Half-values at discontinuities in Riemann's prime counting function

Question

Wikipedia (here) and MathWorld (here) disagree about the behaviour of Riemann's prime counting function $\Pi(x)$ at points of discontinuity. The function is defined by

$$\Pi(x)=\sum _{p \text{ prime} \\ p^{\alpha }<x} \frac{1}{\alpha }$$

In Riemann's original paper, he stipulates that at points of discontinuity the value increases by half the value of the jump; i.e. with $0<\delta<1$,

$$\Pi(x)=\frac{1}{2} \left(\sum _{p^{\alpha }<x-\delta } \frac{1}{\alpha }+\sum _{p^{\alpha }<x+\delta} \frac{1}{\alpha }\right)$$

Wikipedia follows this convention, MathWorld does not (as evidenced by its list of the functions first few values).

What I want to know is, does this matter? Is it simply that the function is otherwise undefined at jump-points and therefore Riemann is following the standard convention, or do the half-values have real significance to the rest of Riemann's paper?

The key finding I am thinking of is

$$\frac{\log (\zeta (s))}{s}=\int_0^{\infty } \Pi (x) x^{-s-1} \, dx$$

(which I found here, eqn. 42). Perhaps, since It's an integral, the value at the discontinuity is irrelevant?

Answer 1

From $\frac{\log (\zeta (s))}{s}=\int_0^{\infty } \Pi (x) x^{-s-1} \, dx$ the Fourier-Laplace-Mellin inversion theorem gives that $$\Pi(x)= \frac1{2i\pi} \int_{2-i\infty}^{2+i\infty} \frac{\log \zeta(s)}{s} x^sds$$ At the points of discontinuity the RHS is the half value. Then (for $x > 1$) some kind of residue theorem expresses the contour integral as a series over the singularities of $\log \zeta(s)$ (at the pole and zeros of $\zeta(s)$, this is called the explicit formula)

Thus you define $\Pi(x)$ as you want, but when needing Mellin inversion and the explicit formula you'll choose the half-value definition.

When dealing with Dirichlet convolutions it is more convenient to choose the $\Pi(x)=\sum_{p^k\le x} 1/k$ definition. Of course this doesn't affect $\frac{\log (\zeta (s))}{s}=\int_0^{\infty } \Pi (x) x^{-s-1} \, dx$.