∑@⁢ERROR(\slimits)⁢@⁢@⁢@_{n≤x}⁢(τ⁢(n))^a=O_a⁢(x⁢(l⁢o⁢gx)^{2^a-1}) for a≥0

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

(

τ

⁢

(

n

)

)

a

=

O

a

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

2

a

-

1

)

for

a

≥

0

Within this entry,

τ

refers to the

divisor function

,

⌊

⋅

⌋

refers to the

floor function

,

log

refers to the

natural logarithm

,

p

refers to a prime, and

k

and

n

refer to

positive integers

.

Theorem

.

For

a

≥

0

,

∑

n

≤

x

(

τ

⁢

(

n

)

)

a

=

O

a

⁢

(

x

⁢

(

log

⁡

x

)

2

a

-

1

)

.

The

O

a

indicates that the constant implied by the definition of

O

depends on

a

. (See

Landau notation

for more details.)

Proof.

Let

a

≥

0

. Since

(

τ

)

a

=

id

a

∘

τ

,

id

is

completely multiplicative

, and

τ

is multiplicative,

(

τ

)

a

is multiplicative. (See

composition of multiplicative functions

for more details.)

For any

y

≥

0

,

∑

p

≤

y

(

τ

⁢

(

p

)

)

a

⁢

log

⁡

p

=

∑

p

≤

y

2

a

⁢

log

⁡

p

=

2

a

⁢

∑

p

≤

y

log

⁡

p

≤

2

a

⁢

y

⁢

log

⁡

4

by this theorem (

http://planetmath.org/UpperBoundOnVarthetan

).

Also,

∑

p

∑

k

≥

2

(

τ

⁢

(

p

k

)

)

a

p

k

⁢

log

⁡

(

p

k

)

=

∑

p

∑

k

≥

2

(

k

+

1

)

a

p

k

⋅

k

⁢

log

⁡

p

≤

∑

p

log

⁡

p

⁢

∑

k

≥

2

(

k

+

1

)

a

+

1

p

k

≤

∑

p

log

⁡

p

p

2

⁢

∑

k

≥

2

(

2

⁢

k

)

a

+

1

p

k

-

2

≤

2

a

+

1

⁢

∑

p

1

p

3

2

⁢

∑

k

≥

2

k

a

+

1

2

k

-

2

≤

2

a

+

3

⁢

ζ

⁢

(

3

2

)

⁢

∑

k

≥

2

k

a

+

1

2

k

, where

ζ

denotes the

Riemann zeta function

.

Since

lim

k

→

∞

⁡

|

(

(

k

+

1

)

a

+

1

2

k

+

1

)

(

k

a

+

1

2

k

)

|

=

lim

k

→

∞

⁡

|

(

k

+

1

)

a

+

1

⁢

2

k

k

a

+

1

⁢

2

k

+

1

|

=

lim

k

→

∞

⁡

(

1

2

)

⁢

(

k

+

1

k

)

a

+

1

=

1

2

⁢

(

lim

k

→

∞

⁡

k

+

1

k

)

a

+

1

=

1

2

,

∑

k

≥

2

k

a

+

1

2

k

converges

by the

ratio test

. Thus, by this theorem (

http://planetmath.org/AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions

),

∑

n

≤

x

(

τ

⁢

(

n

)

)

a

=

O

a

⁢

(

x

log

⁡

x

⁢

∑

n

≤

x

(

τ

⁢

(

n

)

)

a

n

)

. Therefore,

∑

n

≤

x

(

τ

⁢

(

n

)

)

a

=

O

a

⁢

(

x

log

⁡

x

⁢

∑

n

≤

x

(

τ

⁢

(

n

)

)

a

n

)

=

O

a

⁢

(

x

log

⁡

x

⁢

∏

p

≤

x

(

1

+

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

(

τ

⁢

(

p

k

)

)

a

p

k

)

)

=

O

a

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

∑

p

≤

x

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

(

k

+

1

)

a

p

k

)

)

)

=

O

a

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

∑

p

≤

x

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

(

2

⁢

k

)

a

p

k

)

)

)

=

O

a

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

2

a

⁢

∑

p

≤

x

∑

k

=

1

⌊

log

⁡

x

log

⁡

p

⌋

k

a

p

k

)

)

)

=

O

a

⁢

(

x

log

⁡

x

⁢

(

exp

⁡

(

2

a

⁢

(

log

⁡

log

⁡

x

+

O

a

⁢

(

1

)

)

)

)

)

=

O

a

(

x

log

⁡

x

(

exp

(

log

(

log

x

)

2

a

)

)

)

=

O

a

⁢

(

x

log

⁡

x

⁢

(

log

⁡

x

)

2

a

)

=

O

a

⁢

(

x

⁢

(

log

⁡

x

)

2

a

-

1

)

.

∎

Title

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

(

τ

⁢

(

n

)

)

a

=

O

a

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

2

a

-

1

)

for

a

≥

0

Canonical name

displaystylesumnleXtaunaOaxlogX2a1ForAge0

Date of creation

2013-03-22 16:09:53

Last modified on

2013-03-22 16:09:53

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

15

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11N37

Related topic

AsymptoticEstimatesForRealValuedNonnegativeMultiplicativeFunctions

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

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