Function Repository Resource:

ChebyshevTheta

Calculate the value of the first Chebyshev function θ(x)

Contributed by: Jack Heimrath

ResourceFunction["ChebyshevTheta"][x]

returns the value of the first Chebyshev function at x.

Details and Options

The first Chebyshev function θ(x) is equal to the sum of the logarithms of prime numbers less than or equal to x. For x<2, the sum is empty and we set θ(x)=0 by convention.

The Prime Number Theorem is equivalent to the statement that

as x→∞.

θ(x)=ψ(x)+O(1), where ψ is the second Chebyshev function and O(1) is O.

Examples

Basic Examples (5)

Compute θ(30):

In[1]:=

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Obtain a decimal approximation:

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Plot θ(x) from 0 to 30:

In[3]:=

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Compare x, θ(x) and |x-θ(x)|:

In[4]:=

$Plot[{x, ResourceFunction["ChebyshevTheta"][x], Abs[x - ResourceFunction["ChebyshevTheta"][x]]}, {x, 0, 30}, PlotLegends -> {"\!$\* StyleBox[\"x\", \"TI\"]$", "\[Theta](\!$\* StyleBox[\"x\", \"TI\"]$)", "|\!$\* StyleBox[\"x\", \"TI\"]$-\[Theta](\!$\* StyleBox[\"x\", \"TI\"]$)|"}]$