Values of the Euler phi function not divisible by a given odd prime

c 2006 by Institut Mittag-Leffler. All rights reserved

Values of the Euler phi function not divisible by a given odd prime

Blair K. Spearman and Kenneth S. Williams

Abstract. An asymptotic formula is given for the number of integersn≤xfor whichφ(n) is not divisible by a given odd prime.

1. Introduction

We denote the set of natural numbers by N and the set of integers by Z. If a∈Zand b∈Z are not both 0, we denote the greatest common divisor of aand b by (a, b). We letφdenote Euler’s phi function so that forn∈Nwe have

φ(n) := card{m∈N|1≤m≤nand (m, n) = 1}=n

p|n

1−1

p

, (1)

where the product is taken over the distinct primespdividingn. Throughout this paperpdenotes a prime. It is well known that forn∈N,

2φ(n) ⇐⇒ n= 1,2.

We are interested in thosen∈Nfor whichqφ(n), whereqis a fixed odd prime. We set

Eq(x) = card{n≤x|qφ(n)}. (2)

In 1990 Erd˝os, Granville, Pomerance and Spiro gave an upper bound for E_q(x), which is valid for all sufficiently largex, see [1, Equation (4.2) with k=1, p. 191].

Both authors were supported by research grants from the Natural Sciences and Engineering Research Council of Canada.

In this paper we give an asymptotic formula for E_q(x) asx!∞, see the theorem in Section 4. Let 0<ε<1. Forqa fixed odd prime, we show that

Eq(x) =e(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}),

as x!∞, where e(q) is given in Definition 4.1 and the constant implied by the O-symbol depends only on q and ε. In 2002 Luca and Pomerance [2, Lemma 2, p. 114] proved the related result: For some constant c>0, for almost alln, φ(n) is divisible by all prime powersp^a≤clog logn/log log logn.

2. Notation

We denote the sets of real numbers and complex numbers byRandC, respec- tively. As usual Γ denotes the gamma function andγis Euler’s constant. IfKis an algebraic number field we writeh(K) for the class number ofK and R(K) for the regulator ofK, see for example [3, pp. 97, 106]. Throughout this paper qdenotes a fixed odd prime. We set

Kq:=Q(e^2πi/q)⊆C, (3)

so thatKq is a cyclotomic field with [Kq:Q]=φ(q)=q−1. For brevity we set h(q) :=h(K_q) and R(q) :=R(K_q).

(4)

We also let

ω:=e^2πi/(q−1)∈C, (5)

so thatω^q−1=1. The principal characterχ₀(mod q) is defined as follows: forn∈Z we have

χ₀(n) =

1, ifn≡0 (modq), 0, ifn≡0 (modq).

(6)

Let g be a primitive root (mod q). For n∈Z with n≡0 (modq) the index indg(n) ofnwith respect tog is defined moduloq−1 by

n≡g^indg(n)(mod q).

We define a characterχ_g (mod q) as follows: forn∈Zwe set χg(n) =

ω^indgn, ifn≡0 (modq), 0, ifn≡0 (modq).

(7)

There are exactlyφ(q)=q−1 characters (modq). They are χ₀, χ_g, χ²_g, ..., χ^q_g⁻², (8)

where χ^q_g⁻¹=χ₀.

3. The constantC(q)

It is convenient to define the following constant involvingχg.

Definition 3.1. Letqbe an odd prime. Letgbe a primitive root (modq). Let r∈{1,2, ..., q−2}. We define

C(q, r, χ_g) :=

χ_g(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)}

, (9)

where the product is taken over all primespsuch thatχg(p)=ω^r.

Note that the prime q is not included in the product asχ_g(q)=0 by (7). As 1≤(r, q−1)≤¹₂(q−1) forr∈{1,2, ..., q−2}we have

q−1 (r, q−1)≥2 (10)

so that the infinite product in (9) converges. Let h be another primitive root (modq). Then there exists an integers such that

h≡g^s(modq), (s, q−1) = 1.

Lettbe an integer such thatst≡1 (mod q−1). Then, forn∈Nwithn≡0 (modq), we have

indh(n)≡tindg(n) (modq−1) so that

χ_h(n) =ω^indh(n)=ω^tindg(n)= (χ_g(n))^t=χ^t_g(n), that isχ_h=χ^t_g. Hence

q−2 r=1

C(q, r, χ_h)^(r,q−1)=

q−2 r=1

χ_h(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)} (r,q−1)

=

q−2 r=1

χ^t_g(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)} ₍r,q−1)

=

q−2 r=1

χg(p)=ω^rs

1− 1

p^{(q−1)/(r,q−1)} ₍r,q−1)

=

q−2 r=1

χg(p)=ω^rs

1− 1

p^{(q−1)/(rs,q−1)}

₍rs,q−1)

=

q−2 r=1

χg(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)} ₍r,q−1)

=

q−2 r=1

C(q, r, χ_g)^(r,q−1) so that the product

q−2 r=1

C(q, r, χg)^(r,q−1) (11)

does not depend on the choice of primitive rootg. Thus we can make the following definition.

Definition3.2. Letq be an odd prime. We define the constantC(q) by

C(q) :=

q−2 r=1

C(q, r, χg)^(r,q−1). (12)

We take this opportunity to determineC(3). It is convenient to define the constant k_a,b(m) by

k_a,b(m) :=

p≡b(moda)

1− 1

p^m

, (13)

where a∈N and b∈N∪{0} are such that 0≤b<aand (a, b)=1 and m∈N is such that m≥2.

Lemma 3.1. C(3)=k_3,2(2).

Proof. Letq=3. Thenω=−1,r=1,g=2 andχ₂(n)=(−3/n). Hence C(3) =C(3,1, χ₂) =

χ₂(p)=−1

1− 1

p²

=

p≡2 (mod 3)

1− 1

p²

=k_3,2(2),

as asserted.

4. Statement of main result We begin with a definition.

Definition 4.1. Letqbe an odd prime. We define

e(q) :=

(q+1)(q−1)^{(q−2)/(q−1)}Γ 1

q−1

sin π

q−1 2^{(q−3)/2(q−1)}q^{3(q−2)/2(q−1)}π^3/2(h(q)R(q)C(q))^1/(q−1). (14)

Before stating our main result, we give the value ofe(3).

Lemma 4.1.

e(3) =2^7/2

3^9/4k₃,1(2)^1/2. Proof. We have Γ₁

2

=√

π, C(3)=k_3,2(2) and h(3)=R(3)=1, so that Defini- tion 4.1 withq=3 gives

e(3) = 2^5/2 3^3/4πk_3,2(2)^1/2.

As

1−1 3²

k_3,1(2)k_3,2(2) =

p

1−1

p²

= 6 π²

we have

k₃,2(2) = 27 4π²

1

k₃,1(2) and e(3) =2^7/2

3^9/4k₃,1(2)^1/2, as asserted.

Our main result is the following asymptotic formula forE_q(x).

Theorem. Let 0<ε<1. For qan odd prime, we have

E_q(x) =e(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}),

asx!∞, where the constant implied by theO-symbol depends only onqandε, and e(q)is given in Definition 4.1.

This theorem is proved in Section 7 after some preliminary results are given in Sections 5 and 6.

5. Preliminary results The following results will be used in Sections 6 and 7.

Proposition 5.1. Let n∈N and letq be an odd prime. Then

qφ(n) ⇐⇒ n=

p≡1 (modq)

p^a(p) orn=q

p≡1 (modq)

p^a(p),

where the product is taken over all primes p=qwithp≡1 (modq)and the a(p)are non-negative integers.

Proof. If

n=q^a t j=1

p^a_j^j,

where aandt are non-negative integers, thepj are distinct primes=q, and theaj

are non-negative integers, then by (1)

φ(n) =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ t j=1

p^a_j^j−1(p_j−1), ifa=0, q^a−1(q−1)

t j=1

p^a_j^j−1(pj−1), ifa≥1.

Henceqφ(n)⇔a∈{0,1}andqpj−1 (j=1, ..., t), which proves Proposition 5.1.

Next we define the setAby

A={m∈N|p(prime)|m⇒p=qandp≡1 (modq)}. (15)

The functionA(x) is defined forx∈Rby A(x) =

m≤x m∈A

1.

(16)

Proposition 5.2. Forx∈Randq an odd prime we have E_q(x) =A(x)+A

x q

.

Proof. This follows immediately from Proposition 5.1.

Proposition 5.3. (Wirsing’s theorem) Let f:N!R be multiplicative with f(n)≥0 for all n∈N. Suppose that there exist constants c₁ andc₂ with c₁>0 and 0<c₂<2 such that

0≤f(p^k)≤c₁c^k₂,

for all primes p and all k∈N, and also that there is a constant τ with τ >0 such that

p≤x

f(p) =τ x logx+o

x logx

,

asx!∞, then

n≤x

f(n) = e^−γτ

Γ(τ)+o(1) x

logx

p≤x

1+f(p)

p +f(p²) p² +...

,

asx!∞.

Proof. See [7, Satz 1, p. 76].

Proposition 5.4. (Odoni’s theorem) Let f: N!R be multiplicative with f(n)≥0 for all n∈N. Suppose that there exist constants a₁>1 and a₂>1 such that

0≤f(p^k)≤a₁k^a2,

for all primespand all k∈N, and also that there are constantsτ andβ with τ >0 and 0<β <1such that

p≤x

f(p) =τ x logx+O

x (logx)^1+β

,

asx!∞, then there is a constantB >0 such that

n≤x

f(n)n⁻¹=B(logx)^τ+O((logx)^τ−β), asx!∞. Further, for each fixedλ>0, we have

n≤x

f(n)n^λ−1=λ⁻¹Bx^λτ(logx)^τ−1+O(x^λ(logx)^τ−1−β), (17)

asx!∞.

Proof. See [4, Theorem II, p. 205; Theorem III, p. 206; Note added in proof, p. 216].

From Propositions 5.3 and 5.4 we obtain the following corollary.

Proposition 5.5. Let f: N!R be multiplicative with 0≤f(n)≤1 for all n∈N. Suppose that there are constantsτ andβ withτ >0 and 0<β <1such that

p≤x

f(p) =τ x logx+O

x (logx)^1+β

.

Then

xlim!∞

1 (logx)^τ

p≤x

1+f(p)

p +f(p²) p² +...

exists, and

n≤x

f(n) =Ex(logx)^τ−1+O(x(logx)^τ−1−β),

with

E=e^−γτ Γ(τ) lim

x!∞

1 (logx)^τ

p≤x

1+f(p)

p +f(p²) p² +...

.

Proof. The conditions of Odoni’s theorem are met (witha₁=a₂=2) so by (17) withλ=1 there is a constantB >0 such that

n≤x

f(n) =Bxτ(logx)^τ−1+O(x(logx)^τ−1−β).

The conditions of Wirsing’s theorem are also met (withc₁=c₂=1) so that

n≤x

f(n) = e^−γτ

Γ(τ)+o(1) x

logx

p≤x

1+f(p)

p +f(p²) p² +...

.

Equating the two expressions for

n≤xf(n), and dividing byx(logx)^τ−1, we obtain Bτ+O((logx)^−β) =

e^−γτ Γ(τ)+o(1)

(logx)^−τ

p≤x

1+f(p)

p +f(p²) p² +...

.

Letting x!∞we have

xlim!∞(logx)^−τ

p≤x

1+f(p)

p +f(p²) p² +...

=BτΓ(τ)e^γτ.

Thus

n≤x

f(n) =Ex(logx)^τ−1+O(x(logx)^τ−1−β), with

E=Bτ=e^−γτ Γ(τ) lim

x!∞(logx)^−τ

p≤x

1+f(p)

p +f(p²) p² +...

,

as asserted.

Proposition 5.6. Letk∈Nandl∈Nbe such that 1≤l≤kand (k, l)=1. Then

p≤x p≡l(modk)

1 = 1 φ(k)

x logx+O

x (logx)²

, asx!∞.

Proof. This is the prime number theorem for the arithmetic progression{kr+l| r=0,1,2, ...}, see for example [5, p. 139].

Let k∈N. Let χ be a character (modk). Let χ₀ be the principal character (modk). The DirichletL-series corresponding toχ is given by

L(s, χ) = ∞ n=1

χ(n) n^s , (18)

wheres=σ+it∈C. Forχ=χ₀, the series in (18) converges forσ >0 and L(1, χ) =

∞ n=1

χ(n)

n =

p

1−χ(p) p

₋₁

= 0.

(19)

For each characterχ(mod k) we define a completely multiplicative functionk_χ(n) (n∈N) by setting, for primesp,

k_χ(p) =p

1−

1−χ(p) p

1−1

p ₋χ(p)

. (20)

The Dirichlet series corresponding tokχ is given by K(s, χ) =

∞ n=1

kχ(n) n^s , (21)

where s=σ+it∈C. It is shown in [6] that the series in (21) converges absolutely forσ >0 and that

K(1, χ) = ∞ n=1

k_χ(n)

n =

p

1−k_χ(p) p

₋₁

=

p

1−χ(p) p

₋₁ 1−1

p χ(p)

= 0.

Proposition 5.7. Letk∈Nandl∈Nbe such that 1≤l≤kand (l, k)=1. Then

p≤x p≡l(modk)

1−1

p

=A(l, k)(logx)^−1/φ(k)+O((logx)^{−1/φ(k)−1}),

asx!∞, where

A(l, k) =

e^−γ k φ(k)

χ=χ0

K(1, χ) L(1, χ)

χ(l)₁/φ(k)

.

Proof. This proposition is Mertens’ theorem for the arithmetic progression {kr+l|r=0,1,2, ...}, which was first proved by Williams [6] in 1974.

Proposition 5.8. Let k, m, r∈N. Let ω_k be a primitive k-th root of unity.

Then

k−1 j=0

1−ω^jr_k

m

=

1− 1 m^k/(k,r)

₍k,r)

.

Proof. Letk, r∈N. Set h= k

(k, r) and s= r (k, r).

As (h, s)=1 theh-th roots of unity areω_h^js,j=0,1, ..., h−1. Thusω_h^js,j=0,1, ..., k−1 comprise theh-th roots of unity each repeatedk/htimes. Hence

(x^h−1)^k/h=

k−1 j=0

x−ω^js_h .

Takingx=m∈N, and dividing both sides bym^k, we obtain

1− 1 m^h

k/h

=

k−1 j=0

1−ω_h^js

m

=

k−1 j=0

1−ω_k^jr

m

,

which is the asserted result.

6. Estimation of

p≡1p≤x(modq)

(1−1/p)

We begin with the following result.

Proposition 6.1.

q−2 j=1

K 1, χ^j_g

= 1

C(q).

Proof. By Definition 3.1 we have

q−2 r=1

C(q, r, χg)^−(r,q−1)= lim

x!∞

q−2 r=1

p≤x χg(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)}

₋₍r,q−1)

.

Next, as

q−2

j=0

ω^jr=

q−1, ifr=0,

0, ifr=1,2, ..., q−2, we have

q−2 r=1

p≤x χg(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)}

₋₍r,q−1)

=

q−2 r=0

p≤x χg(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)}

₋₍r,q−1) 1−1

p

^q−2_j=0ω^jr

.

By Proposition 5.8 withm=p,k=q−1 andω=ω_q−1 we have

1− 1

p^{(q−1)/(r,q−1)} ₍r,q−1)

=

q−2 j=0

1−ω^jr

p

so that

q−2 r=0

p≤x χg(p)=ω^r

1− 1

p^{(q−1)/(r,q−1)}

₋₍r,q−1) 1−1

p

^q−2_j=0ω^jr

=

q−2 r=0

p≤x χ_g(p)=ω^r

q−2 j=0

1−ω^jr

p ₋₁

1−1 p

ω^jr

=

q−2 j=1

q−2 r=0

p≤x χg(p)=ω^r

1−ω^jr

p ₋₁

1−1 p

ω^jr

=

q−2 j=1

p≤x

1−χ^j_g(p) p

₋₁ 1−1

p χ^j_g(p)

.

Finally by Definition 3.2 we obtain 1

C(q)=

q−2 r=1

C(q, r, χg)^−(r,q−1)=

q−2 j=1

xlim!∞

p≤x

1−χ^j_g(p) p

₋₁ 1−1

p χ^j_g(p)

=

q−2 j=1

p

1−χ^j_g(p) p

₋₁ 1−1

p χ^j_g(p)

=

q−2

j=1

K(1, χ^j_g), as asserted.

Proposition 6.2.

q−2 j=1

L 1, χ^j_g

= 2^(q−3)/2q^−q/2π^(q−1)/2h(q)R(q).

Proof. The cyclotomic fieldKqis a totally complex field which contains exactly 2q roots of unity, namely{±1,±ωq,±ω_q², ...,±ω^q_q⁻¹}. Hence, by the class number formula for abelian fields applied to the cyclotomic fieldK_q, we have

h(q)R(q) = 2q|d(K_q)|^1/22^−(q−1)/2π^−(q−1)/2

q−2 j=1

L 1, χ^j_g

,

where d(K_q) is the discriminant of K_q, see for example [3, Theorem 8.4, p. 436].

Now the discriminant ofKq is given by

d(K_q) = (−1)^q(q−1)/2q^q−2, see for example [3, Theorem 2.9, p. 63]. Hence

q−2 j=1

L(1, χ^j_g) = 2^(q−3)/2q^−q/2π^(q−1)/2h(q)R(q), as asserted.

Proposition 6.3. Let qbe an odd prime. Then

p≤x p≡1 (modq)

1−1

p

=λ(q)(logx)^−1/(q−1)+O((logx)^−q/(q−1)),

asx!∞, where

λ(q) =

e^−γ2^−(q−3)/2q^(q+2)/2π^−(q−1)/2 (q−1)h(q)R(q)C(q)

1/(q−1)

.

Proof. By Propositions 6.1 and 6.2 we obtain

q−2 j=1

K(1, χ^j_g)

L(1, χ^jg) =2^−(q−3)/2q^q/2π^−(q−1)/2 h(q)R(q)C(q) .

By Proposition 5.7 withk=qandl=1, we have

p≤x p≡1 (modq)

1−1

p

=λ(q)(logx)^−1/(q−1)+O((logx)^−q/(q−1)),

where

λ(q) =A(1, q) =

e^−γ q q−1

q−2 j=1

K(1, χ^j_g) L(1, χ^jg)

₁/(q−1)

=

e^−γ q q−1

2^−(q−3)/2q^q/2π^−(q−1)/2 h(q)R(q)C(q)

₁/(q−1)

=

e^−γ2^−(q−3)/2q^(q+2)/2π^−(q−1)/2 (q−1)h(q)R(q)C(q)

₁/(q−1)

,

as asserted.

Proposition 6.4. Let 0<ε<1. Then

A(x) =α(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}), asx!∞, where

α(q) =

(q−1)^{(q−2)/(q−1)}Γ 1

q−1

sin π

q−1

2^{(q−3)/2(q−1)}q^{(q−4)/2(q−1)}π^3/2(h(q)R(q)C(q))^1/(q−1). (The constant implied by theO-symbol depends only onq andε.)

Proof. By (16) we have

A(x) =

n≤x n∈A

1 =

n≤x

f(n),

where

f(n) =

1, ifn∈A, 0, ifn /∈A.

Clearlyf(n) is a multiplicative function by (15). Moreover 0≤f(n)≤1 for alln∈N. By Proposition 5.6 we have

p≤x

f(p) =

p≤x p∈A

1 =

p≤x p≡1 (modq)

1+O(1) =q−2 q−1

x logx+O

x (logx)²

,

as x!∞. Hence, by Proposition 5.5 withτ=(q−2)/(q−1) and β=1−ε, the limit

xlim!∞

1

(logx)^{(q−2)/(q−1)}

p≤x p=q p≡1 (modq)

1−1

p ₋₁

exists, say equal toM(q), and A(x) =e^{−γ(q−2)/(q−1)}

Γ q−2

q−1

M(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}),

as x!∞. Now forx≥q

p≤x p=q p≡1 (modq)

1−1

p

=

p≤x

1−1

p

1−1

q

p≤x p≡1 (modq)

1−1

p .

By Mertens’ theorem we have

p≤x

1−1

p

=e^−γ(1+o(1)) 1 logx, as x!∞, so appealing to Proposition 6.3, we obtain

p≤x p=q p≡1 (modq)

1−1

p

= e^−γ(1+o(1))(logx)⁻¹

1−1 q

λ(q)(1+o(1))(logx)^−1/(q−1)

= qe^−γ

(q−1)λ(q)(1+o(1))(logx)^{−(q−2)/(q−1)},

so that

1

(logx)^{(q−2)/(q−1)}

p≤x p=q p≡1 (modq)

1−1

p ₋₁

=(q−1)e^γλ(q)

q (1+o(1)).

Hence

M(q) =(q−1)e^γλ(q)

q .

Finally

A(x) = e^γ/(q−1) Γ

q−2 q−1

(q−1)

q λ(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}), asx!∞, so that as

Γ 1

q−1

Γ q−2

q−1

= π

sin π q−1 we have

α(q) = e^γ/(q−1) Γ

q−2 q−1

(q−1) q λ(q) =

(q−1)^{(q−2)/(q−1)}Γ 1

q−1

sin π

q−1

2^{(q−3)/2(q−1)}q^{(q−4)/2(q−1)}π^3/2(h(q)R(q)C(q))^1/(q−1). This completes the proof of Proposition 6.4.

7. Proof of the theorem By Propositions 5.2 and 6.4 we have

E_q(x) =A(x)+A x

q

=α(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}) +α(q)x

q

logx q

_−1/(q−1) +O

x q

logx q

₋q/(q−1)+ε

=α(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}) +α(q)

q x((logx)^−1/(q−1)+O((logx)^−q/(q−1)))+O(x(logx)^{−q/(q−1)+ε})

=α(q)

1+1 q

x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε})

=e(q)x(logx)^−1/(q−1)+O(x(logx)^{−q/(q−1)+ε}), as x!∞.

By Lemma 4.1 and the theorem (with q=3), the number of n≤xfor which 3φ(n) is

2^7/2 3^9/4

p≡1 (mod 3)

1− 1

p² ₁/2

x(logx)^−1/2+Oε(x(logx)^−3/2+ε), as x!∞, for anyε>0.

References

1. Erd˝os, P.,Granville, A.,Pomerance, C. andSpiro, C., On the normal behavior of the iterates of some arithmetic functions, inAnalytic Number Theory(Allerton Park, IL, 1989), pp. 165–204, Birkh¨auser, Boston, MA, 1990.

2. Luca, F. andPomerance, C., On some problems of Makowski–Schinzel and Erd˝os concerning the arithmetical functionsφandσ,Colloq. Math.92(2002), 111–

130.

3. Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, Springer, Berlin–Heidelberg–New York, 1990.

4. Odoni, R. W. K., A problem of Rankin on sums of powers of cusp-form coefficients, J. London Math. Soc.44(1991), 203–217.

5. Prachar, K.,Primzahlverteilung, Springer, Berlin–G¨ottingen–Heidelberg, 1957.

6. Williams, K. S., Mertens’ theorem for arithmetic progressions, J. Number Theory 6 (1974), 353–359.

7. Wirsing, E., Das asymptotische Verhalten von Summen ¨uber multiplikative Funktio- nen,Math. Ann.143(1961), 75–102.

Blair K. Spearman

Department of Mathematics and Statistics University of British Columbia Okanagan Kelowna, BC

Canada V1V 1V7

Kenneth S. Williams

School of Mathematics and Statistics Carleton University

Ottawa, Ontario Canada K1S 5B6 Received May 18, 2004

in revised form December 15, 2004 published online August 3, 2006