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This was proved by Leonhard Euler in 1737, [1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).
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 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.
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737.. Like rational numbers, the reciprocals of primes have repeating decimal representations.
Prime number theorem; ... Divergence of the sum of the reciprocals of the primes; V. Vantieghems theorem; Vinogradov's theorem; W. Wilson's theorem; Wolstenholme's ...
The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let π 2 ( x ) {\displaystyle \pi _{2}(x)} denote the number of primes p ≤ x for which p + 2 is also prime (i.e. π 2 ( x ) {\displaystyle \pi _{2}(x)} is the number of twin primes with the smaller at most x ).
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 .
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series , which converges for ℜ ( s ) > 1 {\displaystyle \Re (s)>1} :