E596 -- De summa seriei ex numeris primis formatae 1/3 - 1/5 + 1/7 + 1/11 - 1/13 ... 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 1/3 - 1/5 + 1/7 - 1/11 + 1/13... in which the prime numbers of the form 4n-1 have positive signs, and those of the form 4n+1 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 - (1/3) + (1/5) - (1/7) + (1/9) - ... = p/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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