Euler's theorem (prime numbers)

One of Leonhard Euler's numerous theorems in the mathematical branch of analysis is Euler's theorem on the summation of the reciprocal values of prime numbers . This means that the series formed from these reciprocal values diverges . The proof of this theorem rests essentially on the fundamental theorem of arithmetic and the divergence of the harmonic series .

Formulation of the sentence

The following applies to the sequence of all prime numbers: ${\displaystyle \left(p_{k}\right)_{k\in \mathbb {N} }=\left(2,3,5,7,11,\dotsc \right)}$

${\displaystyle \sum _{k=1}^{\infty }{\tfrac {1}{p_{k}}}={\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+\dotsb =\infty }$

proof

A possible proof using only elementary results of analysis is the following:

It applies to Euler's number

${\displaystyle e=\sup \left\{\left(1+{\tfrac {1}{n}}\right)^{n}\mid n=1,2,3,\dotsc \right\},}$

so that for every prime number the inequality ${\displaystyle p}$

${\displaystyle 1+{\tfrac {1}{p-1}}<e^{\tfrac {1}{p-1}}}$

consists. Hence, using fractions and the natural logarithm , we get :

${\displaystyle \ln \left({\frac {1}{1-{\tfrac {1}{p}}}}\right)=\ln \left(1+{\tfrac {1}{p-1}}\right)<{\tfrac {1}{p-1}}={\tfrac {1}{p}}+{\tfrac {1}{p(p-1)}}\leq 2\cdot {\tfrac {1}{p}}}$

Now let be any natural number and let it be the finite sequence of all prime numbers up to the number . ${\displaystyle N}$ ${\displaystyle {\bigl (}p_{k}{\bigr )}_{k=1,2,\dotsc ,k(N)}}$ ${\displaystyle N}$

The following then applies:

${\displaystyle \ln \left(\prod _{k=1}^{k(N)}{\frac {1}{1-{\tfrac {1}{p_{k}}}}}\right)=\sum _{k=1}^{k(N)}\ln \left({\frac {1}{1-{\tfrac {1}{p_{k}}}}}\right)<2\cdot \sum _{k=1}^{k(N)}{\tfrac {1}{p_{k}}}}$

In connection with the known limit value

${\displaystyle \lim _{x\to \infty }\ln x=\infty }$

It is therefore sufficient to prove the asserted divergence to show that

${\displaystyle \prod _{k=1}^{k(N)}{\frac {1}{1-{\tfrac {1}{p_{k}}}}}}$

with growing also grows across all borders. ${\displaystyle N}$

The latter results from first including the properties of the geometric series and thereby the identity

${\displaystyle \prod _{k=1}^{k(N)}{\frac {1}{1-{\tfrac {1}{p_{k}}}}}=\prod _{k=1}^{k(N)}{\left(1+{\tfrac {1}{p_{k}}}+{\tfrac {1}{{p_{k}}^{2}}}+{\tfrac {1}{{p_{k}}^{3}}}+\dotsb \right)}}$

derives. Since absolutely convergent series always occur in the last product within the brackets on the right-hand side , it is possible to multiply them taking into account the distributive law .

So, because of the fundamental theorem of arithmetic, multiplying the brackets gives the reciprocal of every natural number of the form

${\displaystyle \prod _{k=1}^{k(N)}{p_{k}}^{{\alpha }_{k}}}$

with exactly once. ${\displaystyle {\alpha }_{k}\in \{0,1,2,3,\dotsc \}\,\left(k=1,2,\dotsc ,k(N)\right)}$

This gives you the following identity :

${\displaystyle \prod _{k=1}^{k(N)}{\frac {1}{1-{\tfrac {1}{p_{k}}}}}=\sum _{n=1,2,3,4,\dotsc \atop {\text{Kein Primteiler von n ist größer als N.}}}{\tfrac {1}{n}}}$

However, this implies the inequality

${\displaystyle \prod _{k=1}^{k(N)}{\frac {1}{1-{\tfrac {1}{p_{k}}}}}>\sum _{n=1}^{N}{\tfrac {1}{n}}}$

and since the harmonic series diverges, the evidence is given.

history

From his considerations on the Euler product of the Riemann zeta function , Euler was able to show his theorem in 1737. He also had an idea of ​​the magnitude of the partial sums:

“The sum of the reciprocal series of prime numbers is infinitely large, but infinitely times smaller than the sum of the harmonic series . And the sum of those is practically the logarithm of this sum. " ${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+\mathrm {etc} }$ ${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\mathrm {etc} }$

- Leonhard Euler

For him, his solution was an indicator that the prime numbers must be much closer than the square numbers, since in solving the Basel problem he had proven that the infinite sum of the reciprocal values ​​of all square numbers tends towards a finite limit. However, this argument is only of a heuristic nature - to this day it is not even known whether there is always a prime number between two neighboring square numbers (this question is also known as the Legendre's conjecture ).

Notes and additions

• The sentence goes back to the year 1737.

• It follows immediately from the theorem that there are infinitely many prime numbers.

• With a little more analysis it can even be shown more clearly that for real numbers the inequality ${\displaystyle x\geq 2}$

${\displaystyle \sum _{{\text{p ist Primzahl}} \atop {\text{und nicht größer als x.}}}{\tfrac {1}{p}}\;\;>\;\;\ln(\ln(x))-{\tfrac {1}{2}}}$

consists.

• With deeper methods of analytical number theory it can also be shown that the limit value

${\displaystyle \lim _{x\to \infty }{\bigl (}\;\sum _{{\text{p ist Primzahl}} \atop {\text{und nicht größer als x.}}}{\tfrac {1}{p}}\;-\;\ln(\ln(x))\;{\bigr )}}$

exists, equal to the Meissel-Mertens constant

${\displaystyle M=0{,}2614972128\dotso }$(Follow A077761 in OEIS )

is and that for real numbers there is always the inequality ${\displaystyle x>1}$

${\displaystyle \sum _{{\text{p ist Primzahl}} \atop {\text{und nicht größer als x.}}}{\tfrac {1}{p}}\;\;>\;\;\ln(\ln(x))+M-{\tfrac {1}{2\;{\ln }^{2}(x)}}}$

consists.

• Coupled with the fact that the series of reciprocal values ​​of the square numbers converges and the limit

${\displaystyle \zeta (2)=\sum _{n=1}^{\infty }{\tfrac {1}{n^{2}}}={\tfrac {\pi ^{2}}{6}}}$

has, it follows from the sentence that “there are more prime numbers than square numbers in a well-defined sense.” “Nevertheless, it is an open and apparently very difficult problem whether there is always a prime number between two consecutive square numbers.”

• Potentiates one in the above prime reciprocal number of all prime numbers with an exponent we always wins instead of a divergent a convergent series with finite limit: ${\displaystyle \sigma >1,}$

${\displaystyle \sum _{k=1}^{\infty }{\tfrac {1}{{p_{k}}^{\sigma }}}={\tfrac {1}{2^{\sigma }}}+{\tfrac {1}{3^{\sigma }}}+{\tfrac {1}{5^{\sigma }}}+{\tfrac {1}{7^{\sigma }}}+\dotsb <\infty }$

Leonhard Euler systematically calculated these series values ​​in the Introductio for even whole numbers and specified them with an accuracy of 15 decimal places. So he calls u. a. the following approximations : ${\displaystyle \sigma =2,4,6,8,\dotsc ,36}$

${\displaystyle {\tfrac {1}{2^{2}}}+{\tfrac {1}{3^{2}}}+{\tfrac {1}{5^{2}}}+{\tfrac {1}{7^{2}}}+\dotsb \approx 0{,}452247420041222}$(Follow A085548 in OEIS )

${\displaystyle {\tfrac {1}{2^{4}}}+{\tfrac {1}{3^{4}}}+{\tfrac {1}{5^{4}}}+{\tfrac {1}{7^{4}}}+\dotsb \approx 0{,}076993139764252}$(Follow A085964 in OEIS )

${\displaystyle {\tfrac {1}{2^{6}}}+{\tfrac {1}{3^{6}}}+{\tfrac {1}{5^{6}}}+{\tfrac {1}{7^{6}}}+\dotsb \approx 0{,}017070086850639}$(Follow A085966 in OEIS )

${\displaystyle {\tfrac {1}{2^{8}}}+{\tfrac {1}{3^{8}}}+{\tfrac {1}{5^{8}}}+{\tfrac {1}{7^{8}}}+\dotsb \approx 0{,}004061405366515}$(Follow A085968 in OEIS )

${\displaystyle \vdots }$

${\displaystyle {\tfrac {1}{2^{36}}}+{\tfrac {1}{3^{36}}}+{\tfrac {1}{5^{36}}}+{\tfrac {1}{7^{36}}}+\dotsb \approx 0{,}000000000014551}$

• In the same way, according to the Leibniz criterion for alternating series, it is certain that the series of reciprocal prime numbers weighted with alternating signs also always converges. Here is:

${\displaystyle \sum _{k=1}^{\infty }{\tfrac {(-1)^{k}}{p_{k}}}=-{\tfrac {1}{2}}+{\tfrac {1}{3}}-{\tfrac {1}{5}}+{\tfrac {1}{7}}\pm \dotsb =-0{,}2696063519\dotso }$(Follow A078437 in OEIS )

• On the other hand there is the problem posed by Paul Erdős and - as far as is known today - as yet unsolved problem, whether the alternating series

${\displaystyle \sum _{k=1}^{\infty }{\tfrac {(-1)^{k}\cdot k}{p_{k}}}=-{\tfrac {1}{2}}+{\tfrac {2}{3}}-{\tfrac {3}{5}}+{\tfrac {4}{7}}\pm \dotsb }$

converges or diverges. However, according to Erdős, it is known that the related series

${\displaystyle \sum _{k=2}^{\infty }{\tfrac {(-1)^{k}\cdot k\cdot \ln(k)}{p_{k}}}={\tfrac {2\cdot \ln(2)}{3}}-{\tfrac {3\cdot \ln(3)}{5}}+{\tfrac {4\cdot \ln(4)}{7}}-{\tfrac {5\cdot \ln(5)}{11}}\pm \cdots }$

is divergent.

literature

• Leonhard Euler: Introduction to the Analysis of the Infinite . First part of the Introductio in Analysin Infinitorum . Springer Verlag, Berlin / Heidelberg / New York 1983, ISBN 3-540-12218-4 ( MR0715928. - Reprint of the Berlin 1885 edition).
• Steven R. Finch: Mathematical Constants (=  Encyclopedia of Mathematics and its Applications . Volume 94 ). Cambridge University Press, Cambridge (et al.) 2003, ISBN 0-521-81805-2 ( MR2003519. ).
• Richard K. Guy : Unsolved Problems in Number Theory (=  Problem Books in Mathematics ). 3. Edition. Springer-Verlag, New York 2004, ISBN 0-387-20860-7 ( MR2076335. ).
• Friedrich Ischebeck: Invitation to number theory . BI-Wissenschaftsverlag, Berlin ( inter alia) 1992, ISBN 3-411-15451-9 ( MR1182988. ).
• Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences . Volume 2 ). 5th, corrected edition. Springer Verlag, Berlin ( inter alia) 1964, ISBN 3-540-03138-3 ( MR0183997. ).
• J. Barkley Rosser , Lowell Schoenfeld: Approximate formulas for some functions of prime numbers . In: Illinois J. Math . tape 6 , 1962, pp. 64-94 ( projecteuclid.org ). MR0137689.
• József Sándor , Dragoslav S. Mitrinović, Borislav Crstici: Handbook of Number Theory. I . Springer Verlag, Dordrecht 2006, ISBN 1-4020-4215-9 ( MR2119686. ).
• Alexander Schmidt : Introduction to algebraic number theory . Springer Verlag, Berlin (among others) 2007, ISBN 978-3-540-45973-6 .

References and comments

1. ↑ Euler: Introduction ... (§ 273) . S. 226-227 .
2. ^ Schmidt: Introduction ... p. 5-6 .
3. ↑ Ischebeck: invitation ... S. 38-39 .
4. ↑ Knopp: Theory ... (§§ 17, 58) . S. 146-147, 461 .
5. ^ Leonhard Euler : Variae observationes circa series infinitas. April 25, 1737, Commentarii academiae scientiarum imperialis Petropolitanae 9, 1744, pp. 160–188 (Latin; Euler product as “Theorema 19” on p. 187 f. ). German translation (PDF) by Alexander Aycock.
6. ↑ Knopp: p. 461.
7. ↑ Ischebeck: pp. 38–39.
8. ^ Rosser, Schoenfeld: Approximate formulas for some functions of prime numbers . In: Illinois. J. Math. Band 6 , p. 64 ff .
9. ↑ Sándor Mitrinović-Crstici: Handbook ... S. 257-258 .
10. ^ Schmidt: p. 6.
11. ↑ Ischebeck: p. 40.
12. ↑ Because the associated Zeta series is a convergent majorante . ${\displaystyle \zeta (\sigma )}$
13. ↑ Euler: Introduction ... (§ 282) . S. 237 .
14. ↑ Finch: Mathematical Constants (Section 2.2) . S. 94 ff., 96 .
15. ↑ Finch: p. 96.
16. ↑ Guy: Unsolved Problems ... (Section E7) . S. 316 .