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:
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Originally published in Opuscula Analytica 2, 1785, pp. 240-256.
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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
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