 1, . . . , n. By recursion, we obtain the following chain of congruence relations:

LiaP,i= [i]Li 1aP,i 1⌘ Li 1aP,i 1= [i 1]Li 2aP,i 1

⌘( 1)²L_{i 2}a_{P,i 2}⌘· · ·⌘( 1)ⁱL₀a_P,0(modP²).

For alli= 1, . . . , n 1, we haveLi6⌘0(modP) and so aP qC(a, P) =⇢P 1(a) =a^|P| a+

n 1

X

i=0

aP,ia^qi⌘a^|P| a+P

n 1

X

i=0

( 1)ⁱ L_i a^qi

⌘aP q(a, P) +

n 1

X

i=0

Sia^{qi 1}

!

(modP²).

Dividing both sides of the congruence byaP gives q_C(a, P)⌘q(a, P) +

nX1 i=0

S_ia^{qi 1}(modP)

Eq.(4)

⌘ X

deg(R)<n

(R,P)=1

1 aR

aR

P + (R)

R a^{qdeg(R) 1}(modP)

⌘ X

deg(R)<n

(R,P)=1

1 aR

✓aR

P + (R)a^qdeg(R)

◆

(modP),

where is defined as (R) = 1 ifR is monic and (R) = 0 otherwise.

In [2] and [14], a c-Wieferich prime is defined to be any prime P satisfying

⇢P 1(1) ⌘0(mod P²). This is equivalent to saying thatqC(1, P)⌘0(mod P). So Lerch’s formula gives a criterion to check for c-Wieferich primes in A. Calculations for c-Wieferich primes are simplified by the fact thatq(↵, P) = 0 for any ↵2F^⇤q.

For each m 2A {0}, the set ⇤_m :={ 2F : ⇢_m( ) = 0} denotes the Carlitz m-torsion points. An element 2⇤misprimitiveif it generates⇤mas anA-module.

Definition 15. Letm2A+. The Carlitzm-cyclotomic polynomial is defined as

m(x) := Y

2⇤m:primitive

(x ).

m(x) has integer coefficients, degree'(m) and is irreducible overk. It satisfies nice relations, for example, factorization and composition identities, see [4] and [3].

Fixd|q 1. The c-Fermat quotient of degreedto baseais defined as qC,d(a, P) :=⇢P(a^1d) (_P^a)da^1d

a^1dP = ^P(a^1d) (_P^a)d

P .

This is related to the multiplicative Fermat quotient of degreedto baseaas follows:

q_C,d(a, P) = a^|P|d1 (_P^a)d

P + 1

P

nX1 i=0

a_P,ia^qid1 =q_d(a, P) + 1 P

n 1

X

i=0

a_P,ia^qid1,

wheren= deg(P). Moreover, if d= 1, thenqC,1(a, P) =qC(a, P).

Theorem 16. Leta2AandP be a prime in A. IfaandP are coprime, then

q_C,d(a, P)⌘ X

deg(R)<deg(P)

(R,P)=1

1 a^1dR

"

a^1dR P

#

+ (R)

R a^{qdeg(R)d 1}

!

(mod P),

where (R) = 1 ifR is a monic polynomial in Aand (R) = 0otherwise.

Proof. Letb2A be such thatb^d =a. Then

q_C,d(a, P) =q_C,d(b^d, P) =q_d(b^d, P) + 1 P

1+deg(P)X

i=0

a_P,ib^{qi 1}

=q(b, P) + 1 P

1+deg(P)X

i=0

aP,ib^{qi 1}=qC(b, P).

By Theorem 14, we have qC(b, P)⌘ X

deg(R)<deg(P)

(R,P)=1

1 bR

✓bR

P + (R)b^qdeg(R)

◆

(modP)

⌘ X

deg(R)<deg(P)

(R,P)=1

1 a^1dR

"

a^1dR P

#

+ (R) R a^q

deg(R) 1 d

!

(modP),

which completes the proof.

Theorem 17. Leta, b2AandP be a prime in A. We have that q_C(·, P)satisfies q_C(ab, P)⌘q(a, P) +q(b, P) + X

deg(f)<deg(P)

(f,P)=1

(f)

f (ab)^{qdeg(f) 1}(modP),

q_C(a+bP, P)⌘q_C(a, P) b

a(mod P), andq_C(↵, P)⌘ X

deg(f)<deg(P)

(f,P)=1

(f)

f (mod P), where (f) = 1 if f a monic polynomial inA and (f) = 0 otherwise.

Proof. We have that

q_C(ab, P) =⇢_{P 1}(ab)

abP = 1

abP 0

@(ab)^qdeg(P) ab+

1+deg(P)X

i=0

a_P,i(ab)^qi 1 A

=(ab)^qdeg(P) ab

abP + 1

abP 0

@

1+deg(P)X

i=0

aP,i(ab)^qi 1 A

⌘q(ab, P) + 0

@

1+deg(PX ) i=0

( 1)ⁱ

Li (ab)^{qi 1} 1

A(modP)

⌘q(a, P) +q(b, P) + X

deg(f)<deg(P)

(f,P)=1

(f)

f (ab)^{qdeg(f) 1}(mod P).

For the second congruence, we have that qC(a+bP, P) = ⇢P 1(a+bP)

(a+bP)P = ⇢P 1(a)

(a+bP)P + ⇢P 1(bP) (a+bP)P

⌘ ⇢_{P 1}(a)

aP +⇢_{P 1}(bP)

aP ⌘q_C(a, P) b

a(modP).

The last congruence follows from the first one utilising the fact that, for each↵2F^⇤q, q_C(↵, P)⌘q(↵, P) + X

deg(f)<deg(P)

(f,P)=1

(f)

f ↵^{qdeg(f) 1}⌘ X

deg(f)<deg(P)

(f,P)=1

(f)

f (modP).

To extendq_C(·, P) to a c-Euler quotient, we use Theorem 18 below.

Theorem 18 (Carlitz-Euler Totient Theorem of type I). If (a, m) = 1, then

m(a)⌘1(modm).

For the proper flow of the paper, we postpone the proof of Theorem 18 to after Theorem 23. Now since Theorem 18 is analogous to the Euler Totient Theorem, we define

q_C(a, P) := ^P(a) 1

P and q_C(a, m) := ^m(a) 1

m ,

as thec-Fermat andc-Euler quotients of type I respectively.

Theorem 19. Leta, m2A. Ifaandmare coprime, then

q_C(a, m) = ^m(a) 1

m ⌘ X

deg(R)<deg(m)

(R,m)=1

1 aR

✓aR m

◆ + 1

m

'(m) 1X

i=0

c_m,iaⁱ(mod m),

wherec_m,i is the coefficient of xⁱ in _m(x).

Proof. Leta, m2A. Ifaandmare coprime, then q_C(a, m) = ^m(a) 1

m = a^'(m) 1

m + 1

m

'(m) 1X

i=0

c_m,iaⁱ

⌘ X

deg(R)<deg(m)

(R,m)=1

1 aR

✓aR m

◆ + 1

m

'(m) 1X

i=0

cm,iaⁱ(modm).

Proposition 20. The sum of coefficients of _m(x)is congruent to1modulom.

Proof. Since (a, m) = 1, we have _m(a)⌘1( mod m), by Theorem 18. Seta= 1.

To define another analogue of the Fermat and Euler quotients in the Carlitzian context, we introduce the function _⇤. This is the map _⇤ : A₊ ! A defined by

⇤(m) =P

deg(D)<deg(m)(D, m). This is anFq[t]-analogue of the Pillai function.

Proposition 21. _⇤ is a multiplicative function.

Proof. By grouping the terms according to gcd, _⇤(m) = P

deg(a)<deg(m)(a, m) = P

D|m'(^m_D)(D, m). The result follows from the multiplicativity of the gcd map.

By definition, we have _⇤(1) = 1, _⇤(P^s) =P^{s 1}(P 1). For anya, b2A, ifais coprime tob, then _⇤(ab) = _⇤(a) _⇤(b), this is the multiplicativity property of _⇤. Proposition 22. Equivalently,

m=X

D|m

⇤(D), and _⇤(m) = X

D|m

Dµ⇣m D

⌘ . Proof. For the second formula, _⇤(m) =P

deg(a)<deg(m)(a, m) =P

D|m'(^m_D)(D, m) = P

D|mDµ ^m_D . LetIbe a map defined as I(m) = 1 for eachm2A. Iis completely multiplicative, so by the Mobius inversion formula, we getm=P

D|m ⇤(D).

With _⇤(·), we get the second (or additive) version of the c-Euler Totient Theorem.

Theorem 23 (Carlitz-Euler Totient Theorem of type II). Let a, m2A {0}. Then⇢ _⇤(m)(a)⌘0(modm).

Proof. Letn:= deg(P). Since _⇤is a multiplicative function, it suffices to check that

⇢ _⇤(P^s₎(a)⌘0(mod P^s). So ⇢_P^{s 1}(a) =Pn(s 1)

i=0 a_P^{s 1}_,ia^qi. It is not hard to show that vP(a_P^{s 1}_,i) = s 1 b_nⁱc for i = 0,1, . . . , n(s 1). If a =P g, g 2 A, then v_P(a_Ps 1,ia^qi) =s 1 b_nⁱc+qⁱ s, sinceqⁱ> ifori 0. Sov_P(⇢_Ps 1(P g)) s,

⇢ _⇤(P^s₎(a)

P^{s 1} = ⇢_P^{s 1}_{(P 1)}(a)

P^{s 1} =⇢_Ps 1(⇢_{P 1}(a))

P^{s 1} = ⇢_Ps 1(P g)

P^{s 1} ⌘0(modP).

It follows by the Chinese Remainder Theorem that⇢ _⇤(m)(a)⌘0(modm).

Proof of Theorem 18. To prove this result, there are three cases we need to consider.

1. m= P, a prime polynomial in A. Sincea and P are coprime, dividing both sides of the congruence⇢_P(a)⌘a(mod P) byagives _P(a)⌘1(modP).

2. m = P^s, where s 2 Z>1. Then ⇢ _⇤(P^s₎(a) ⌘ 0(mod P^s). It follows that

⇢_P(a) ⌘ a 6⌘ 0(mod P) and ⇢_Pi(a) 6⌘ 0(mod Pⁱ). By [2, Corollary 2.2.5]

together with the congruence⇢_P^s(a)⌘⇢_Ps 1(a)6⌘0(modP^s), we have that

P^s(a) = ⇢P^s(a)

⇢_P^{s 1}(a)⌘1(modP^s).

3. m has at least two prime factors. Here, it suffices to show that _m(a) ⌘ 1( mod P^s) for every prime factor ofm, withP^skm, wheres 1. Ifm=N P^s, then

m(a) = ^N(⇢_P^s(a))

N(⇢_Ps 1(a))= ^N(⇢ _⇤(P^s₎(a) +⇢_Ps 1(a))

N(⇢_Ps 1(a)) ⌘1(modP^s).

We define the c-Fermat and Euler quotients of type II as follows:

qC_⇤(a, P) :=⇢ _⇤(P)(a)

aP and qC_⇤(a, m) :=⇢ _⇤(m)(a) am . Theorem 24. Leta, m2A. Ifa, m2Aare coprime, then

q_C⇤(a, m)⌘ X

deg(R)deg(m)

06=R2A+

0

@µ⇣m R

⌘^deg(R)X

i=0

a_R,i m a^{qi 1}

1

A(modm),

whereµ(·)is the extended M¨obiusµfunction. The extended M¨obius function is defined asµ ^m_R =µ(C), wherem=CR for someC2Aand0if R-m.

Proof. We have

q_C⇤(a, m) = ⇢ _⇤(m)(a) am

Prop 22

⌘ X

D|m

µ⇣m D

⌘⇢D(a) am

⌘ X

deg(R)deg(m)

06=R2A+

µ⇣m R

⌘^deg(R)X

i=0

aR,i

m a^{qi 1}(mod m),

whereaR,iis the coefficient ofxⁱ in⇢R(x) andµ(·) is the extended M¨obius map.

Remark 25. The c-Fermat quotients of type I and II are the same.

AcknowledgementI thank my advisor, A. Keet, for having read and improved the drafts of this document. I also thank the editor for pointing out the many errors in the earlier manuscript.

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