Second Chebyshev Function

The second Chebyshev function is a function which plays an important role in the prime number theorem. It counts prime powers with a particular weighting, which is generally easier to work with than the true prime counting function.

Definition

The second Chebyshev function is defined as

ψ(x)=∑n≤xΛ(n)

where Λ is the von Mangoldt function.

While we take this as the definition of the function, there are two commonly used alternative expressions for it (sometimes taken as the definition).

Theorem
ψ(x)=∑k=1⌊log2⁡(x)⌋∑p primepk≤xln⁡(p).
Proof

Noting that pk≤x⟹k≤logp⁡(x)≤log2⁡(x), we have

∑n≤xΛ(n)=∑n≤x{ln⁡(p)n=pk0otherwise}=∑k=1∞∑pk≤xln⁡(p)=∑k=1⌊log2⁡(x)⌋∑pk≤xln⁡(p).
Theorem
ψ(x)=∑p≤xlog⁡p⌊log⁡xlog⁡p⌋
Proof
∑pn≤xlog⁡p=∑p≤x(log⁡p)⋅#{n∈Z>0:pn≤x}=∑p≤x(log⁡p)⋅#{n∈Z>0:logp⁡(x)≤n}=∑p≤x(log⁡p)⋅#{n∈Z>0:log⁡xlog⁡p≤n}=∑p≤xlog⁡p⌊log⁡xlog⁡p⌋

We can also relate the definition of this function to the first Chebyshev function.

Theorem
ψ(x)=∑n=1∞θ(x1n)

where θ is the first Chebyshev function.

Proof
∑n=1∞θ(x1n)=∑n=1∞∑p≤x1np=∑n=1∞∑pn≤xp=∑n≤x{ln⁡(p)n=pk0otherwise}=∑n≤xΛ(n).