On the sum of the series formed from the prime numbers where the prime numbers of the form \$4n-1\$ have a positive sign and those of the form \$4n+1\$ a negative sign
This is an English translation of the Latin original "De summa seriei ex
numeris primis formatae
\$1/3-1/5+1/7+1/11-1/13-1/17+1/19+1/23-1/29+1/31-\$ etc. ubi
numeri primi formae \$4n-1\$ habent signum positivum formae autem \$4n+1\$ signum
negativum" (1775). E596 in the Enestrom index.
Let \$\chi\$ be the nontrivial character modulo 4. Euler wants to know what
\$1-\sum\_p \chi(p)/p\$ is, either an exact expression or an approximation. He
looks for analogies to the harmonic series and the series of reciprocals of the
primes.
Another reason he is interested in this is that if this series has a finite
value (which is does, the best approximation Euler gets is 0.3349816 in section
27) then there are infinitely many primes congruent to 1 mod 4 and infinitely
many primes congruent to 3 mod 4.
In section 15 Euler gives the Euler product for the L(chi,1).
%0 Generic
%1 citeulike:3036250
%A Euler, Leonhard
%D 2007
%K Vor1800 available-in-tex-format mathematics number-theory pre1800
%T On the sum of the series formed from the prime numbers where the prime numbers of the form \$4n-1\$ have a positive sign and those of the form \$4n+1\$ a negative sign
%U http://arxiv.org/abs/0708.2564
%X This is an English translation of the Latin original "De summa seriei ex
numeris primis formatae
\$1/3-1/5+1/7+1/11-1/13-1/17+1/19+1/23-1/29+1/31-\$ etc. ubi
numeri primi formae \$4n-1\$ habent signum positivum formae autem \$4n+1\$ signum
negativum" (1775). E596 in the Enestrom index.
Let \$\chi\$ be the nontrivial character modulo 4. Euler wants to know what
\$1-\sum\_p \chi(p)/p\$ is, either an exact expression or an approximation. He
looks for analogies to the harmonic series and the series of reciprocals of the
primes.
Another reason he is interested in this is that if this series has a finite
value (which is does, the best approximation Euler gets is 0.3349816 in section
27) then there are infinitely many primes congruent to 1 mod 4 and infinitely
many primes congruent to 3 mod 4.
In section 15 Euler gives the Euler product for the L(chi,1).
@misc{citeulike:3036250,
abstract = {This is an English translation of the Latin original "De summa seriei ex
numeris primis formatae
\${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-\$ etc. ubi
numeri primi formae \$4n-1\$ habent signum positivum formae autem \$4n+1\$ signum
negativum" (1775). E596 in the Enestrom index.
Let \$\chi\$ be the nontrivial character modulo 4. Euler wants to know what
\$1-\sum\_p \chi(p)/p\$ is, either an exact expression or an approximation. He
looks for analogies to the harmonic series and the series of reciprocals of the
primes.
Another reason he is interested in this is that if this series has a finite
value (which is does, the best approximation Euler gets is 0.3349816 in section
27) then there are infinitely many primes congruent to 1 mod 4 and infinitely
many primes congruent to 3 mod 4.
In section 15 Euler gives the Euler product for the L(chi,1).},
added-at = {2009-08-02T17:14:35.000+0200},
archiveprefix = {arXiv},
author = {Euler, Leonhard},
biburl = {https://www.bibsonomy.org/bibtex/27fb16bbdbb8172b730577ba03b687020/rwst},
citeulike-article-id = {3036250},
citeulike-linkout-0 = {http://arxiv.org/abs/0708.2564},
citeulike-linkout-1 = {http://arxiv.org/pdf/0708.2564},
description = {my bookmarks from citeulike},
eprint = {0708.2564},
interhash = {616b760a1934e38d97a28835e6a1a484},
intrahash = {7fb16bbdbb8172b730577ba03b687020},
keywords = {Vor1800 available-in-tex-format mathematics number-theory pre1800},
month = Aug,
posted-at = {2008-07-23 08:36:51},
priority = {2},
timestamp = {2009-08-05T17:09:47.000+0200},
title = {On the sum of the series formed from the prime numbers where the prime numbers of the form \$4n-1\$ have a positive sign and those of the form \$4n+1\$ a negative sign},
url = {http://arxiv.org/abs/0708.2564},
year = 2007
}