Function Repository Resource:

ChebyshevPsi

Calculate the value of the second Chebyshev function ψ(x)

Contributed by: Jack Heimrath

ResourceFunction["ChebyshevPsi"][x]

returns the value of the second Chebyshev function at x.

Details and Options

The second Chebysev function ψ(x) is calculated by summing the MangoldtLambda function from 2 to the greatest integer less than or equal to x. If x<2, the defining sum is empty and we set ψ(x)=0 by convention.

The Prime Number Theorem is equivalent to the statement that

as x→∞.

Examples

Basic Examples (5)

Compute ψ(30):

In[1]:=

Out[1]=

Obtain a decimal approximation:

In[2]:=

Out[2]=

Plot ψ(x) from 0 to 30:

In[3]:=

Out[3]=

Compare x, ψ(x), and |x-ψ(x)|:

In[4]:=

$Plot[{x, ResourceFunction["ChebyshevPsi"][x], Abs[x - ResourceFunction["ChebyshevPsi"][x]]}, {x, 0, 30}, PlotLegends -> {"x", "\[Psi](x)", "|x-\[Psi](x)|"}]$