Erdos–Borwein constant
Encyclopedia
The Erdős–Borwein constant
is the sum of the reciprocals of the Mersenne number
s. It is named after Paul Erdős
and Peter Borwein
.
By definition it is:
which is approximately 1.60669 51524 15291 763… .
It can be proven that the following forms are equivalent to the former:
where σ0(n) = d(n) is the divisor function
, a multiplicative function
that equals the number of positive divisor
s of the number n. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resummed as such.
Erdős in 1948 showed that the constant
E is an irrational number
.
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
is the sum of the reciprocals of the Mersenne number
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
s. It is named after Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...
.
By definition it is:
which is approximately 1.60669 51524 15291 763… .
It can be proven that the following forms are equivalent to the former:
where σ0(n) = d(n) is the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
, a multiplicative function
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
that equals the number of positive divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
s of the number n. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resummed as such.
Erdős in 1948 showed that the constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
E is an irrational number
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
.