I am working through a problem set in an analytic number theory course, and the following problem was included:
Let $I(y,T) = \int_{c-iT}^{c+iT} \frac{y^s}{s} ds.$
(1) If $y = 1$, $c>0$ and $T>0$, that $|I(1,T)-\frac{1}{2}| \le \frac{c}{T}.$
(2) If $y>1$, $c>0$ and $T>0$, that $|I(y,T)-1| \le y^c \min (1, \frac{1}{T|\log y|}).$ Also prove for 0
(3) Comparing (1) and (2), where does the discontinuity come from when first $T \to \infty$, and then $y$ is moved? Change variables $y = e^u$ and interpret this result in terms of a Fourier transform. Explain the role of the requirement $c>0$ (versus, say, c < 0). Also, what would happen if for finite $T$ fixed we let $y \to \infty$ (resp., let $y \to 0$)? Compare the three answers
WARNING: The professor writing these problems has an unfortunate habit of TeX-ing problems up incorrectly! Hence part of the "fun" for students taking the course is to figure out if the statement of the problem itself is correct, and if not, to figure out how to modify the statement to make it workable.
I am wondering if anyone visiting would be able to tell whether the problem as stated is right (and if so, suggest a strategy for proving it); if the statement is false, I am curious to know if anyone could either suggest how to modify the statement to be workable, or even point me in the direction of a text that contains a correct statement.