Recent content by riemannian

greetings , we have the following integral : I(x)=\lim_{T\rightarrow \infty}\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\sin(n\pi s)\frac{x^{s}}{s}ds n is an integer . and \gamma >1 if x>1 we can close the contour to the left . namely, consider the contour : C_{a}=C_{1}\cup C_{2}\cup...

after some manipulation , the integral reduces to : \frac{1}{2\pi i }\int_{1}^{\infty}\frac{\left(\pi i +\ln(-e^{2\pi i x}) \right )}{x(x^{s}-1)}dx

greetings . the following integral appears in some references on analytic number theory . i am really intrigued by it . and would love to understand it . \int_{1}^{\infty}\frac{\left \{x \right \}}{x}\left(\frac{1}{x^{s}-1}\right)dx \Re(s)>1 , \left \{x \right \} is the fractional ...

i think i got my answer . \Pi _{0}(x)=\sum_{n=1}^{\infty}\frac{1}{n}\pi_{0}(x^{1/n}) and \Pi_{0}(x) does change by 1 at primes . now i am intrigued by the terms \sum \rho^{k} , it seems to me we don't need to know \rho themselves to evaluate \Pi_{0}(x) , we just need to evaluate...

greetings . i have a couple of questions about the prime counting function . when \pi _{0}(x) changes by 1, then it's logical to assume that it should happen at a prime argument . meaning : \lim_{\xi\rightarrow 0}\pi _{0}(x+\xi )-\pi _{0}(x-\xi )=1 implies that x is a prime . is this a true...