∑@⁢ERROR(\slimits)⁢@⁢@⁢@_{n≤x}⁢y^{Ω⁢(n)}=O⁢({x⁢(l⁢o⁢gx)^{y-1}}/{2-y}) for 1≤y<2

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

y

Ω

⁢

(

n

)

=

O

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

y

-

1

2

-

y

)

for

1

≤

y

<

2

Within this entry,

Ω

refers to the number of (nondistinct)

prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

),

μ

refers to the Möbius function,

log

refers to the

natural logarithm

,

p

refers to a prime, and

d

,

k

,

m

, and

n

refer to

positive integers

.

Theorem

.

For

1

≤

y

<

2

,

∑

n

≤

x

y

Ω

⁢

(

n

)

=

O

⁢

(

x

⁢

(

log

⁡

x

)

y

-

1

2

-

y

)

.

Proof.

Let

g

be a function such that

y

Ω

=

1

*

g

. Then

g

is

multiplicative

and

g

=

μ

*

y

Ω

. Thus:

∑

n

≤

x

y

Ω

⁢

(

n

)

=

∑

d

≤

x

∑

m

≤

x

d

g

⁢

(

d

)

by the

convolution method

=

O

⁢

(

∑

d

≤

x

g

⁢

(

d

)

⋅

x

d

)

=

O

⁢

(

x

⁢

∏

p

≤

x

(

1

+

∑

k

g

⁢

(

p

k

)

p

k

)

)

=

O

⁢

(

x

⁢

∏

p

≤

x

(

1

+

∑

k

μ

⁢

(

1

)

⁢

y

k

+

μ

⁢

(

p

)

⁢

y

k

-

1

p

k

)

)

=

O

⁢

(

x

⁢

∏

p

≤

x

(

1

+

y

-

1

p

⁢

∑

k

(

y

p

)

k

-

1

)

)

=

O

⁢

(

x

⁢

∏

p

≤

x

(

1

+

y

-

1

p

⋅

1

1

-

y

p

)

)

=

O

⁢

(

x

⁢

∏

p

≤

x

(

1

+

y

-

1

p

-

y

)

)

=

O

⁢

(

x

⁢

(

1

+

y

-

1

2

-

y

)

⁢

∏

3

≤

p

≤

x

(

1

+

y

-

1

p

-

y

)

)

=

O

⁢

(

x

⁢

(

2

-

y

+

y

-

1

2

-

y

)

⁢

∏

3

≤

p

≤

x

exp

⁡

(

y

-

1

p

-

y

)

)

=

O

⁢

(

x

2

-

y

⁢

(

exp

⁡

(

∑

3

≤

p

≤

x

y

-

1

p

-

y

)

)

)

=

O

⁢

(

x

2

-

y

⁢

(

exp

⁡

(

∑

3

≤

p

≤

x

1

p

-

y

)

)

y

-

1

)

=

O

⁢

(

x

2

-

y

⁢

(

exp

⁡

(

log

⁡

log

⁡

x

+

O

⁢

(

1

)

)

)

y

-

1

)

=

O

⁢

(

x

2

-

y

⁢

(

c

⁢

e

log

⁡

log

⁡

x

)

y

-

1

)

for some

c

>

0

=

O

⁢

(

x

2

-

y

⁢

(

max

⁡

{

1

,

c

}

⁢

log

⁡

x

)

y

-

1

)

=

O

⁢

(

x

2

-

y

⁢

(

max

⁡

{

1

,

c

}

)

2

-

1

⁢

(

log

⁡

x

)

y

-

1

)

=

O

⁢

(

x

⁢

(

log

⁡

x

)

y

-

1

2

-

y

)

.

∎

Note that a result for

y

=

2

(and therefore for

y

≥

2

), such as

∑

n

≤

x

2

Ω

⁢

(

n

)

=

O

⁢

(

x

⁢

log

⁡

x

)

, is unobtainable, as evidenced by this theorem (

http://planetmath.org/DisplaystyleXlog2xOleftsum_nLeX2OmeganRight

). On the other hand, the

asymptotic estimates

∑

n

≤

x

2

ω

⁢

(

n

)

=

O

⁢

(

x

⁢

log

⁡

x

)

and

∑

n

≤

x

τ

⁢

(

n

)

=

O

⁢

(

x

⁢

log

⁡

x

)

are true.

Title

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

y

Ω

⁢

(

n

)

=

O

⁢

(

x

⁢

(

l

⁢

o

⁢

g

x

)

y

-

1

2

-

y

)

for

1

≤

y

<

2

Canonical name

displaystylesumnleXYOmeganOleftfracxlogXy12yrightFor1leY2

Date of creation

2013-03-22 16:09:15

Last modified on

2013-03-22 16:09:15

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

18

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11N37

Related topic

AsymptoticEstimate

Related topic

ConvolutionMethod

Related topic

DisplaystyleXlog2xOleftsum_nLeX2OmeganRight

Related topic

DisplaystyleSum_nLeXYomeganO_yxlogXy1ForYGe0

Related topic

DisplaystyleSum_nLeXTaunaO_axlogX2a1ForAGe0

Related topic

2omeganLeTaunLe2Omegan

Generated on Fri Feb 9 15:39:59 2018 by

LaTeXML