 # Question about the analytic continuation of a Laplace transform

2 posts

 In my previous thread you said you wanted more number theory. I will deliver. I've been reading [this](https://people.math.gatech.edu/~mbaker/pdf/pnt2.pdf) proof of the prime number theorem. My question is [here](https://www.overleaf.com/read/vzjfnyyfrxgc) Some context: analytic number theory is the application of complex analysis to number theory. The part I'm working on is about estimating growth rates of functions. An example is the prime-counting function pi(x), which is the number of primes <= x. The theorem that I'm asking about is used in a step in the proof of the prime number theorem, which states that pi(x) is approximately x / ln(x). More precisely, the limit of the quotient of pi(x) and x/ln(x) is 1. To prove this fact, we use the Chebyshev function ϑ(x), where instead of adding 1 for every prime like the prime-counting function does, we instead add ln(p) for every prime. We use techniques involving integrals related to these functions to prove this is approximately x. That is: the limit ϑ(x)/x as x -> infinity is 1. ![](https://images-wixmp-ed30a86b8c4ca887773594c2.wixmp.com/f/7c3de594-ffb1-4879-8b74-26cdba19a38c/dc5z09w-0e405888-984b-47d7-a30a-e8afc29a619d.png?token=eyJ0eXAiOiJKV1QiLCJhbGciOiJIUzI1NiJ9.eyJzdWIiOiJ1cm46YXBwOjdlMGQxODg5ODIyNjQzNzNhNWYwZDQxNWVhMGQyNmUwIiwiaXNzIjoidXJuOmFwcDo3ZTBkMTg4OTgyMjY0MzczYTVmMGQ0MTVlYTBkMjZlMCIsIm9iaiI6W1t7InBhdGgiOiJcL2ZcLzdjM2RlNTk0LWZmYjEtNDg3OS04Yjc0LTI2Y2RiYTE5YTM4Y1wvZGM1ejA5dy0wZTQwNTg4OC05ODRiLTQ3ZDctYTMwYS1lOGFmYzI5YTYxOWQucG5nIn1dXSwiYXVkIjpbInVybjpzZXJ2aWNlOmZpbGUuZG93bmxvYWQiXX0.Wa_y3FovTlmCvXCMCIjObcwsNIONOVPpHlzIYOPT0X4)