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Harmonic Series of Primes

Like the entire harmonic series, the harmonic series

sum_(k=1)^infty1/(p_k)=infty
(1)

taken over all primes p_k also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.

Its asymptotic behavior is given by

sum_(p prime)^x1/p=lnlnx+B_1+o(1),
(2)

where

B_1=0.2614972128...
(3)

(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).

See also

Harmonic Series, Prime Sums

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References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The Sequence of Primes." §1.2 and 1.4 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 1-4, 17, 22, and 251, 1979.Havil, J. "Harmonic Series of Primes." §3.2 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 28-31, 2003.Nagell, T. Introduction to Number Theory. New York: Wiley, 1951.Sloane, N. J. A. Sequences A046024 and A077761 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 41, 1986.

Referenced on Wolfram|Alpha

Harmonic Series of Primes

Cite this as:

Weisstein, Eric W. "Harmonic Series of Primes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmonicSeriesofPrimes.html

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