Chebyshev function
![The Chebyshev function ψ(x), with x < 50](/Images/godic/202501/04/ChebyshevPsi5454.png")
![The function ψ(x) − x, for x < 10,000](/Images/godic/202501/04/Chebyshev.svg5454.png")
![The function ψ(x) − x, for x < 10 million](/Images/godic/202501/04/Chebyshev-big.svg5454.png")
![The difference of the smoothed Chebyshev function and x2/2
for x < 106](/Images/godic/202501/04/Chebyshev-smooth.svg5454.png")
In mathematics, the Chebyshev function is either of two related functions. The first Chebyshev function ϑ(x) or θ(x) is given by
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: