Omega theorems related to the general Euler totient function

J. Math. Anal. Appl. 412 (2014) 401–415

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Omega theorems related to the general Euler totient function ✩ Jerzy Kaczorowski a,b,∗ , Kazimierz Wiertelak a a b

Adam Mickiewicz University, Faculty of Mathematics and Computer Science, 61-614 Pozna´ n, Poland Institute of Mathematics of the Polish Academy of Sciences, 00-956 Warsaw, Poland

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 25 June 2013 Available online 21 October 2013 Submitted by B.C. Berndt

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new. © 2013 Elsevier Inc. All rights reserved.

Keywords: Omega estimates Generalized Euler totient function Riemann zeta function Dirichlet L-functions Newforms L-functions

1. Introduction Following [3] we define the general Euler totient function associated to a polynomial Euler product

F (s) =



F p (s) =

p

d   p

j =1

1−

α j ( p) ps

−1 ,

where p runs over primes and |α j ( p )|  1 for all p and 1  j  d as follows

ϕ (n, F ) = n



F p (1)−1 .

p |n

We assume here that d is chosen as small as possible, i.e. that there exists at least one prime number p 0 such that d 

α j ( p 0 ) = 0.

j =1

Then d is called the Euler degree of F . Of course every polynomial Euler product is also a Dirichlet series

✩

*

Both authors were supported in part by the grant no. N N201 6059400 from the National Science Centre (Poland). Corresponding author. E-mail addresses: [email protected] (J. Kaczorowski), [email protected] (K. Wiertelak).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.10.050

(1.1)

402

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

F (s) =

∞  a F (n)

ns

n =1

σ = (s) > 1.

which is absolutely convergent for Let



and

1 2

(1.2)

F p (1)



C (F ) =



1

γ ( p) = p 1 −

1−

γ ( p)

 (1.3)

.

p2

p

It was proved in [3, Theorem 1.1] that for a polynomial Euler product F of degree d and x  1 we have



 

d 

ϕ (n, F ) = C ( F )x2 + O x log(2x)

.

nx

Let us denote the corresponding error term by

E (x, F ) =



ϕ (n, F ) − C ( F )x2 .

(1.4)

nx

As in [3] let S0 denote the set of all polynomial Euler products F (s) belonging to the Selberg class S such that F (s) = 0 for

σ >1−

c0 ( F )

(s = σ + it , −∞ < t < ∞),

log(|t | + 10)

where c 0 ( F ) denotes a positive constant depending on F . We refer to [2,4,8,9] for basic definitions and results on the Selberg class. Let us remark that most probably S = S0 but it has not been proven yet. It is convenient to introduce a more general class of L-functions as follows. We say that F ∈ S˜0 if there exists F ∗ ∈ S0 such that F and F ∗ differ by a finite number of local factors, i.e. there exists a finite set of primes T such that

F (s) F ∗ (s)

=

∂p  

 β j ( p )  j ( p)

1−

ps

p ∈ T j =1

,

where |β j ( p )|  1 and  j ( p ) ∈ {−1, 1} for all p ∈ T and 1  j  ∂ p . It is known that for F ∈ S0 we have

E (x, F ) = Ω(x)

(1.5)

as x → ∞ (see [3, Corollary 1.4]). The principal goal of the present paper is to improve on this estimate. To formulate our main result we need the following auxiliary notation. For every prime number p and  = ±1 we put

    ξ p ( F ,  ) = arg −γ ( p ) −π < ξ p ( F ,  )  π

(1.6)

and for every positive integer k let

Ψk (x, F ,  ) =

 p x

|a F ( p )| p

cos ξ p ( F ,  ).

(1.7)

|ξ p ( F , )|π /2 p ≡ (mod k)

Theorem 1.1. Let F (s) be a polynomial Euler product such that F (s + i λ) ∈ S˜0 for certain real λ. Suppose that C ( F ) = 0 (see (1.3)). Then, there exists a positive constant C which may depend on F such that for all integers k > 2 and arbitrary  = ±1 we have









E (x, F ) = Ω x exp Ψk C ϕ (k) log x, F , 

 .

(1.8)

Obviously Ψk (x, F ,  )  0 for all x and hence we reprove (1.5) for a slightly larger class of L-functions but under additional but mild assumption C ( F ) = 0. However, in many concrete cases we have Ψk (x, F ,  ) → ∞ as x → ∞ for suitably chosen k and  . In such cases we get an improvement of (1.5), see Theorems 2.1–2.3 below.

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

403

2. Some applications 2.1. Euler functions related to the Riemann zeta function Euler functions related to the Riemann zeta function are defined as follows

ϕ (n, λ) = n



1−

p |n



1

,

p 1+ i λ

where λ is a fixed real number. The corresponding error term equals

E (x, λ) =



ϕ (n, λ) −

nx

1 2ζ (2 + i λ)

x2 .

The  case λ = 0 is classical. The best known omega estimate due to H.L. Montgomery [7] gives oscillations of size Ω(x log log x ), see also [5]. This is exactly what also follows from our Theorem 1.1. As far as we know the case λ = 0 has not been treated in the literature so far. Theorem 2.1. For λ = 0 we have



1



E (x, λ) = Ω x(log log x) 2π . The exponent 1/(2π ) here comes as a surprise, instead one would expect something depending on λ and tending to 1/2 as λ → 0. 2.2. Euler functions related to Dirichlet L-functions For a Dirichlet character

E (x, χ , λ) =



χ (mod q) and a real number λ the error term we are interested in looks as follows

ϕ (n, χ , λ) −

nx

1 2L (s + i λ, χ )

x2 ,

where

ϕ (n, χ , λ) = n



1−

p |n

χ ( p)

 .

p 1+ i λ

The case of λ = 0 and real χ was treated in a recent paper [6]. It was proved there (Corollary 1.4 of [6]) E (x, χ , 0) = Ω± (x(log log x)1/4 ) (χ -real). Now we are able to treat the general case. Theorem 2.2. Let χ be a Dirichlet character and let h denote its order. Then, for x → ∞ we have





E (x, χ , 0) = Ω x(log log x)η(χ ) , where

η(χ ) =

⎧1 ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

if h = 1, 1

4h sin(π /(2h)) 1 2h sin(π /h) 1 cot πh 2h

if 2  h, h > 1, if 2 h,

(2.1)

if 4|h.

For λ = 0 we have



1



E (x, χ , λ) = Ω x(log log x) 2π .

(2.2)

Observe that η(χ ) → 1/(2π ) when h → ∞. This is in accordance with the intuition that a Dirichlet character of large order mimics continuous character n → ni λ , λ = 0.

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2.3. Euler functions related to modular forms Let φ denote a new form of weight K and level q with the normalized Fourier expansion at infinity as follows

φ( z) =

∞ 

K −1 2

λφ (n)n

e (nz)

( z ∈ H).

n =1

The corresponding normalized L-function defined for

L (s, φ) =

∞  λφ (n)

ns

n =1

σ = (s) > 1 by the Dirichlet series

,

belongs to the class S 0 and has a polynomial Euler product of degree two. Local factors have the following form

 L p (s, φ) = 1 −

α( p)

 −1 

ps

1−

β( p )

 −1

ps

,

where



α ( p ) , β( p )  1.

(2.3)

In particular, it is possible that one or both of these coefficients vanish. But this can happen for a finite number of primes only. For primes coprime to the level we have |α ( p )| = |β( p )| = 1. Clearly λφ ( p ) = α ( p ) + β( p ) and hence in particular



λφ ( p )  2

(2.4)

for all primes p. We define the Euler function corresponding to L λ (s, φ) = L (s + i λ, φ), where λ ∈ R, and the corresponding error term according to general rules (see (1.1) and (1.4))

ϕ (n, φ, λ) = n

 p |n

and

E (x, φ, λ) =



1 L p (1 + i λ, φ)

ϕ (n, φ, λ) − C (φ, λ)x2 ,

nx

where



C (φ, λ) =

1 2

1−

p

λφ ( p ) p 2+ i λ

+

α ( p )β( p ) p 3+2i λ

 (2.5)

.

The case λ = 0 was essentially settled in [5, Proposition 1.6], and the result is





E (x, φ, 0) = Ω± x(log log x)1/8 , provided λφ (n) ∈ R for all n. We complete this theorem with the following one. Theorem 2.3. Let φ be a new form with real Fourier coefficients. Then, for every real λ = 0 we have





E (x, φ, λ) = Ω x(log log x)1/(8π ) . 3. A general Ω -result By A we denote the set of all arithmetic functions

α (n) satisfying the following conditions:

(i) α (n) is multiplicative. (ii) There exists a positive real number θ < 1 such that

α (n)  nθ .

(3.1)

(iii) We have ∞  α (n) n =1

n2

= 0.

(3.2)

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

405

(iv) For every N  1 we have N 

α (n)  N g ( N )

(3.3)

n =1

for certain non-decreasing function g ( N ) such that 1  g ( N )  N ε for every (v) The series

ε > 0.

∞  α (n)

n

n =1

converges. Let {x} denote the fractional part of a real number x, and let s(x) be the saw-tooth function

 s(x) = For

1 2

− {x} if x ∈ / Z,

0

(3.4)

otherwise.

α ∈ A we write f (x, α ) =

  ∞  α (d) x s

d

d =1

d

(3.5)

,

where s(x) is defined in (3.4) and



α (n)

(x  1), n

R (x, α ) = sup

(3.6)

y x n> y

R ∗ (x, α ) =



√

R ( x, α ) +

1 x

(3.7)

.

It is evident that R ∗ (x, α ) is positive, monotonic and according to (v) R ∗ (x, α ) → 0 as x → ∞. Moreover, we put

ρ (x, α ) = R

∗





x g (x)

, α g (x)

and for a positive integer k and



Φk (x, α ,  ) =

(3.8)

 = ±1 we write |α ( p )| p

p x p ≡ (mod k) |ξ p |π /2

cos ξ p ,

where

  ξ p = ξ p (α ,  ) = arg α ( p ) (−π < ξ p  π ).

(3.9)

Theorem 3.1. Let α ∈ A and let b0 (x) = b0 (x, α ) and b1 (x) = b1 (x, α ) be positive and monotonic functions satisfying the following inequalities



R ∗ b0 (x)

ρ (x, α ) b0 (x)

x g (x)

 , α  b0 (x),

(3.10)

 b1 (x)

(3.11)

for sufficiently large x, where ρ (x, α ) is as in (3.8). Suppose that b0 (x) = o(1) and b1 (x) = o(1) as x → ∞. Then, for every integer k > 2 and arbitrary  = ±1 we have





f (x, α ) = Ω exp Φk as x → ∞.



ϕ (k) 3

log

1 b1 (x2/3 , α )

 , α, 

(3.12)

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4. Lemmas Lemma 4.1. Let α ∈ A. For y  max(xR ∗ (x, α ),

f (x, α ) =

 α (d) d

d y

 

s

x

d

√

x ) we have

  + O R ∗ (x, α ) .

Proof. This is Lemma 3.1 in [5] where it has been proven for α (n) belonging to an another class A of arithmetic functions, but the proof applies to our situation without any changes. 2 Lemma 4.2. Let b and r > 0 be relatively prime integers, and let β be a real number. Then, for any positive N, N   b

s n

n =1

r

 N + β = s(r β) + O (r ). r

Proof. This is Lemma 3 in [7]. Note that there is a misprint in (13) of [7], and “” there should read as “=”. Lemma 4.3. Let satisfying

q0  q 

2

α ∈ A and let b0 be as in Theorem 3.1. Then there exists a positive integer q0 = q0 (α ) such that for square-free q 1 2

 min

√

N,



b0 ( N )

(4.1)

ρ (N , α)

and arbitrary 0 < a < q we have N 



f (nq + a) = c (q, a, α ) N + O N

3θ +1 4



τ (q) + b0 ( N ) N + N 3/4 g ( N ) ,

n =1

where τ (q) denotes the familiar divisor function and

c (q, a, α ) =

 e |q

s(a/e )  e

α (e f 1 )

 

f 12

f 1 |e ∞

α( f 2) f 22

( f 2 ,q)=1

(4.2)

.

The notation f 1 |e ∞ means that all the prime divisors of f 1 divide e. Proof. We can assume that N is sufficiently large √ since otherwise there is nothing to prove. We apply Lemma 4.1 with y = b0 ( N ) N / g ( N ) + N 3/4 and x = nq + a. Since q  N /2 we have

 nq + a  ( N + 1)q 

√



1√ 2

N ( N + 1)  N 3/4  y .

Moreover, if nq + a  b0 ( N ) N / g ( N ) then

(nq + a) R ∗ (nq + a, α )  b0 ( N )

N

y

g (N )

since R ∗ (nq + a, α )  1 if q  q0 . If b0 ( N ) N / g ( N ) < nq + a  N / g ( N ) we have using (3.10)

(nq + a) R ∗ (nq + a, α ) 

N g (N )



R ∗ b0 ( N )

 N , α  b0 ( N )  y. g (N ) g (N ) N

Finally, if nq + a > N / g ( N ), n  N, using (4.1) we have

(nq + a) R ∗ (nq + a, α )  ( N + 1)qR ∗  ( N + 1)  b0 ( N ) We see that in all cases



N g (N )



,α

b0 ( N )

1

2

R ∗ ( N / g ( N ), α ) g ( N )

N g (N )

 y.

R∗



N g (N )

 ,α

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

max

√

407



nq + a, (nq + a) R ∗ (nq + a, α )  y

and we can apply Lemma 4.1. It yields N 

f (nq + a) =

n =1

  N   α (d) nq + a

=

s

d

n =1 d  y

d

 +O

N 

 ∗

R (nq + a, α )

n =1

  N  α (d)    q/(d, q) a + O b0 ( N ) N s n + d d/(d, q) d

(4.3)

n =1

d y

since N 

R ∗ (nq + a, α ) =



n =1

 nq+ab0 ( N ) N / g ( N )

 b0 ( N )

N g (N )



+ 



R ∗ (nq + a, α )

b0 ( N ) N / g ( N )
+ N R ∗ b0 ( N )

N g (N )

 , α  b0 ( N ) N .

Now we apply Lemma 4.2 with b = q/(d, q), r = d/(d, q) and β = a/d to the inner sum in (4.3), and then use (3.3) getting N 

f (nq + a) =

n =1

  α (d)  (q, d)  a    + O (d) + O b0 ( N ) N N s d d (q, d)

d y

 α (d)

=N

d2

d y

 α (d)

=N

d2

d y

 α (d)

=N

d2

d y

 (q, d)s

a

 +O

(q, d) 

(q, d)s

a



(q, d) 

(q, d)s

a



(q, d)



   |α (d)| + O b0 ( N ) N

n y

    + O yg ( y ) + O b0 ( N ) N   + O N 3/4 g ( N ) + b0 ( N ) N .

We replace the last sum by the series over all d  1. Using (3.1) it is easy to see that this induces an error of size

N

 e |q

N

 e |q

e

 |α (d)| d2

d> y (q,d)=e



e θ −1

 1  |α (de )| e |q

1

d> y /e

N

d2−θ

e

d> y /e (q,d)=1

 N y θ −1d(q)  N

d2 3θ +1 4

d(q).

Since an easy computation shows that ∞  α (d) d =1

d2

 (q, d)s

the proof is complete.

a

(q, d)

 = c (q, a, α ),

2

Lemma 4.4. Let α ∈ A. Then there exists a positive constant b1 = b1 (α ) such that for every integer q we have



α (n)

 b1 .

n2 (n,q)=1

Proof. This is Lemma 3.4 in [5]. As in the case of Lemma 4.1, it was proved in [5] for arithmetic functions, but the proof applies to our situation without any changes. 2

α (n) belonging to another class A of

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J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

5. Proof of Theorem 3.1 Let us fix θ such that

0 < θ <

1−θ

(5.1)

2

where θ is the exponent in (3.1). For every sufficiently large x  1 let



q = q(x, α ,  ) =

p,

c (x) p x |ξ p |π /2

(5.2)

|α ( p )|> p −θ p ≡ (mod k)

where ξ p is defined in (3.9) and c (x) is sufficiently large but bounded as x → ∞ and chosen in such a way that q ≡ 1 (mod k). Let



1

N := min m  1:

2

 min

√

b0 (m)

ρ (m, α )



, m

 q ,

where b0 has the same meaning as in Lemma 4.3. Let us observe that N is well defined since m → ∞. According to Lemma 4.3 for every 0 < a < q we have N 

(5.3)

ρ (m, α ) = o(b0 (m)) as

f (nq + a, α ) = c (q, a, α ) N + o( N ),

n =1

where c (q, a, α ) is given by (4.2). Hence

max

1x( N +1)q



f (x, α )  c (q, a, α ) + o(1).

(5.4)

To end the proof we show that with a proper choice of a, |c (q, a, α )| is large. For p |q we have using (3.1) ∞  α ( p l +1 )

p 2l

l =0



= α( p) + O 

∞  |α ( pl+1 )|



p 2l

l =1

= α ( p ) + O p −2(1−θ )



= α ( p ) + ϑ p p −(1−θ ) for certain |ϑ p |  1 if p is large enough which is the case provided c (x) in (5.2) is chosen sufficiently large. Since

|α ( p )| > p −θ for p |q and using (5.1) we obtain ∞  α ( p l +1 ) l =0

p 2l



 = α ( p ) 1 + ϑ p p −θ

for certain |ϑ p |  1. Hence, for e |q we have

 α (e f 1 ) f 1 |e ∞

f 12

=

∞  α ( p l +1 ) p |e l=0

= α (e )



p 2l



1 + ϑ p p −θ .

p |e

Thus, recalling Lemma 4.4, we have



 s(a/e )

 −θ

c (q, a, α )  α (e) 1 + ϑp p

.

e e |q

p |e

Let now a = q/k. Then, since q is square-free, q ≡ 1 (mod k) and p ≡  (mod k) for every prime p |q, we have

 

s

a e

 

a

=  ω(e) =  ω(e)

s

e



1 2

−

1 k



(5.5)

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

409

for every e |q. Consequently recalling (5.5) we have



  ω(e)  −θ

c (q, a, α )  α ( e ) 1 + ϑ p p

e e |q

p |e

iξp    −θ

1 + |α ( p )|e 1 + ϑ p = p

p p |q



=

|c p |,

p |q

say. Easy computation shows that

|c p |2 = 1 + 2

|α ( p )| p

and thus

1

log |c p | =

2

 log 1 + 2

|α ( p )|

=

p

|α ( p )|

=

p



cos ξ p + O

|α ( p )| p

p



1+θ

 cos ξ p + O



cos ξ p + O

 cos ξ p + O

|α ( p )|

|α ( p )| p

1+θ

|α ( p )| p 1+θ

|α ( p )| p 1+θ





+O



|α ( p )|2





p2

because of (3.1) and (5.1). Since, using (3.3) it is easy to see that

 |α ( p )| p |q

p

1+θ

∞  |α (n)|



n =1

n1+θ

 1,

we have

 

|α ( p )|

c (a, q, α )  exp cos ξ p 

p |q

p

  exp Φk (x, α ,  ) .

(5.6) 4

By the Prime Number Theorem in arithmetic progressions we have q  exp( 3ϕ (k) x) for large x and thus, recalling (5.3), (3.11) and using

1/ρ ( N − 1, α )  ( N − 1)/ g ( N − 1)2  N − 1, we obtain

x



3ϕ (k) 4

ϕ (k) 3

 min log

log

b 0 ( N − 1) 2ρ ( N − 1, α )

b 0 ( N − 1)

ρ ( N − 1, α )



ϕ (k) 3

√ , log log

N −1



2 1

b 1 ( N − 1)

(5.7)

if N is sufficiently large. On the other hand, we know that b1 (t ) is monotonic and

√ 1 ( N + 1)q  ( N + 1) N − 1  ( N − 1)3/2 2

for N  5. Thus taking into account (5.4), (5.6) and (5.7) we obtain (3.12), and the result follows. 6. Proof of Theorem 1.1 For a positive integer n we put

α (n) = μ(n)



γ ( p ),

p |n

where

μ(n) denotes the familiar Möbius function and γ ( p ) is defined in (1.2). We have

(6.1)

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J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415



α (n)  τd+1 (n) where

(6.2)

τd+1 (n) denotes the well-known divisor function of order d + 1 (see [3, Lemma 2.4]). For σ > 1 we have ∞  α (n)

ns

n =1

H (s)

=

F (s)

,

where H (s) is a Dirichlet series which is absolutely convergent for complex integration method it is easy to see that



σ > 1/2 (see [3, Lemma 2.3]). Thus applying the known



α (n)  x exp(−c log x )

(6.3)

nx

for certain positive c = c ( F ). The proof is standard and we skip it. Using the above properties and also the fact that ∞  α (n)

n2

n =1



=

γ ( p)

1−

 = 2C ( F ) = 0,

p2

p

α belongs to the class A defined in Section 3, and that   c R ∗ (x, α )  exp − log x

it is easy to check that

(6.4)

4

and

g (x) = logd (3x). Consequently,



ρ (x, α )  exp −

c 5

 log x .

(6.5)

By Lemma 2.5 from [3] we have



ϕ (n, F ) =

nx

x2  α (n) 2

where

R (x, α ) =

1 2

n2

nx

+x

 α (n)  x 

 

α (n)

nx

x

n

s

n

nx

n

− R (x, α ),

(6.6)

  1−

x

(6.7)

.

n



We extend the range of summations in (6.6) to infinity. By (6.3) √ the induced √ error term is  x exp(−c log√ x ). Moreover, splitting the range of summation in (6.7) into the initial part [1, [ x ]] and  x disjoint subintervals I ⊂ ([ x ], x] where the function n → { nx }(1 − { nx }) is monotonic, estimating the sum over the initial segment trivially by

 

√

α (n)  τd+1 (n)  x logd x √

√

n x

n x



and using partial summation combined with (6.3) in every subinterval I , we see that R (x, α )  x exp(−c log x ) for certain positive c < c. Finally, we see that



E (x, F ) = xf (x, α ) + O xe −c



log x



,

c where f (x, α ) is defined in (3.5). We apply Theorem 3.1 with b0 (x) = b1 (x) = exp(− 11 easy to verify (3.10) and (3.11) for x large enough. Hence, for arbitrary  = ±1 we have

   1 ϕ (k) E (x, F ) = Ω x exp Φk , α,  log 3 b1 (x2/3 , α )      = Ω x exp Φk C ϕ (k) log x, α ,  ,



log x ). Using (6.4) and (6.5) it is



where

C=

c 33



2 3

.

(6.8)

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

411

Observe that for primes p we have

  1

α ( p ) = −a F ( p ) + O

p

and due to (1.6), (3.9), (6.1) and (1.2) also

ξ p (α ,  ) = ξ p ( F ,  ). Thus

Φk (x, α ,  ) = Ψk (x, F ,  ) + O (1). Inserting this into (6.8) we obtain (1.8). The proof is complete. 7. Proof of Theorem 2.1 We apply Theorem 1.1 to F (s) = ζ (s + i λ) with parameters equals



Ψ (x, λ) =

1

p x p ≡3 (mod 4)

p

 = −1 and k = 4. The corresponding Ψ -function (see (1.7))

cos+ (λ log p ),

where, as usual, cos+ (t ) = max(0, cos t ) for all real t. We have

Ψ (x, λ) = =

11 2 1 2

p x

cos+ (λ log p ) −

p

S1 −

1 2

1 2

p x

χ4 ( p ) p

cos+ (λ log p )

(7.1)

S2,

χ4 denotes the unique non-principal Dirichlet character (mod 4). We have the following Fourier expansion  cos x = a(ν )e i ν x , (7.2)

say, where

+

ν ∈Z

where

a(ν ) =

⎧ 1 ⎪ ⎨4 ⎪ ⎩

if |ν | = 1,

(−1)(ν /2)+1

π

1

ν 2 −1

0

(7.3)

if 2|ν , otherwise.

In particular we have a(0) = 1/π . Lemma 7.1. For every positive constant c 0 and any real number |λ|  c 0 we have

 p x

1 p 1+ i λ

   log log |λ| + 10 ,

where the implied constant depends only on c 0 . For a non-principal Dirichlet character χ we have

 χ ( p) p x

p 1+ i λ

   log log |λ| + 10

uniformly for all real λ. The implied constant depends on χ . This can be proved using basic analytic properties of Dirichlet L-functions by a standard application of the complex integration method (along the lines of the proof of Satz 5.3 in [10] for instance). We skip details. Inserting (7.2) into (7.1) and using (7.3) together with Lemma 7.1 we obtain

S1 =



a(ν )

ν ∈Z

= =

p x

1 1

π 1

π

p x



p

1 p 1− i ν λ

+ Oλ

 ν 1

log log x + O λ (1),

1

ν2

log log



ν |λ| + 10

 

412

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

and similarly S 2 λ 1. Thus

Ψ (x, λ) =

1 2π

log log x + O λ (1)

and the result follows from (1.8) after some trivial calculations. 8. Proof of Theorem 2.2 Let us adopt the following convention. For two Dirichlet characters χ1 (mod q1 ) and χ2 (mod q2 ) we write χ1 = χ2 when the corresponding induced characters mod [q1 , q2 ] are equal. Hence, if χ1 = χ2 the values χ1 ( p ) and χ2 ( p ) still can differ for a finite number of primes p. Lemma 8.1. For k ∈ {3, 4} let we have

χk denote the unique non-principal Dirichlet character (mod k). Then, for every Dirichlet character χ

χ ν = χ3 for all integers ν  1

(8.1)

χ ν = χ4 for all integers ν  1.

(8.2)

or

χ ν = χ3 and χ μ = χ4 for certain positive integers ν and μ. Then, for d = (ν , μ), d = mν + nμ we have ⎧ if 2|n, ⎨ χ3 if 2|m, χ d = χ3m χ4n = χ4 (8.3) ⎩ χ3 χ4 if 2  nm.

Proof. Suppose that

Observe that the case 2|n and 2|m cannot occur since otherwise we would have a contradiction. Suppose that χ d = χ3 Then μ

μ χ4 = χ μ = χ d d = χ3d =



χ d = χ0 and thus also χ3 = (χ d )ν /d = χ0 ,

χ3 if 2  μd , χ0 otherwise.

Since χ4 = χ0 and χ4 = χ3 in both cases we obtain a contradiction. In a similar way we exclude other possibilities in (8.3) and the lemma follows. 2 Now we can give a proof of Theorem 2.2. We apply Theorem 1.1 to F (s) = L (s + i λ, χ ) with parameters  = −1 and k ∈ {3, 4}, where k is chosen in such a way that (8.1) or (8.2) of Lemma 8.1 holds. The corresponding Ψ -function (see (1.7)) equals



Ψk (x, χ , λ) =

1

p x p ≡−1 (mod k)

p





cos+ ξ p (χ ) ,

where

e i ξ p (χ ) =

χ ( p) p iλ

.

We proceed as in the proof of Theorem 2.1. Using Fourier expansion (7.2) we obtain

Ψk (x, χ , λ) =

 ν ∈Z

=

1 2



a(ν )

p x p ≡−1 (mod k)

a(ν )

ν ∈Z

χ ν ( p) p 1+ i ν λ

 χ ν ( p) p x

p 1+ i ν λ

−

1 2

ν ∈Z

a(ν )

 χ ν ( p )χk ( p ) p x

p 1+ i ν λ

.

Since χ ν χk = χ0 for all ν  1, arguing as in the proof of Theorem 2.1 we see that the second summand contributes at most O (1) in both cases λ = 0 and λ = 0. Hence

Ψk (x, χ , λ) =

1 2

ν ∈Z

a(ν )

 χ ν ( p) p x

p 1+ i ν λ

+ O (1).

(8.4)

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

Let λ = 0. Using Lemma 7.1 again we see that the total contribution of all summands with

1

Ψk (x, χ , λ) = a(0) 2

p x

1

=

1 p

413

ν = 0 is O (1) and hence

+ O (1)

log log x + O (1).

2π

A direct application of Theorem 1.1 yields (2.2) and the proof in the case λ = 0 is complete. Let now λ = 0. In this case also other summands in (8.4) can provide a non-trivial contribution. They are exactly these with h|ν . We have

Ψk (x, χ , 0) = A (χ ) log log x + O (1), where

A (χ ) =

1  2

a(ν ) =

ν ∈Z h|ν

1 2π



+

a(ν ).

(8.5)

ν 1 h|ν

Now an application of Theorem 1.1 gives





E (x, χ , 0) = Ω x(log log x) A (χ ) , and to conclude the proof it suffices to show that A (χ ) = η(χ ), see (2.1). Case 1: 2  h. Suppose in addition that h > 1. Then, using (8.5) and (7.3) we obtain

A (χ ) =

=

1 2π

+

1 

π



1

ν 1

1+4

2π

(−1)ν 1 − 4h2 ν 2

∞ 

ν =1

1 1 − 16h2 ν 2

−2

∞ 

ν =1

1

 .

1 − 4h2 ν 2

Using the following well-known formula valid for complex z ∈ / πZ

cot( z) =

1 z

+ 2z

∞ 

1 z2 − ν 2 π 2

ν =1

,

we see after some simple computations that

A (χ ) =

1



4h



cot



π



− cot

4h



π 2h

=

1 4h sin(π /(2h))

.

This proves the second case in (2.1). When h = 1 we have one additional term corresponding to

A (χ ) =

1

1

+ a(1) = ,

4 sin(π /2)

2

and the first case in (2.1) follows as well. Case 2: 2 h. Now we have

A (χ ) =

1 2π

+

∞ 1  (−1)ν

π

ν =1

1 − h2 ν 2

and this is the same sum as in Case 1 but with h replaced by h/2. Hence

A (χ ) =

1 2h sin(π /h)

,

and the third case in (2.1) follows. Case 3: 4|h. Now

A (χ ) =

=

1 2π 1 2h



1+2

∞ 

ν =1

cot

π h

,

and the proof is complete.

1 1 − h2 ν 2



ν = 1. Then

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J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

9. Proof of Theorem 2.3 We shall use the following auxiliary result. Lemma 9.1. For a cusp form φ we have

 |λφ ( p )|2 p

p x

= log log x + O φ (1),

(9.1)

and

 λφ ( p ) p x

p 1+ i λ

  φ log log |λ| + 10 ,

 |λφ ( p )|2 χ4 ( p ) p 1+ i λ

p x

(9.2)

  φ log log |λ| + 10

(9.3)

uniformly for all real λ. Moreover, for every fixed positive b0 we have

 |λφ ( p )|2 p x

p 1+ i λ

  φ log log |λ| + 10 ,

(9.4)

uniformly for |λ|  b0 . We skip the proof for the same reasons as in the case of Lemma 7.1. Required analytic properties of L (s, φ), L (s, φ ⊗ χ4 ), L (s, φ ⊗ φ) and L (s, φ ⊗ φ ⊗ χ4 ) are either easy to establish or are explicitly stated in [1, Chapter 5]. In order to prove Theorem 2.3 we apply Theorem 1.1 with ( , k) = (−1, 4). Using (2.3) and (2.4) we trivially check that



1 − λφ ( p ) + α ( p )β( p )  1 − 2 − 1 > 0

2 + i λ 3 + 2i λ p2 p3 p p for all primes p and real numbers λ. Thus, recalling (2.5) we have C (φ, λ) = 0, and Theorem 1.1 can be applied. In our case the corresponding Ψ -function equals



Ψ (x, φ, λ) =

|λφ ( p )| p

p x p ≡−1 (mod 4)





cos+ ξ p (φ, λ) ,

(9.5)

where

e i ξ p (φ,λ) =

sgn λφ ( p )

.

p iλ

We use the following trivial identity cos+ (t ) = two summands Ψ1 + Ψ2 , say. We have

Ψ1 =



1 4 1

p x p ≡−1 (mod 4)



=  4

λφ ( p )

p x

p 1+ i λ

λφ ( p ) p 1+ i λ

+

+

1 4

1 2

cos(t ) + 12 | cos(t )| and split the sum on the right hand side of (9.5) into



λφ ( p )

p x p ≡−1 (mod 4)

p 1− i λ

 λφ ( p )χ4 ( p )  p x

p 1+ i λ

,

where, as before, χ4 denotes the non-principal Dirichlet character (mod 4). According to (9.2) applied to φ and φ ⊗ χ4 we see that Ψ1 = O (1). Now we estimate Ψ2 . Recalling (2.4) and using



cos(x) = a(ν )e i ν x , 2|ν

where coefficients a(ν ) are the same as in (7.3), we have

J. Kaczorowski, K. Wiertelak / J. Math. Anal. Appl. 412 (2014) 401–415

Ψ2 

=

=



1 4



a(ν )

2|ν

p

p x

+



|λφ ( p )|2 (sgn λφ ( p ))ν p 1+ i ν λ

p x p ≡−1 (mod 4)

1  |λφ ( p )|2 8π



cos ξ p (φ, λ)

p

p x p ≡−1 (mod 4)

1 4

|λφ ( p )|2

415

1  |λφ ( p )|2 χ4 ( p ) 8π

p x

p

+

1  4

2|ν =0



a(ν )

p x

|λφ ( p )|2 p 1+ i ν λ

+

 |λφ ( p )|2 χ4 ( p )  . 1+ i ν λ p x

p

Formula (9.1) shows that the first sum is log log x + O (1). Using (9.2)–(9.4) we see that all the remaining sums contribute O (1). Thus

Ψ (x, φ, λ) 

1 8π

log log x + O (1),

and the result follows. References [1] H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53, American Mathematical Society, Providence, RI, 2004. [2] J. Kaczorowski, Axiomatic theory of L-functions: the Selberg class, in: A. Perelli, C. Viola (Eds.), Analytic Number Theory, C.I.M.E. Summer School, Cetraro, Italy, 2002, in: Lecture Notes in Math., vol. 1891, Springer, 2006, pp. 133–209. [3] J. Kaczorowski, On a generalization of the Euler totient function, Monatsh. Math. 170 (1) (2013) 27–48. [4] J. Kaczorowski, A. Perelli, The Selberg class: a survey, in: K. Györy, et al. (Eds.), Number Theory in Progress, Proc. Conf. in Honor of A. Schinzel, de Gruyter, 1999, pp. 953–992. [5] J. Kaczorowski, K. Wiertelak, Oscillations of the remainder term related to the Euler totient function, J. Number Theory 130 (2010) 2683–2700. [6] J. Kaczorowski, K. Wiertelak, On the sum of the twisted Euler function, Int. J. Number Theory 8 (7) (2012) 1741–1761. [7] H.L. Montgomery, Fluctuations in the mean of Euler’s phi function, Proc. Indian Acad. Sci. Math. Sci. 97 (1987) 239–245. [8] A. Perelli, A survey of the Selberg class of L-functions, part I, Milan J. Math. 73 (2005) 19–52. [9] A. Perelli, A survey of the Selberg class of L-functions, part II, Riv. Mat. Univ. Parma (7) 3* (2004) 83–118. [10] K. Prachar, Primzahlverteilung, Grundlehren Math. Wiss., vol. 91, Springer-Verlag, Berlin–New York, 1978.