Divergence of the sum of the reciprocals of the primes

This article uses technical mathematical notation for logarithms. All instances of

log(x)

without a subscript base should be interpreted as a natural logarithm, also commonly written as

ln(x)

or

log e (x)

.

The sum of the reciprocals of all prime numbers diverges; that is:

\sum _{p{\text{ prime}}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty

This was proved by Leonhard Euler in 1737,^[1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).

There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that

\sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}\geq \log \log(n+1)-\log {\frac {\pi ^{2}}{6}}

for all natural numbers

n

. The double natural logarithm (

log log

) indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant.

The harmonic series

First, we describe how Euler originally discovered the result. He was considering the harmonic series

\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots =\infty

He had already used the following "product formula" to show the existence of infinitely many primes.

\sum _{n=1}^{\infty }{\frac {1}{n}}=\prod _{p}\left(1+{\frac {1}{p}}+{\frac {1}{p^{2}}}+\cdots \right)=\prod _{p}{\frac {1}{1-p^{-1}}}

Here the product is taken over the set of all primes.

Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

Proofs

Euler's proof

Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the Taylor series expansion for $log x$ as well as the sum of a converging series:

{\begin{aligned}\log \left(\sum _{n=1}^{\infty }{\frac {1}{n}}\right)&{}=\log \left(\prod _{p}{\frac {1}{1-p^{-1}}}\right)=-\sum _{p}\log \left(1-{\frac {1}{p}}\right)\\&=\sum _{p}\left({\frac {1}{p}}+{\frac {1}{2p^{2}}}+{\frac {1}{3p^{3}}}+\cdots \right)\\&=\sum _{p}{\frac {1}{p}}+{\frac {1}{2}}\sum _{p}{\frac {1}{p^{2}}}+{\frac {1}{3}}\sum _{p}{\frac {1}{p^{3}}}+{\frac {1}{4}}\sum _{p}{\frac {1}{p^{4}}}+\cdots \\&=A+{\frac {1}{2}}B+{\frac {1}{3}}C+{\frac {1}{4}}D+\cdots \\&=A+K\end{aligned}}

for a fixed constant $K < 1$ . Then he invoked the relation

\sum _{n=1}^{\infty }{\frac {1}{n}}=\log \infty ,

which he explained, for instance in a later 1748 work,^[2] by setting $x = 1$ in the Taylor series expansion

\log \left({\frac {1}{1-x}}\right)=\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}.

This allowed him to conclude that

A={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+\cdots =\log \log \infty .

It is almost certain that Euler meant that the sum of the reciprocals of the primes less than $n$ is asymptotic to $log log n$ as $n$ approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874.^[3] Thus Euler obtained a correct result by questionable means.

Zdroj:https://en.wikipedia.org?pojem=Divergence_of_the_sum_of_the_reciprocals_of_the_primes
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

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Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok.
Podrobnejšie informácie nájdete na stránke Podmienky použitia.

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