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The sum of the reciprocal of the primes increasing without bound. The x axis is in log scale, showing that the divergence is very slow. The red function is a lower bound that also diverges. The sum of the reciprocals of all prime numbers diverges; that is:
The sum of the reciprocals of the primes of the form 4n + 1 is divergent. By Fermat's theorem on sums of two squares , it follows that the sum of reciprocals of numbers of the form a 2 + b 2 , {\displaystyle \ a^{2}+b^{2}\ ,} where a and b are non-negative integers, not both equal to 0 , diverges, with or without repetition.
In the limit, the sum of the reciprocals of the primes < n and the function ln(ln n) are separated by a constant, the Meissel–Mertens constant (labelled M above). The Meissel-Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as the Mertens constant, Kronecker's constant, Hadamard-de la Vallée-Poussin constant, or the prime reciprocal constant, is a ...
Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture. The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem .
Divergence of the sum of the reciprocals of the primes; V. Vantieghems theorem; Vinogradov's theorem; W. Wilson's theorem; Wolstenholme's theorem This page was ...
Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. [13]
The convergence to Brun's constant. In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B 2 (sequence A065421 in the OEIS).
For example, the sum of 1/n where n has at most one 9, is a convergent series. But the sum of 1/n where n has no 9 is convergent. Therefore, the sum of 1/n where n has exactly one 9, is also convergent. Baillie [11] showed that the sum of this last series is about 23.04428 70807 47848 31968.