prime harmonic series

prime harmonic series

\PMlinkescapephrase

square-free part

\PMlinkescapephrase

set of primes

\PMlinkescapephrase

harmonic series

The prime

1

1

ℙ

denotes the set of
primes.

harmonic series (also known as

series of reciprocals of primes

) is the infinite sum

∑

p

∈

ℙ

1

p

. The following result was originally proved by Euler (using the

Euler product

of the

Riemann Zeta function

) but the following extremely elegant proof is due to Paul Erdős

[

2

]

.

Theorem

.

The series

∑

p

∈

P

1

p

diverges

.

Proof.

Assume that this series is

convergent

. If so, then, for a certain

k

∈

ℕ

, we have:

∑

i

>

k

1

p

i

<

1

2

,

where

p

i

is the

i

th

prime. Now, we define

λ

k

(

n

)

:=

#

{

m

≤

n

:

p

i

∣

m

⇒

i

≤

k

}

, the
number of

integers

less than

n

divisible

only by the first

k

primes. Any of these numbers can be expressed as

m

=

p

1

ω

1

⁢

⋯

⁢

p

k

ω

k

⋅

s

2

,

ω

i

∈

{

0

,

1

}

(i.e. a square multiplied by a

square-free number

). There are

2

k

ways to chose the square-free part and clearly

s

≤

n

, so

λ

k

⁢

(

n

)

≤

2

k

⁢

n

. Now, the number of integers divisible by

p

i

less than

n

is

⌊

n

p

i

⌋

, so the number of integers less than

n

divisible by primes bigger than

p

k

(which we shall denote by

Λ

k

⁢

(

n

)

) is bounded above as follows:

Λ

k

⁢

(

n

)

≤

∑

i

>

k

⌊

n

p

i

⌋

≤

∑

i

>

k

n

p

i

<

n

2

.

However, by their

definitions

,

λ

k

⁢

(

n

)

+

Λ

k

⁢

(

n

)

=

n

for all

n

∈

ℕ

and so it is

sufficient

to find an

n

such that

λ

k

⁢

(

n

)

≤

n

2

for a

contradiction

and, using the previous bound for

λ

k

⁢

(

n

)

, which is

λ

k

⁢

(

n

)

≤

2

k

⁢

n

, we see that

n

=

2

2

⁢

k

+

2

works.
∎

The series is in some ways

similar

to the Harmonic series (

http://planetmath.org/HarmonicSeries

)

H

n

:=

∑

k

=

1

n

1

n

. In fact, it is well known that

H

n

=

log

⁡

n

+

γ

+

o

⁢

(

1

)

, where

γ

=

0.577215664905

⁢

…

is Euler’s

constant

, and this series obeys the similar

asymptotic

relation

∑

k

=

1

n

1

p

k

=

log

⁡

log

⁡

n

+

C

+

o

⁢

(

1

)

, where

C

=

0.26149721

⁢

…

and is sometimes called the

Mertens constant

. Its

divergence

, however, is extremely slow: for example, taking

n

as the biggest currently known prime, the

42

nd

Mersenne prime

M

25964951

, we get

log

⁡

log

⁡

(

2

25964951

-

1

)

+

C

≈

16.9672

(while

H

M

25964951

≈

1.79975

⋅

10

17

which is enormous considering

H

n

’s also slow divergence).

References

1

M. Aigner & G. M. Ziegler

:

Proofs from THE BOOK

, 3

rd

edition (2004), Springer-Verlag, 5–6.

2

P. Erdős

:

Über die Reihe

∑

1

p

, Mathematica, Zutphen B 7 (1938).

Title

prime harmonic series

Canonical

name

PrimeHarmonicSeries

Date of creation

2013-03-22 15:07:16

Last modified on

2013-03-22 15:07:16

Owner

Cosmin (8605)

Last modified by

Cosmin (8605)

Numerical id

17

Author

Cosmin (8605)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 11A41

Synonym

series of reciprocals of primes

Related topic

Prime

Related topic

HarmonicSeries

Related topic

HarmonicNumber

Related topic

Series

Related topic

EulersConstant

Related topic

HarmonicSeriesOfPrimes

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