x⁢l⁢o⁢g^2x=O⁢(∑@⁢ERROR(\slimits)⁢@⁢@⁢@_{n≤x}⁢2^{Ω⁢(n)})

x

⁢

l

⁢

o

⁢

g

2

x

=

O

⁢

(

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

2

Ω

⁢

(

n

)

)

Within this entry,

Ω

refers to the number of (nondistinct)

prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

),

τ

refers to the

divisor function

,

μ

refers to the Möbius function,

⌊

⋅

⌋

refers to the

floor function

,

log

refers to the

natural logarithm

,

p

refers to a prime, and

d

,

k

,

ℓ

,

m

, and

n

refer to

positive integers

.

Theorem

.

x

⁢

log

2

⁡

x

=

O

⁢

(

∑

n

≤

x

2

Ω

⁢

(

n

)

)

Proof.

∑

n

≤

x

2

Ω

⁢

(

n

)

=

∑

2

k

⁢

m

≤

x

m

⁢

is odd

2

Ω

⁢

(

2

k

⁢

m

)

=

∑

2

k

≤

x

2

Ω

⁢

(

2

k

)

⁢

∑

m

≤

x

2

k

m

⁢

is odd

2

Ω

⁢

(

m

)

since

2

Ω

is

multiplicative

≥

∑

k

≤

log

⁡

x

log

⁡

2

2

k

⁢

∑

m

≤

x

2

k

m

⁢

is odd

τ

⁢

(

m

)

≥

∑

k

≤

log

⁡

x

log

⁡

2

2

k

⁢

∑

d

≤

x

2

k

d

⁢

is odd

∑

ℓ

≤

x

2

k

⁢

d

ℓ

⁢

is odd

1

by the

convolution method

≥

∑

k

≤

log

⁡

x

log

⁡

2

2

k

⁢

∑

d

≤

x

2

k

d

⁢

is odd

x

2

k

+

2

⁢

d

≥

x

4

⁢

∑

k

≤

log

⁡

x

log

⁡

2

∑

d

≤

x

2

k

d

⁢

is odd

1

d

≥

x

4

⁢

∑

k

≤

log

⁡

x

log

⁡

2

(

1

x

⁢

∑

d

≤

x

2

k

d

⁢

is odd

1

-

∫

1

x

2

k

-

1

t

2

⁢

(

∑

d

≤

t

d

⁢

is odd

1

)

⁢

𝑑

t

)

by

summation by parts

(

http://planetmath.org/AbelsLemma

)

≥

x

4

⁢

∑

k

≤

log

⁡

x

log

⁡

2

(

1

x

⋅

x

2

k

+

2

+

∫

1

x

2

k

1

t

2

⋅

t

2

⁢

𝑑

t

)

≥

x

4

⁢

∑

k

≤

log

⁡

x

log

⁡

2

(

1

2

k

+

2

+

1

2

⁢

∫

1

x

2

k

1

t

⁢

𝑑

t

)

≥

x

4

⁢

∑

k

≤

log

⁡

x

log

⁡

2

(

1

2

k

+

2

+

1

2

⁢

log

⁡

(

x

2

k

)

)

≥

x

16

⁢

∑

k

≤

log

⁡

x

log

⁡

2

(

1

2

k

+

log

⁡

x

-

k

⁢

log

⁡

2

)

≥

x

16

⁢

(

1

2

⁢

(

1

-

(

1

2

)

⌊

log

⁡

x

log

⁡

2

⌋

+

1

1

-

1

2

)

+

log

⁡

x

⁢

⌊

log

⁡

x

log

⁡

2

⌋

-

log

⁡

2

⁢

(

⌊

log

⁡

x

log

⁡

2

⌋

2

+

⌊

log

⁡

x

log

⁡

2

⌋

2

)

)

≥

x

16

⁢

⌊

log

⁡

x

log

⁡

2

⌋

⁢

(

log

⁡

x

-

1

2

⁢

log

⁡

2

⁢

(

⌊

log

⁡

x

log

⁡

2

⌋

+

1

)

)

≥

x

16

⁢

⌊

log

⁡

x

log

⁡

2

⌋

⁢

(

log

⁡

x

-

1

2

⁢

log

⁡

2

⁢

(

log

⁡

x

log

⁡

2

+

1

)

)

≥

x

16

⁢

⌊

log

⁡

x

log

⁡

2

⌋

⁢

(

log

⁡

x

-

1

2

⁢

log

⁡

x

-

1

2

⁢

log

⁡

2

)

≥

x

32

⁢

⌊

log

⁡

x

log

⁡

2

⌋

⁢

log

⁡

(

x

2

)

Since, for

x

sufficiently large,

log

⁡

x

=

O

⁢

(

log

⁡

(

x

2

)

)

and

log

⁡

x

=

O

⁢

(

⌊

log

⁡

x

log

⁡

2

⌋

)

, it follows that

x

⁢

log

2

⁡

x

=

O

⁢

(

∑

n

≤

x

2

Ω

⁢

(

n

)

)

.
∎

Title

x

⁢

l

⁢

o

⁢

g

2

x

=

O

⁢

(

∑

@

⁢

\slimits

⁢

@

⁢

@

⁢

@

n

≤

x

⁢

2

Ω

⁢

(

n

)

)

Canonical name

displaystyleXlog2xOleftsumnleX2Omeganright

Date of creation

2013-03-22 16:09:18

Last modified on

2013-03-22 16:09:18

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11N37

Related topic

AsymptoticEstimate

Related topic

ConvolutionMethod

Related topic

DisplaystyleYOmeganOleftFracxlogXy12YRightFor1LeY2

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