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

TL;DR

This paper explores the sum of a series formed from primes with signs determined by their residue class modulo 4, discussing convergence, implications for prime distribution, and providing a modern proof of convergence.

Contribution

It provides an analysis of a prime series with signs based on residue classes, including convergence proof and connections to prime distribution theorems.

Findings

The series converges to approximately 0.335.

Infinitely many primes are congruent to 1 mod 4 and 3 mod 4.

Provides a modern proof of series convergence using Davenport's methods.

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 $\sum_p \chi(p)/p$ 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). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $\sum_p \frac{\chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshev's estimate for the second Chebyshev function (summing the von Mangoldt function).