prime harmonic series
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square-free part \PMlinkescapephraseset of primes \PMlinkescapephraseharmonic series
The prime11ℙ denotes the set of
primes. harmonic series (also known as series of reciprocals of primes) is the infinite sum ∑p∈ℙ1p. 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∈P1p diverges.
Proof.
Assume that this series is convergent. If so, then, for a certain k∈ℕ, we have:
∑i>k1pi<12, |
where pi is the ith prime. Now, we define λk(n):=#{m≤n:pi∣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=pω11⋯pωkk⋅s2,ωi∈{0,1} (i.e. a square multiplied by a square-free number). There are 2k ways to chose the square-free part and clearly s≤√n, so λk(n)≤2k√n. Now, the number of integers divisible by pi less than n is ⌊npi⌋, so the number of integers less than n divisible by primes bigger than pk (which we shall denote by Λk(n)) is bounded above as follows:
Λk(n)≤∑i>k⌊npi⌋≤∑i>knpi<n2. |
However, by their definitions, λk(n)+Λk(n)=n for all n∈ℕ and so it is sufficient to find an n such that λk(n)≤n2 for a contradiction and, using the previous bound for λk(n), which is λk(n)≤2k√n, we see that n=22k+2 works.
∎
The series is in some ways similar to the Harmonic series (http://planetmath.org/HarmonicSeries) Hn:=∑nk=11n. In fact, it is well known that Hn=logn+γ+o(1), where γ=0.577215664905… is Euler’s constant, and this series obeys the similar asymptotic relation ∑nk=11pk=loglogn+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 42nd Mersenne prime
M25964951, we get loglog(225964951-1)+C≈16.9672 (while HM25964951≈1.79975⋅1017 which is enormous considering Hn’s also slow divergence).
References
- 1 M. Aigner & G. M. Ziegler: Proofs from THE BOOK, 3rd edition (2004), Springer-Verlag, 5–6.
- 2 P. Erdős: Über die Reihe ∑1p, 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 |