RIMS Kôkyûroku Bessatsu
B34 (2012), 81–102
Operator norms
and the mean values of multiplicative functions
By
Peter D. T. A. Elliott∗
Abstract
Elementary functional analysis is employed to give necessary and sufficient conditions for
the asymptotic mean value of logarithmically sized multiplicative functions.
§ 1.
Introduction
In the present paper I establish a weighted version of the Turán–Kubilius inequality
and apply its dual to the study of multiplicative functions.
In general outline the arguments follow those in the relevant parts of my Cambridge
Tract, [6]. However, the underlying spaces are no longer of finite dimension and the
structure of their associated measures does not readily admit of a restriction that would
yield the classical inequality of Turán and Kubilius.
A typical positive integer will be denoted by n, a prime by p and a prime-power by
q. f (x) ¿ g(x) will denote that |f (x)| ≤ cg(x), for some constant c, holds uniformly on
a specified set of x-values.
Theorem 1.1.
Let β be a non-negative real-valued arithmetic function that sat∑
isfies β(1) = 1, β(qn) ≤ β(q)β(n) if (q, n) = 1 and for which the series
β(n), taken
over all positive integers, converges.
Received February 15, 2011. Accepted January 19, 2012.
2000 Mathematics Subject Classification(s): 11N37, 11N56, 11N64.
Key Words: Multiplicative Functions, Power Mean Values, Generalised Turan-Kubilius Inequality
∗ University of Colorado at Boulder, Boulder, CO 80309-0395, USA.
e-mail: pdtae@euclid.colorado.edu
c 2012 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
°
82
Peter D. T. A. Elliott
If α ≥ 2 and f is a complex-valued additive function, then
¯
¯α )1/α
(
(
) ∞
¯
¯
∑
∑
∑
¯
¯
exp −
β(q)
β(n) ¯f (n) −
f (q)β(q)¯
¯
¯
q
q
n=1
¿
(
∑
)1/α
|f (q)| β(q)
α
+
(
∑
q
)1/2
|f (q)| β(q)
2
q
provided the series in the upper bound converge. The implicit constant depends at most
upon α.
If 1 ≤ α < 2, then there is a similar inequality with upper bound

1/α 
1/2 




∑
∑
α
2
inf 
|f (q)| β(q)
+
|f (q)| β(q)
.
η>0 

 |f (q)|>η

|f (q)|≤η
Each of the inequalities with α > 1 in Theorem 1.1 has a dual. For ease of presentation only I confine myself to the cases 1 < α ≤ 2; cf. [6], Chapter 3.
Theorem 1.2.
If α ≥ 2 and the function β is everywhere positive, then
¯
¯α 1/α

¯ ∞
¯
∞
¯
∑
∑ ¯¯ 
∑
¯
β(q)1−α ¯
an − β(q)
an ¯ 

¯
¯
q
n=1
n=1
¯ ∼
¯
n=o(q)
+ (a similar expression with α = 2)
(
¿
(
exp (α − 1)
∑
)
β(q)
q
∞
∑
)1/α
β(n)1−α |an |α
n=1
provided the series in the upper bound that involves the complex numbers an , converges.
The condition n ∼
= o(q) indicates that q exactly divides n: q | n and (nq −1 , q) = 1.
I give two applications of these results.
§ 1.1.
Theorem 1.3.
tion, g,
First application.
Let τ > 0, α > 1. If for the complex-valued multiplicative funclim sup(x(log x)τ
α
)
x→∞
lim (x(log x)τ −1 )−1
x→∞
∑
−1 −1
|g(n)|α
n≤x
∑
n≤x
g(n)
Operator norms and the mean values of multiplicative functions
83
exist and the second is non-zero, then the series
∑
p−1 (g(p) − τ )
p
∑
∑
p−1 |g(p) − τ |2 ,
|g(p)−τ |≤τ /2
p−1 |g(p) − τ |α ,
|g(p)−τ |>τ /2
converge and no sum
1+
∞
∑
∑
p−m |g(pm )|α
p,m≥2
g(pm )p−1
m=1
vanishes.
Moreover, if τ ≥ 1, then the converse holds.
Theorem 1.3 exemplifies the results established in a recent extensive work of Zhang,
[7].
The case τ = 1 of Theorem 1.3, in which no logarithmic factors occur, was established by Elliott, [2], [4] and Daboussi, [1]. A considerable study of related theorems,
with emphasis on method, may be found in the author’s volume, [6].
For its own interest and to meet a challenge of Zhang, [7] §4, see also the review MR
2452645 (2010), I shall illustrate with an appropriate weighted version of the Turán–
Kubilius inequality and its dual, that the general approach of that volume applies equally
well to the cases τ 6= 1.
Other weighted versions of the Turán–Kubilius inequality may be found in the
author’s paper on compositions of arithmetic operators, [5].
The second application is concerned only with inequalities.
§ 1.2.
Second application
Theorem 1.4.
Let τ > 1, α > 1. In order that for positive constants cj , j =
1, 2, 3, the non-negative multiplicative function, g, satisfy the inequalities
∑
α
g(n)α ≤ c1 x(log x)τ −1
n≤x
c2 (log x)τ ≤
∑
n−1 g(n) ≤ c3 (log x)τ
n≤x
for all sufficiently large values of x, it is necessary and sufficient that the series
∑
∑
∑
p−1 |g(p) − τ |2 ,
p−1 |g(p) − τ |α ,
p−m |g(pm )|α
|g(p)−τ |≤τ /2
|g(p)−τ |>τ /2
converge and that the partial sums
∑
p−1 (g(p) − τ ),
p≤y
p,m≥2
y ≥ 2,
84
Peter D. T. A. Elliott
be uniformly absolutely bounded.
If the condition on the mean-value of g is replaced by
∑
α
n−1 g(n)α ≤ c1 (log x))τ ,
n≤x
then with no further change the theorem holds for τ > 0.
§ 2.
Background Functional Analysis
Let Q be the set of prime-powers. We define a measure, µ, on the power set of Q
∑
by µ(F ) = β(q), the sum taken over the prime-powers in the set F . Under the initial
hypothesis of Theorem 1.1, Q itself has finite measure.
A function f : Q → C may be viewed as a point (f (q)) in the space CQ . For α ≥ 1
there is an attached norm
(
)1/α (∫
)1/α
∑
α
α
||f ||α =
|f (q)| β(q)
=
|f | dµ
.
q
Lα (Q) will denote the space of functions f with well defined norm ||f ||α .
With N the set of positive integers we define a measure
(
)
∑
∑
ν(G) = exp −
β(n)
β(q)
q
n∈G
on the power set of N . Lemma 3.1 which follows will show that ν assigns to N a measure
of at most 1.
For α ≥ 1 and with (h(n)) a typical point in CN , there is a corresponding norm
(
||h||α =
(
exp −
∑
q
)
β(q)
∞
∑
)1/α
|h(n)| β(n)
α
(∫
=
)1/α
|h| dν
α
.
n=1
M α (N ) will denote the space of functions h with well defined norm ||h||α .
To each complex-valued additive arithmetic function f (n) there corresponds a map
A : CQ → CN given by


(
)
∫
∑
∑
f (q)β(q) = f (n) − f dµ .
(f (q)) −→ 
f (q) −
q||n
q
The first part of Theorem 1.1 asserts that for α ≥ 2, and in an obvious notation,
the map
Aα
L2 ∩ Lα (Q) −→ M α (N )
Operator norms and the mean values of multiplicative functions
85
has norm ||Aα || bounded in terms of α. The domain of Aα is the intersection of spaces
L2 (Q) and Lα (Q), with associated norm || ||2 + || ||α .
0
We may dualise the diagram, identifying the dual space (M α (N ))0 with M α (N ),
where 1/α + 1/α0 = 1:
A0
0
α
(L2 ∩ Lα (Q))0 ←−
M α (N ).
A standard result in functional analysis guarantees that ||A0α || = ||Aα ||, hence ||A0α || ¿
1.
For 1 < α ≤ 2 there is a corresponding map
0
(L2 ∩ Lα (Q))0 −→ M α (N )
Aα
with dual
A0
0
0
α
(L2 ∩ Lα (Q))00 ←−
M α (N ).
Since Q has finite measure we may identify the second dual in the final target space
0
with the original space L2 ∩ Lα (Q). Once again kAα k ¿ 1, which is the second part of
Theorem 1.1, and kA0α k (= kAα k) ¿ 1, which is the content of Theorem 1.2.
For α ≥ 2, the composition
Aα
A0α0
L ∩ L (Q) −→ M (N ) −→ L2 ∩ Lα (Q)
2
α
α
corresponds to the map T in the author’s volume, [6]. There is a similar composition
0
between the spaces (L2 ∩ Lα (Q))0 if 1 < α ≤ 2.
Note that without severe restrictions upon the function β, A0α0 Aα will not be the
identity map.
§ 3.
Proof of Theorem 1.1
In order that various infinite series arising in the argument should converge absolutely, it is convenient to establish Theorem 1.1 with β(n)n−σ in place of β(n), uniformly
for 0 < σ ≤ 2, and let σ approach zero. The notation of Theorem 1.1 will otherwise
remain in force.
∑
For an arithmetic function t, let E(t) denote the sum
t(q)β(q)q −σ , taken over
the prime-powers, q.
§ 3.1.
Lemma 3.1.
∞
∑
Large values of f
β(n) ≤ exp
n=1
provided one of the two series converges.
(
∑
q
)
β(q)
86
Peter D. T. A. Elliott
Proof of Lemma 3.1. Argument by induction shows that
(
)
∞
∞
∑
∑
∏
β(n) ≤
1+
β(pm ) ,
p
n=1
m=1
and the asserted inequality follows from applications of the inequality eb ≥ 1 + b, valid
for all real non-negative b.
As a paradigm for later procedures we may replace the excursion through an Euler
∑∞
product by the notion of dominance. The function J(σ) = h=1 β(n)n−σ , defined for
real positive σ, is differentiable and satisfies
0
−J (σ) =
∞
∑
β(n)n
−σ
log n =
n=1
=
∑
q
∞
∑
β(n)n−σ
n=1
∞
∑
log q
∑
log q
qkn
β(mq)(mq)−σ ≤
∑
β(q)q −σ log q · J(σ),
q
m=1
(m,q)=1
the condition (m, q) = 1 having been abandoned. Hence
(− log J(y))0 ≤
∑
β(q)q −y log q
q
and, integrating over the half-line y ≥ σ,
log J(σ) ≤
∑
β(q)q −σ .
q
We let σ approach zero to complete the proof.
Define
D=
(
∑
)1/2
|f (q)|2 β(q)q −σ
≥ 0.
q
For complex z, w define the additive function h = zf1 +wf¯1 , where f1 is the additive
function
∑
f (q).
f1 (n) =
qkn
|f (q)|≤D
Define
ψ(z, w) =
∞
∑
n=1
β(n)n−σ exp(D−1 (h(n) − E(h))).
Operator norms and the mean values of multiplicative functions
Lemma 3.2.
ψ(z, w) ¿ exp
(
∑
87
)
β(q)q −σ
q
uniformly for |z| ≤ 1, |w| ≤ 1, 0 < σ ≤ 2.
Proof of Lemma 3.2. Define the multiplicative function g(n) = exp(D−1 h(n)).
Within the constraints of Lemma 3.2, |g(q)| = exp(D−1 Re(h(q))) ≤ e2 . By Lemma
3.1
(
)
∞
∑
∑
|g(n)|β(n)n−σ ¿ exp
|g(q)|β(q)q −σ .
q
m=1
(
Thus
ψ(z, w) exp −
∑
)
β(q)q
−σ
q
¿ exp
(
∑
)
β(q)q −σ {|g(q)| − 1 − Re(D−1 h(q))} .
q
In view of the inequality |es − 1 − s| ≤ |s|2 max(1, eRe(s) ), uniform in complex s, the
sum in the bounding exponential is
¿
∑
β(q)q −σ D−2 |Re(h(q))|2 ¿ D−2
q
∑
β(q)q −σ |f (q)|2 ¿ 1.
q
Lemma 3.2 is established.
Lemma 3.3.
∞
∑
For α > 0,
β(n)n−σ |f1 (n) − E(f1 )|α ¿ Dα exp
(
∑
)
β(q)q −σ
.
q
n=1
Proof of Lemma 3.3. Application of Hölder’s inequality shows that it will suffice
to establish the bound for integral values 2k of α. In view of Lemma 3.2, the desired
inequality follows from the representation
D−2k
∞
∑
β(n)n−σ |f1 (n) − E(f1 )|2k
n=1
2 −1
= −(k!) (4π )
2
∫
∫
|w|=1
ψ(z, z −1 w)z −1 w−k−1 dzdw.
|z|=1
This completes our treatment of the large values of f .
88
Peter D. T. A. Elliott
§ 3.2.
Small values of f
Lemma 3.4.
If the non-negative real arithmetic function t satisfies t(q) ≤ 1 and
t(qm) ≤ t(q) + t(m) when (m, q) = 1, then
∞
∑
(
t(n) β(n) ≤
k
k+
∑
)k
t(q)β(q)
q
n=1
∞
∑
β(n)
n=1
for each non-negative integer, k.
Proof of Lemma 3.4. The proof goes by induction on k for all such functions, t.
The case k = 0 is clear.
Assume that the proposition holds for non-negative integers up to r. Then
∞
∑
t(n)r+1 β(n) ≤
n=1
=
∞
∑
t(n)r β(n)
∑
n=1
∑
∑
t(q)
qkn
t(n) β(n) ≤
r
n∼
=o(q)
q
t(q)
∑
t(q)β(q)
q
∞
∑
t(qm)r β(m).
m=1
(m,q)=1
Here t(qm)r ≤ (t(q)+t(m))r ≤ (1+t(m))r , which we expand. Suppressing the condition
(q, m) = 1, we apply the induction hypothesis to each of the resulting sums. The upper
bound is at most
(
)r ∞
∑
∑
∑
t(q)β(q) r + 1 +
t(q)β(q)
β(m),
q
q
m=1
from which the validity of the proposition with k = r + 1 follows.
Corollary 3.5.
The function
t(n) =
∑
1
qkn,|f (q)|>D
satisfies
∞
∑
α
t(n) β(n)n
−σ
≤ (α + 2)
n=1
α+1
∞
∑
β(n)n−σ
n=1
uniformly for α ≥ 0, σ ≥ 0.
A further application of Hölder’s inequality shows that it will suffice to establish
the corollary for α a positive integer k, with an upper bound factor (k + 1)k . Since
∑
∑
|f (q)|2 β(q)q −σ = 1,
β(q)q −σ ≤ D−2
|f (q)|>D
q
Operator norms and the mean values of multiplicative functions
89
we may apply Lemma 3.4 with β replaced by the function n 7→ β(n)n−σ .
To complete the proof of Theorem 1.1 when α ≥ 2, define
∑
f2 (n) =
f (q).
qkn,|f (q)|>D
If α ≥ 1, then
Lemma 3.6.
∞
∑
β(n)n−σ |f2 (n) − E(f2 )|α ¿
∑
|f2 (q)|α β(q)q −σ
q
n=1
∞
∑
β(n)n−σ
n=1
uniformly for σ ≥ 0.
Proof of Lemma 3.6. With the above definition of t,
∑
|f2 (n)|α ≤
|f2 (q)|α t(n)α−1 ,
qkn
and
∞
∑
β(n)n−σ |f2 (n)|α ≤
∑
q
n=1
∑
|f2 (q)|α β(q)q −σ
t(qm)α−1 β(m)m−σ .
(m,q)=1
Again t(qm)α−1 ≤ (1 + t(m))α−1 ¿ 1 + t(m)α−1 . We omit the condition (m, q) = 1
and apply the corollary to Lemma 3.4.
Moreover, an application of Hölder’s inequality with α−1 + γ −1 = 1, suitably interpreted, yields
1/γ
(
)1/α 
∑
∑

|E(f2 )| ≤
|f2 (q)|α β(q)q −σ
β(q)q −σ 
|f (q)|>D
q
with a final sum that does not exceed 1.
The proof of Lemma 3.6 is complete.
Since f = f1 + f2 , from Lemma 3.3 and Lemma 3.6 combined with Lemma 3.1
(
(
)1/α
) ∞
∑
∑
exp −
β(q)q −σ
β(n)n−σ |f (n) − E(f )|α
q
¿
(
∑
q
n=1
)1/α
|f (q)|α β(q)q −σ
+
(
∑
)1/2
|f (q)|2 β(q)q −σ
q
uniformly for 0 < σ ≤ 2, and we may let σ approach zero.
For the cases 1 ≤ α < 2 of Theorem 1.1 the following modification suffices.
90
Peter D. T. A. Elliott
Lemma 3.7.

The function
1/α
∑
λ(y) = 
|f (q)|α β(q)q −σ 

∑
+
|f (q)|>y
1/2
|f (q)|2 β(q)q −σ 
|f (q)|≤y
defined for real y > 0, satisfies λ(λ(y)) ≤ 2λ(y).
Proof of Lemma 3.7. If λ(y) = w ≥ y, then

1/α 
1/2
∑
∑
λ(w) ≤ 
|f (q)|α β(q)q −σ 
+
|f (q)|2 β(q)q −σ 
|f (q)|>y

+
|f (q)|≤y
1/2
∑
|f (q)|2 β(q)q −σ 
y<|f (q)|≤w

≤ w + w1−α/2 
1/2
∑
|f (q)|α β(q)q −σ 
≤ 2w.
|f (q)|>y
There is a similar argument if w < y.
This completes the proof of Lemma 3.7.
In particular, for any value, w, of λ(y),
∑
∑
|f (q)|α β(q)q −σ ≤ w−α λ(w)α ≤ 2α ,
β(q)q −σ ≤ w−α
|f (q)|>w
|f (q)|>w
∑
|f (q)|2 β(q)q −σ ≤ λ(w)2 ≤ 4w2 .
|f (q)|≤w
We may replace D by λ(η) and follow the argument for the cases α ≥ 2.
This completes the proof of Theorem 1.1.
§ 3.3.
Remarks
If 0 < α < 1, then for any η > 0 the argument supporting Theorem 1.1 delivers the
inequality
¯
¯α 1/α

(
) ∞
¯
¯
∑
∑
∑
¯
¯
¯
exp −
β(n) ¯f (n) −
β(q)
f (q)β(q)¯¯ 
¯
¯
q
n=1
|f (q)|≤η

¿
∑
|f (q)|>η
1/α
|f (q)|α β(q)

+
∑
|f (q)|≤η
1/2
|f (q)|2 β(q)
.
Operator norms and the mean values of multiplicative functions
91
the implicit constant depending at most upon α. Since they do not satisfy the triangle
inequality, the sums involving α are not norms. This inequality is guided rather by an
aesthetic from the theory of probability.
Over the range 0 < α < 1 similar modifications should be made to the inequalities
of [6], Theorem 2.1, [5], Theorems 1 and 2 and therefore [6], Theorems 34.1 and 34.2.
This does not affect any of the applications in those works since only inequalities with
α > 1 are employed.
For 0 < α < 1 other variants are possible, cf. [3].
With modest assumptions inequalities of Turán–Kubilius type may be extended to
normed freely generated commutative semigroups, as is sketched in exercises 14–17 of
the author’s volume [6], Chapter 34.
This ends the remarks.
§ 4.
Proof of Theorem 1.3
I shall concentrate on the necessity of the conditions involving the primes. I shall
apply Theorem 1.2 with β(n) = τ ω(n) n−σ , where ω(n) denotes the number of distinct
prime divisors of n and 1 < σ ≤ 2. For ease of presentation only, I confine myself to the
cases α ≥ 2. Theorem 1.2 then delivers the estimate
¯
¯α 1/α

¯ ∑
¯
∞
∑
∑
¯
¯
(α−1)σ
−σ
−σ
−σ
¯

q
an n − τ q
an n ¯¯ 
¯
¯n ∼
¯
q
n=1
=o(q)
+ (a similar expression with α = 2)
(
¿ (σ − 1)−τ /γ
)1/α
∞
∑
|an |α τ −(α−1)ω(n) n−σ
n=1
with 1/α+1/γ = 1, valid for all complex numbers an for which the final series converges.
§ 4.1.
∑
General remarks
Let 0 < w < 1/2. Provided 0 < δ < min(α−1 , (α − 1)w/2), the uniform bound on
|g(n)|α , n ≤ x, guarantees uniform bounds g(q) ¿ q 1−δ and
∑
|g(q)| ¿ x1−δ .
q≤x,|g(q)|>q w
∑
An integration by parts shows the series
|g(q)|q −1 , taken over the prime-powers for
∑
which |g(q)| > q w , to converge, consequently the series
|g(pm )|p−m , taken over the
prime-powers with m ≥ 2, also to converge.
92
Peter D. T. A. Elliott
For each prime p,
∞
∑
|g(p )|p
m
−m
¿
m=1
∞
∑
p−δm < 1/2,
m=1
provided p is sufficiently large.
The function
∞
∑
F (σ) =
α −σ
|g(n)| n
∫
=σ
∫
∞
F (σ) ¿ σ
y −σ−1
1
n=1
satisfies
∞
y −σ (log y)τ
α
−1
∑
|g(n)|α dy
n≤y
dy ¿ Γ(τ α )(σ − 1)−τ , 1 < σ ≤ 2,
α
1
the last step after a change of variables y = exp(u(σ − 1)−1 ).
Similarly,
G(σ) =
∞
∑
g(n)n−σ = (1 + o(1))AΓ(τ )(σ − 1)−τ
as
σ → 1+,
n=1
where A is the value of the limit implicit in the hypothesis on the sum
and is not zero.
§ 4.2.
∑
g(n), n ≤ x,
Necessity, Case 0 < τ ≤ 1
We apply the dual Turán–Kubilius inequality to |g(n)|. Since 1 + ta ≤ (1 + a)t if
a ≥ 0, t ≥ 1,
(
)
∞
∏
∑
−(α−1)
−(α−1)
m α −mσ
1+τ
|g(p )| p
≤ F (σ)τ
¿ (σ − 1)−τ ,
p
hence
m=1
¯
¯α 1/α
¯
¯
∞
∑
∑
∑
¯
¯
−σ
−σ
−σ

q ¯¯|g(q)|
|g(n)|n − τ
|g(n)|n ¯¯ 
¯
¯
q
n=1
(n,q)=1

+ (a similar expression with α = 2)
¿ ζ(σ)τ /γ (σ − 1)−τ /α ¿ (σ − 1)−τ ,
uniformly for 1 < σ ≤ 2. Noting that
∞
∑
n=1
|g(n)|n−σ
¯∞
¯
¯∑
¯
¯
¯
≥¯
g(n)n−σ ¯ À (σ − 1)−τ ,
¯
¯
n=1
Operator norms and the mean values of multiplicative functions
93
we derive the convergence of the series
∑
q −1 ||g(q)|θ(q)−1 − τ |α
∑
and
q
q −1 ||g(q)|θ(q)−1 − τ |2 ,
q
∑∞
where θ(q) = 1 + m=1 |g(q0m )|q0−m and q0 is the prime of which q is a power.
∑ −1
From our general remarks the θ(q) are uniformly bounded, the series
q |θ(q) −
∑ −1
α
2
1| and
q |θ(q) − 1| converge, and so do the series
∑
q −1 ||g(q)| − τ |α
∑
and
q
q −1 ||g(q)| − τ |2 .
q
∑
In particular, for any ε, 0 < ε < 1, the series
|g(q)|q −1 , taken over the primepowers for which |g(q)| > τ + ε, converges. Therefore
(
)
∞
∞
∑
∏
∑
1+
|g(n)n−σ =
|g(pm )|p−mσ
p
n=1
≤ exp
m=1
(
∑
)
|g(q)|q −σ
(
¿ exp (τ + ε)
q
∑
)
q −σ
q
¿ ζ(σ)τ +ε ¿ (σ − 1)−(τ +ε)
holds uniformly for 1 < σ ≤ 2.
If now
φ(σ) = 1 +
∞
∑
g(pm )p−mσ
m=1
vanishes for σ = 1, then since the series representing φ0 (σ) converges absolutely, φ(σ) ¿
σ − 1 as σ → 1+. Thus
(
)
∞
∏
∑
(σ − 1)−τ ¿ |G(σ)| ¿ |φ(σ)|
1+
|g(`m )|`−mσ ¿ (σ − 1)1−(τ +ε) ,
` prime
`6=p
m=1
and as σ → 1+ a contradiction ensues.
We can now reapply the earlier argument with g(n) in place of |g(n)| and obtain
the convergence of the series
∑
q
q −1 |g(q) − τ |α and
∑
q −1 |g(q) − τ |2 .
q
As before, the only lower bound that we require is |G(σ)| À (σ − 1)−τ , 1 < σ ≤ 2.
∑ −1
Only the convergence of the series
p (g(p) − τ ) remains to be assured.
94
Peter D. T. A. Elliott
Employing its Euler product representation, as σ → 1+
(
(
))
∞
∑
∑
m −mσ
log 1 +
g(p )p
G(σ) = (1 + o(1))B1 exp
p>p0
= (1 + o(1))B2 exp
(
∑
m=1
)
g(p)p−σ
,
q
where B1 , B2 are non-zero constants and p0 is a suitably large fixed prime. Combining this with the asymptotic estimate G(σ) = (1 + o(1))AΓ(τ )(σ − 1)−τ = (1 +
o(1))AΓ(τ )ζ(σ)τ , we obtain the existence of
∑
lim
(g(p) − τ )p−σ .
σ→1+
q
A simple tauberian argument now suffices.
∑
Let 0 < ε < 1. We denote the sum function of the series (g(p) − τ )p−σ by θ(σ),
set y = exp(ε(σ − 1)−1 ) and employ the decomposition
∑
∑
∑
θ(σ) −
(g(p) − τ )p−1 =
(g(p) − τ )p−σ +
(g(p) − τ )(p−σ − p−1 ).
p>y
p≤y
p≤y
An application of the Cauchy–Schwarz inequality together with the well-known Chebyshev bound π(x) ¿ x(log x)−1 shows the second of these sums to be
(
¿
∑
)1/2
2 −1
|g(p) − τ | p
= o(1) as
σ → 1+,
y → ∞.
p>y
Moreover, p−σ − p−1 ¿ (σ − 1)p−1 log p uniformly for p ≤ y. Hence the third sum is
similarly

1/2 (
)1/2
∑
∑
2
−1
2
−1
2
(log p) p 
|g(p) − τ | p
¿ ε,
¿ (σ − 1)
p
p≤y
the implied constant absolute.
Altogether,
¯
¯
¯∑
¯
¯
¯
−1
lim sup ¯¯ (g(p) − τ )p − θ(σ)¯¯ ¿ ε
σ→1+ ¯
¯
p≤y
and the desired convergence is obtained.
§ 4.3.
Necessity Case, τ > 1
The inequality 1 + ta ≤ (1 + a)t is not immediately available.
95
Operator norms and the mean values of multiplicative functions
Define the multiplicative function h by h(p) = g(p) if |g(p)| ≤ pw , where w is
the parameter appearing in the general remarks, 2wα < 1, and h = 0 on all other
prime-powers. Then
∞
∑
∏
|h(n)|α n−σ =
(1 + |g(p)|α p−σ ) ≤ F (σ) ¿ (σ − 1)−τ
α
|g(p)|≤pw
n=1
is available. Moreover
(σ − 1)−τ ¿ |G(σ)| ¿ exp
(
∑
)
|g(p)|p−σ
p
¿
∏
−σ
(1 + |g(p)|p
|g(p)|≤pw
)¿
∞
∑
|h(n)|n−σ ,
n=1
∑
since
|g(p)|p−1 , |g(p)| > pw , converges.
The argument of the previous section then yields the convergence of the series
∑
||h(p)| − τ |α p−1
and
∑
||h(p)| − τ |2 p−1 ,
hence that of the series
∑
|g(p) − τ |α p−1
and
∑
|g(p) − τ |2 p−1
taken over those primes p for which |g(p)| ≤ pw , with the concomitant non-vanishing of
∑∞
the sums 1 + m=1 g(pm )p−m .
From our general remarks there is a non-zero constant B3 for which
(
)
∑
G(σ) = (1 + o(1))B3 exp
h(p)p−σ
as σ → 1 + .
p
Combined with the asymptotic estimate G(σ) = (1+o(1))AΓ(τ )(σ−1)−τ this guarantees
the existence of
∑
lim
(h(p) − τ )p−σ
σ→1+
and the convergence of the series
pw may then be omitted.
Lemma 4.1.
∑
(g(p) − τ )p−1 , |g(p)| ≤ pw . The restriction |g(p)| ≤
If α > 0, |w − 1| ≤ 1/2, then
|w|α = 1 + Re(α(w − 1)) + O(|w − 1|2 ).
96
Peter D. T. A. Elliott
Proof of Lemma 4.1. With the principal value of the logarithm
|w|α = exp(Re{α log(1 + w − 1)}) = exp(Re(α(w − 1)) + O(|w − 1|2 ))
= exp(Re{α(w − 1)}) + O(|w − 1|2 )
and the result is evident.
Replacing w by w/τ , τ > 0,
|w|α − τ α = ατ α−1 Re (w − τ ) + O(|w − τ |2 )
uniformly for |w − τ | ≤ τ /2. Thus
∑0
(|g(p)|α − τ α )p−σ = ατ α−1 Re
+O
(∑ 0
{∑0
(g(p) − τ )p−σ
2 −σ
|g(p) − τ | p
}
)
,
all sums over primes for which |g(p)−τ | ≤ τ /2. Combined with what we have established
so far
∑
∑
|h(p)|α p−σ = τ α
p−σ + O(1)
p
p
uniformly for 1 < σ ≤ 2.
In particular,
(
)
∏
∑
(1 + |h(p)|α p−σ ) = exp
{|h(p)|α p−σ + O(|h(p)|2α p−2σ )}
p
p
= (1 + o(1))B4 (σ − 1)−τ
α
as
σ → 1+,
with B4 6= 0. As a consequence
(
)
∞
∏
∑
α
1+
|g(pm )|α p−mσ ¿ (σ − 1)τ F (σ) ¿ 1, 1 < σ ≤ 2.
|g(p)|>pw
m=1
The product over the primes p with |g(p)| > pw is uniformly bounded above and the
series
∞
∑ ∑
|g(pm )|α p−m
|g(p)|>pw m=1
converges.
This achieves the convergence of the series
∑
∑
|g(p) − τ |α p−1 ,
|g(p)−τ |>τ /2
|g(p)−τ |≤τ /2
|g(p) − τ |2 p−1
Operator norms and the mean values of multiplicative functions
and
97
∑
(g(p) − τ )p−1 .
In this case we are left to establish the convergence of the series
∑
|g(pm )|α p−m .
p,m≥2
For those primes p with |g(p)| ≤ pw we apply the elementary inequality (1 − b)(1 +
a + b) ≥ 1 + a/4, valid for 0 ≤ b ≤ 1/2, a ≥ 4b2 . With
a = ap =
∞
∑
|g(pm )|α p−mσ ,
b = |h(p)|α p−σ ,
m=2
and provided p ≥ 5,
exp(−|h(p)|α p−σ )(1 + |h(p)|α p−σ + ap ) ≥ 1 + ap /4
∏
The product (1 + ap /4), taken over those primes for which |g(p)| ≤ pw and ap ≥
4(|h(p)|α p−σ )2 , is not more than
∏
exp(−|h(p)|α p−σ )(1 + |h(p)|α p−σ + ap )
|g(p)|≤pw
≤ exp
(
∑
)
−|h(p)|α p−σ
F (σ) ¿ (σ − 1)τ (σ − 1)−τ ¿ 1,
α
α
p
uniformly for 1 < σ ≤ 2. As a consequence the series
∞
∑∑
|g(pm )|α p−mσ ,
p m=2
restricted to the same primes, is bounded uniformly for 1 < σ ≤ 2.
The corresponding sum over the remaining primes does not exceed
∑
∑
(|g(p)|α p−σ )2 ≤ 4
p−2(1−αw) ¿ 1,
4
|g(p)|≤pw
p
and is likewise uniformly bounded.
The sum
∑
|g(pm )|α p−mσ
p,m≥2
is bounded uniformly for 1 < σ ≤ 2 and we may let σ → 1+.
§ 4.4.
Sufficiency
I shall be brief. The argument rests upon the following two results.
98
Peter D. T. A. Elliott
Lemma 4.2.
If the real non-negative arithmetic function β satisfies β(1) = 1,
∑
β(qm) ≤ β(q)β(m) when (q, m) = 1 and y −1 q≤y β(q) log q ≤ ∆ uniformly for 1 ≤
y ≤ x, then


∑
∑
β(n) ¿ ∆x exp  (β(q) − 1)q −1  ,
2≤n≤x
q≤x
the implied constant absolute.
Proof of Lemma 4.2 This result is essentially a special case of [5], Lemma 1. See,
also, [6], Lemma 2.2.
Lemma 4.3.
With β as in Lemma 4.2 and f a complex-valued additive function
¯2
¯
¯
¯
∑
¯
¯
f (q)β(q)q −1 ¯¯
β(n) ¯¯f (n) −
¯
¯
n≤x
q≤x
∑

¿ ∆x exp 
∑

(β(q) − 1)q −1 
∑
|f (q)|2 β(q)q −1 ,
q≤x
q≤x
the implied constant absolute.
Proof of Lemma 4.3. One may proceed along the lines of the proof of the present
Theorem 1.1, bearing in mind the argument of [5] and [6], Chapter 2. The term with
n = 1 may be restored to the first summation since f (1) = 0,
¯
¯2
¯∑
¯
∑
∑
¯
¯
−1 ¯
2
−1
¯
f
(q)β(q)q
|f
(q)|
β(q)q
β(q)q −1 ,
≤
¯
¯
¯q≤x
¯
q≤x
q≤x
β(q) log q ≤ q∆ uniformly for q ≤ x and
∑
q≤x
β(q)q −1 ¿ ∆
∑
(log q)−1 ¿ ∆x(log x)−1 .
q≤x
As for Theorem 1.1, there are versions of Lemma 4.3 involving mean α-powers with
α ≥ 1.
To begin with I shall assume that the function q satisfies |g(q) − τ | ≤ τ /2 on all
prime-powers q. As before, q0 will denote the prime of which q is a power.
For r > 0, define the multiplicative function gr (n) by gr (q) = g(q) if q0 ≤ r,
gr (q) = τ otherwise.
Operator norms and the mean values of multiplicative functions
99
In view of the inequality |ez − 1| ≤ |z| max(eRe z , 1), valid for all complex z,
¯
¯
¯
¯
¯∑
¯
¯
¯
−1 ¯
¯
|g(n) − gr (n)| ≤ ¯
log(g(q)τ )¯ max(|g(n)|, |gr (n)|),
¯ qkn
¯
¯q0 >r
¯
with the principal value of the logarithm.
We note that by Lemma 4.2
∑

τ −ω(n) |g(n)|2 ¿ x(log x)−1 exp 
n≤x
∑

τ −1 |g(q)|2 
q≤x
¿ x(log x)τ −1 ,
since |g(q)|2 = |g(q) − τ |2 + 2τ Re(g(q) − τ ) + τ 2 . Likewise
∑
τ −ω(n) |gr (n)|2 ¿ x(log x)τ −1 ,
n≤x
uniformly for x ≥ 2, r ≥ 2.
With the additive function f defined by f (q) = log(g(q)τ −1 ) if q0 > r, f (q) = 0
otherwise, an application of the Cauchy–Schwarz inequality delivers the bound
1/2

∑
∑
τ ω(n) |f (n)|2  .
|g(n) − gr (n)| ¿ x(log x)τ −1
n≤x
n≤x
Since τ f (q) = g(q) − τ + O(|g(q) − τ |2 ), Lemma 4.3 shows that
∑
∑
|g(q) − τ |2 q −1 ,
τ ω(n) |f (n) − L|2 ¿ x(log x)τ −1
n≤x
where
L=
q≤x
q0 >r
∑
τ f (q)q −1 =
q≤x
+O
∑
(g(p) − τ )p−1
r<p≤x
(
∞
∑∑
)
p−m
(
+O
p>r m=2
Hence
lim sup lim sup(x(log x)τ −1 )−1
r→∞
x→∞
∑
)
|g(q) − τ |2 q −1
.
q0 >r
∑
|g(n) − gr (n)| = 0.
n≤x
The function gr (n) is τ
on those integers n not divisible by primes p ≤ r. A
classical argument shows that for each r
∑
Y (r) = lim x−1 (log x)−(τ −1)
gr (n)
ω(n)
x→∞
n≤x
100
Peter D. T. A. Elliott
exists and an inspection of Euler products shows that lim Y (r), r → ∞, exists.
This establishes the mean-value of g subject to our temporary hypothesis on the
size of g(q).
The bound
∑
α
|g(n)|α ¿ x(log x)τ −1
n≤x
follows from an application of Lemma 4.2 together with Lemma 4.1.
In the general case we first apply the argument to the multiplicative function g+
defined by g+ (q) = g(q) if |g(q) − τ | ≤ τ /2, and g+ (q) = τ otherwise. We define
the multiplicative function h by Dirichlet convolution, h ∗ g+ = g, and the remaining
∑∞
argument hinges on the absolute convergence of the series m=1 h(m)m−1 . A similar
situation arises in [6], Chapter 11, pp. 103–104, and I suppress the details.
Granted this,
∑
∑
∑
g(n) =
h(m)
g+ (t)
n≤x
m≤x
t≤x/m
= (1 + o(1))Ax(log x)τ −1 ,
x → ∞,
and an examiniation of the Euler product representation of A shows it not to vanish.
Alternatively, we may define a multiplicative function t by


0 if p ≤ τ + 1,



t(pm ) = g(pm ) if p > τ + 1 and |g(p) − τ | ≤ τ /2,



 τ if p > τ + 1 and |g(p) − τ | > τ /2.
Then g = t ∗ h where the multiplicative function h has associated Euler product
( ∞
)
∏
∑
∏
g(pm )p−mσ
(1 + (τ − 1)p−σ )−1 ·
p≤τ +1
m=0
p>τ +1
|g(p)−τ |>τ /2
(
· (1 − p−σ )
∞
∑
)
g(pm )p−mσ
m=0
∑
and the series
h(n)n−1 is absolutely convergent. It will suffice to treat the function
t.
For r > τ + 1 we define tr (pm ) = t(pm ) if m ≥ 2,

 t(p) if p ≤ r,
tr (p) =
 τ if p > r.
Operator norms and the mean values of multiplicative functions
101
We represent a typical integer n as a product uv where p | u if and only if pkn. Then
∑
∑
∑
|tr (n) − t(n)| =
|t(v)|
|tr (u) − t(u)|,
n≤x
v≤x
u≤x/v
where u runs through squarefree integers and v those for which every prime divisor
∑
occurs multiply. In particular,
|t(v)|v −1 converges.
The treatment of tr (u) (= µ(n)2 tr (n)) now follows that for gr (n).
This completes the proof of Theorem 1.3.
§ 5.
Proof of Theorem 1.4
I confine myself to the following remark.
A non-negative arithmetic function h satisfies
∑
(log x)t ¿
h(n)n−1 ¿ (log x)t
n≤x
for some t > 0 and all x ≥ 2 if and only if the corresponding Dirichlet series H(σ) =
∑∞
−σ
satisfies
n=1 h(n)n
(σ − 1)−t ¿ H(σ) ¿ (σ − 1)−t
uniformly for 1 < σ ≤ 2.
Assuming only the upper bound on h, an integration by parts shows that
∫ ∞
∑
H(σ) = (σ − 1)
h(n)n−1 dy ¿ (σ − 1)−t , 1 < σ ≤ 2.
y −σ
1
n≤y
Conversely, with σ0 = 1 + (log x)−1 ,
∑
h(n)n
−1
≤e
∞
∑
h(n)n−σ0 ¿ (σ0 − 1)−1 ¿ (log x)t .
n=1
n≤x
Adjoining the lower bound condition on h ensures that (σ − 1)−t ¿ H(σ) with the
same uniformity in σ.
Moreover, if both upper and lower bounds are satisfied by H(σ), then with w =
exp(A(σ − 1)−1 )
∫ ∞
∫ ∞
∑
−σ
1−σ
t
−t
h(n)n ¿ (σ − 1)
y
(log y) dy ¿ (σ − 1)
ut e−u du,
w
n>w
and if A is fixed at a sufficiently large value
∑
h(n)n−σ À (σ − 1)−t ,
n≤w
A
1 < σ ≤ 2.
102
Peter D. T. A. Elliott
Specialising σ to 1 + A(log x)−1 ,
∑
h(n)n−1 À (log x)t ,
n≤x
the implied constant depending upon A.
This ends the remark.
§ 6.
Concluding Remarks
The foregoing is a formal representation, with adjoined details, of the lecture under
the same title that I gave during the International Conference: Functions in Number
Theory and Their Probabilistic Aspects, held in the Research Institute for Mathematical
Sciences, Kyoto University, Japan, December 13–17, 2010.
It gives me great pleasure to thank the organizers Akiyama S., Fukuyama K.,
Matsumoto K., Nakada H., Sugita H., Takahashi Y., Tamagawa A., for their kind
invitation to speak and for their financial support.
I thank all my Japanese colleagues for their warmly welcoming hospitality and their
many kindnesses to my wife and myself.
References
[1] Daboussi, H., Sur les fonctions multiplicatives ayant une valeur moyenne non nulle, Bull.
Soc. Math. France 109 (1981), 183–205.
[2] Elliott, P.D.T.A., A mean value theorem for multiplicative functions, Proc. London Math.
Soc. 31 (1975), 418–438.
[3] Elliott, P.D.T.A., High power analogues of the Turán–Kubilius inequality and an application to number theory, Canadian J. Math. 32 (1980), 893–907.
[4] Elliott, P.D.T.A., Mean value theorems for multiplicative functions bounded in mean αpower, α > 1, J. Australian Math. Soc. (Series A) 29 (1980), 177–205.
[5] Elliott, P.D.T.A., The norms of compositions of arithmetic operators, Bull. London Math.
Soc. 19 (1987), 522–530.
[6] Elliott, P.D.T.A., Duality in Analytic Number Theory, Cambridge Tracts in Math. Number
122, Camb. Univ. Press, 1997.
[7] Zhang, Wen-Bin, Mean-value theorems and extensions of the Elliott–Daboussi theorem on
Beurling’s generalized integers (I), Math. Zeit. 261 (2009), 201–232.