Equation (13), p.17 is just

which we saw derived earlier.

After proving the PNT, Ingham goes on to explain:

"The above proof of the prime number theorem is based on the fact that has no zeros on the line = 1, and it is natural to ask whether the truth of the theorem depends essentially on this property of . We shall show that this is so, by proving that the property in question can be deduced at once from the prime number theorem..."

Equation (17), p. 18 is just

which follows from Abel's identity.