(pp. 189-216)

In his paper Riemann considered another weighted prime counting function, which we will write as$\Pi (x)$, related to the harmonic series and defined by

$\Pi (x)\; = \;\sum\limits_{\scriptstyle {p^r} < x, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $,

which again reveals a bit more about itself if we look at a couple of examples:

$\Pi (20) = \sum\limits_{\scriptstyle {p^r} < 20, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $

$ = \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right) + \;\left( {\frac{1}{1}\; + \;\frac{1}{2}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)$,

where the bracketing is by the primes 2, 3, 5, . . . , 19, and

$\Pi (30) = \sum\limits_{\scriptstyle {p^r} < \,30, \atop \scriptstyle p\;{\rm{prime}}} {\frac{1}{r}} $

$ = \left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}} \right)\; + \;\left( {\frac{1}{1} + \frac{1}{2} + \frac{1}{3}} \right)\; + \;\left( {\frac{1}{1}\; + \;\frac{1}{2}} \right) + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)\; + \;\left( {\frac{1}{1}} \right)$,

where the bracketing is by the primes 2, 3, 5, . . . , 29.

These can be rewritten as

$\Pi (20) = \left( {\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}} \right) + \;\frac{1}{2}\left( {\frac{1}{1}\; + \;\frac{1}{1}} \right)\; + \;\frac{1}{3}\,\left( {\frac{1}{1}} \right)\; + \;\frac{1}{4}\;\left( {\frac{1}{1}} \right)$

and

$\Pi (30) = \left( {\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}\; + \;\frac{1}{1}} \right) + \;\frac{1}{2}\left( {\frac{1}{1} + \frac{1}{1} + \frac{1}{1}} \right) + \frac{1}{3}\left( {\frac{1}{1} + \frac{1}{1}} \right) + \frac{1}{4}\left( {\frac{1}{1}} \right)$.

The first bracket just counts the primes less than the number, the second those less than...