Understanding and Expanding the 2nd Chebyshev Function: Calculation Help

[camilus]

I'm a bit confused as to how to calculate the 2nd chebyshev function.

I know \psi (x) = \sum_{p^k \le x} \ln p

but can someone show me how to expand it? Like, do I use for 2^k, 2, 4, 8, 16, 32... for all the p^k \le x, same with 3, and 4 and so on?

MW gives the example:

\psi (10) = \ln (2520) = 3\ln2+2\ln3+\ln5+\ln7

how would say, \psi (30) be written?

 

[g_edgar]

\spi(30)
Let's see...
2,4,8,16 < 30
3,9,27 < 30
5,25 < 30
7 < 30
11 < 30
13 < 30
17 < 30
19 < 30
23 < 30
29 < 30
so ... \psi(30) = 4\log 2 + 3\log 3 + 2\log 5 + \log 7 + \log 11 + \log 13 + \log 17 + \log 19 + \log 23 + \log 29