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
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).