Definition
The von Mangoldt function, conventionally written as Λ(n), is defined as
which is a sequence starting:
oeis sequence A014963
It is an example of an important arithmetic function that is neither multiplicative nor additive.
The von Mangoldt function satisfies the identity
that is, the sum is taken over all integers d which divide n. This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to 0.
- For instance, let n=12. Recall the prime factorization of 12, 12=22·3, which will turn up in the example.
- Take the summation over all distinct positive divisors d of n:
-
-
- This provides an example of how the summation of the von Mangoldt function equals log (n).
The summatory von Mangoldt function, ψ(x), also known as the Chebyshev function, is defined as
von Mangoldt provided a rigorous proof of an explicit formula for ψ(x) involving a sum over the non-trivial zeros of the Riemann zeta function. This was an important part of the first proof of the prime number theorem.
Read more about this topic: Von Mangoldt Function
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)