
Psi function can refer to
the Dedekind psi function ψ(n)
the Chebyshev function ψ(x)
the polygamma function ψ^m(z) or its special cases
the digamma function ψ(z)
the trigamma function ψ^1(z)
In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by
ψ(n)=n∏_pn(1+1/p),
where the product is taken over all primes p dividing n (by convention, ψ(1) is the empty product and so has value 1). The function was introduced by Richard Dedekind in connection with modular functions.
The value of ψ(n) for the first few integers n is:
1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24 ... .
ψ(n) is greater than n for all n greater than 1, and is even for all n greater than 2. If n is a squarefree number then ψ(n) = σ(n).
The ψ function can also be defined by setting ψ(pn) = (p+1)pn1 for powers of any prime p, and then extending the definition to all integers by multiplicitivity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is
∑ψ(n)/n^s=ζ(s)ζ(s1)/ζ(2s).
This is also a consequence of the fact that we can write as a Dirichlet convolution of ψ=n*ε_2 where ε_2 is the characteristic function of the squares.
The generalization to higher orders via ratios of Jordan's totient is
ψ_k(n)=J_2k(n)/J_k(n)
with Dirichlet series
∑_n\ge1ψ_k(n)/n^s=ζ(s)ζ(sk)/ζ(2s).
It is also the Dirichlet convolution of a power and the square
of the Mobius function,
ψ_k(n)=n^k*μ^2(n).
If
ε_2=1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0…
is the characteristic function of the squares, another Dirichlet convolution
leads to the generalized σfunction,
ε_2(n)*ψ_k(n)=σ_k(n).
In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by
θ(x)=∑_p\lex\logp
with the sum extending over all prime numbers p that are less than or equal to x.
The second Chebyshev function ψ(x) is defined similarly, with the sum extending over all prime powers not exceeding x:
ψ(x)=∑_p^k\lex\logp=∑_n\leqxΛ(n)=∑_p\lex\lfloor\log_px\rfloor\logp,
where Λ is the von Mangoldt function. The Chebyshev function is often used in proofs related to prime numbers, because it is typically simpler to work with than the primecounting function, π(x). Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem.
Both functions are named in honour of Pafnuty Chebyshev.
The second Chebyshev function can be seen to be related to the first by writing it as
ψ(x)=∑_p\lexk\logp
where k is the unique integer such that pk ≤ x but pk+1 > x. A more direct relationship is given by
ψ(x)=∑_n=1^∞θ(x^1/n).
Note that this last sum has only a finite number of nonvanishing terms, as
θ(x^1/n)=0forn>\log_2x.
The second Chebyshev function is the logarithm of the least common multiple of the integers from 1 to n.
lcm(1,2,˙s,n)=e^ψ(n).
Values of lcm(1,2,˙s,n) for the integer variable n is given at .