Prime Numbers and the Riemann Hypothesis

This book is heavy on numerical analysis and intuition. I especially like its use of Fourier analysis. I would like to know more about the following question.

How the first function in Chapter 30 in page 99 is constructed? From this function, the spectrum of prime numbers is calculated. Why the spectrum calculated from this function is called the spectrum of the prime numbers. We can justify it by using the spectrum to construct the distribution of prime numbers, as in Chapter 35. However, I would like to learn more direct argument. Chapter 19 and Figure 19.1 in page 63 provide some intuition.  But I would like to learn some more formal argument.

I shall read more about the derivative of zeta function in page 132, Chapter 37. It should contain more information about the construction of the function in page 99.

Overall, I shall learn more about the spectrum problems.