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)


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.

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