## E596 -- De summa seriei ex numeris primis formatae $$\frac{1}{3} - \frac{1}{5} + \frac{1}{7} + \frac{1}{11} - \frac{1}{13} - \frac{1}{17} + \frac{1}{19} + \frac{1}{23} - \frac{1}{29} + \frac{1}{31}\; \text{etc.}$$ ubi numeri primi formae $$4n-1$$ habent signum positivum, formae autem $$4n+1$$ signum negativum

(On the sum of the series of numbers of the form $$\frac{1}{3} - \frac{1}{5} + \frac{1}{7} + \frac{1}{11} - \frac{1}{13} - \frac{1}{17} + \frac{1}{19} + \frac{1}{23} - \frac{1}{29} + \frac{1}{31}\; \text{etc.}$$ in which the prime numbers of the form $$4n-1$$ have positive signs, and those of the form $$4n+1$$ have negative signs)

Summary:

First, Euler notes that the sum of the reciprocals of the primes diverges, as does the "logarithmic sum" or harmonic series. Then, his derivations start with the "Leibniz" series, $$A = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} \cdots = \frac{\pi}{4}$$.

According to the records, it was presented to the St. Petersburg Academy on October 2, 1775.

Publication:
• Originally published in Opuscula Analytica 2, 1785, pp. 240-256.
• Opera Omnia: Series 1, Volume 4, pp. 146 - 162
• Reprinted in Commentat. arithm. 2, 1849, pp. 116-126 [E596a]
Documents Available:
• Original Publication: E596
• English Translation (Jordan Bell): E596