Divergence of the sum of the reciprocals of the primes information
Theorem
This article uses technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, commonly notated as ln(x) or loge(x).
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:
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).
There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that
for all natural numbers n. The double natural logarithm (log log) indicates that the divergence might be very slow, which is indeed the case. See Meissel–Mertens constant.
^Euler, Leonhard (1737). "Variae observationes circa series infinitas" [Various observations concerning infinite series]. Commentarii Academiae Scientiarum Petropolitanae. 9: 160–188.
and 23 Related for: Divergence of the sum of the reciprocals of the primes information
=\ln(2)} Thereciprocalsofprime numbers produce a divergent series (so the set ofprimes is "large"; see divergenceofthesumofthereciprocalsofthe primes):...
expanded out gives a sum consisting of those terms n−s where n is a product ofprimes less than or equal to q. The inequality results from the fact that therefore...
mathematics, the Fibonacci sequence is a sequence in which each number is thesumofthe two preceding ones. Numbers that are part ofthe Fibonacci sequence...
include: The constant factor rule ( a f ) ′ = a f ′ {\displaystyle (af)'=af'} Thesum rule ( f + g ) ′ = f ′ + g ′ {\displaystyle (f+g)'=f'+g'} The difference...
anything ofthe form 1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} for any finite number of terms. Such numbers are infinite, and their reciprocals are infinitesimals...
Gaussian rational Gauss sum, an exponential sum over Dirichlet characters Elliptic Gauss sum, an analog of a Gauss sum Quadratic Gauss sum Gaussian quadrature...
characteristic ofthe right Riemann sum employed to prevent the integral from degenerating into the harmonic series, thereby averting divergence. Here, − 1...
/\varepsilon _{0}} where ρ is the total charge density and ∇ ⋅ {\textstyle \mathbf {\nabla } \cdot } denotes thedivergence. The concept of electric potential is...
{\displaystyle Q} is the product ofthe 1-dimensional marginal distributions. In the notation ofthe Kullback–Leibler divergence section of this article, Σ...
(differential) entropy or the other way around. In the numerical example above, the Kullback divergence measures the inefficiency of assuming that the distribution...
Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources...
quasi-probability distribution Wigner semicircle distribution Wike's law of low odd primes Wilcoxon signed-rank test Will Rogers phenomenon WinBUGS – software...
case ofthe four-parameter Beta distribution. The Irwin–Hall distribution is the distribution ofthesumof n independent random variables, each of which...
Global Information
