asymptotic estimates for real-valued nonnegative multiplicative functions

asymptotic estimates for real-valued nonnegative multiplicative functions

Note that, within this entry,

p

always refers to a prime,

k

,

m

, and

n

always refer to

positive integers

, and

log

always refers to the

natural logarithm

.

Theorem

.

Let

f

be a real-valued nonnegative

multiplicative function

such that the two following conditions are satisfied:

1.

There exists

A

≥

0

such that, for every

y

≥

0

,

∑

p

≤

y

f

⁢

(

p

)

⁢

log

⁡

p

≤

A

⁢

y

.

2.

There exists

B

≥

0

such that

∑

p

∑

k

≥

2

f

⁢

(

p

k

)

⁢

log

⁡

(

p

k

)

p

k

≤

B

.

Then for all

x

>

1

,

∑

n

≤

x

f

⁢

(

n

)

≤

(

A

+

B

+

1

)

⁢

x

log

⁡

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

.

Proof.

log

⁡

x

⁢

∑

n

≤

x

f

⁢

(

n

)

=

∑

n

≤

x

f

⁢

(

n

)

⁢

(

log

⁡

x

-

log

⁡

n

+

log

⁡

n

)

=

∑

n

≤

x

f

⁢

(

n

)

⁢

log

⁡

(

x

n

)

+

∑

n

≤

x

f

⁢

(

n

)

⁢

log

⁡

n

≤

∑

n

≤

x

f

⁢

(

n

)

⁢

(

x

n

)

+

∑

n

≤

x

f

⁢

(

n

)

⁢

∑

p

k

∥

n

log

⁡

(

p

k

)

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

p

k

≤

x

log

⁡

(

p

k

)

⁢

∑

n

≤

x

and

p

k

∥

n

f

⁢

(

n

)

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

p

k

≤

x

log

⁡

(

p

k

)

⁢

∑

n

≤

x

and

p

k

∥

n

f

⁢

(

p

k

)

⁢

f

⁢

(

n

p

k

)

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

p

k

≤

x

log

⁡

(

p

k

)

⁢

∑

m

≤

x

p

k

f

⁢

(

p

k

)

⁢

f

⁢

(

m

)

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

p

≤

x

f

⁢

(

p

)

⁢

log

⁡

p

⁢

∑

m

≤

x

p

f

⁢

(

m

)

+

∑

p

≤

x

∑

k

≥

2

f

⁢

(

p

k

)

⁢

log

⁡

(

p

k

)

⁢

x

p

k

⁢

∑

m

≤

x

p

k

f

⁢

(

m

)

m

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

m

≤

x

f

⁢

(

m

)

⁢

∑

p

≤

x

m

f

⁢

(

p

)

⁢

log

⁡

p

+

x

⁢

∑

m

≤

x

f

⁢

(

m

)

m

⁢

∑

p

≤

x

∑

k

≥

2

f

⁢

(

p

k

)

⁢

log

⁡

(

p

k

)

p

k

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

∑

m

≤

x

f

⁢

(

m

)

⁢

(

A

⁢

x

m

)

+

x

⁢

∑

m

≤

x

f

⁢

(

m

)

m

⁢

B

≤

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

A

⁢

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

+

B

⁢

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

≤

(

A

+

B

+

1

)

⁢

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

Dividing the

inequality

log

⁡

x

⁢

∑

n

≤

x

f

⁢

(

n

)

≤

(

A

+

B

+

1

)

⁢

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

by

log

⁡

x

yields the desired result.
∎

The theorem has an obvious corollary:

Corollary

.

If

f

the conditions of the theorem, then for all

x

>

1

,

∑

n

≤

x

f

⁢

(

n

)

=

O

⁢

(

x

log

⁡

x

⁢

∑

n

≤

x

f

⁢

(

n

)

n

)

.

Title

asymptotic estimates for real-valued nonnegative multiplicative functions

Canonical name

AsymptoticEstimatesForRealvaluedNonnegativeMultiplicativeFunctions

Date of creation

2013-03-22 16:08:42

Last modified on

2013-03-22 16:08:42

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11N37

Related topic

AsymptoticEstimate

Related topic

DisplaystyleSum_nLeXTaunaO_axlogX2a1ForAGe0

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