My Math Forum What is Relationship between $\zeta(s)$ and Simple Prime-Power Counting Function?

 Number Theory Number Theory Math Forum

 November 5th, 2016, 04:41 PM #1 Newbie   Joined: Sep 2016 From: Tempe, Arizona Posts: 6 Thanks: 1 What is Relationship between $\zeta(s)$ and Simple Prime-Power Counting Function? Assume the following definitions: $\zeta(s)$ - Riemann zeta function $\zeta'(s)$ - first-order derivative of the Riemann zeta function $\vartheta(s)$ - first Chebyshev function $\vartheta'(s)$ - first-order derivative of the first Chebyshev function $\psi(s)$ - second Chebyshev function $\psi'(s)$ - first-order derivative of the second Chebyshev function $\pi(x)$ - fundamental prime counting function which takes a unit step at each prime $\pi'(x)$ - first-order derivative of the fundamental prime counting function $J(x)$ - Riemann's prime-power counting function which takes a step of $1/n$ at each prime-power $x=p^n$ $J'(x)$ - first-order derivative of Riemann's prime-power counting function $K(x)$ - simple prime-power counting function which takes a unit step at each prime-power $K'(x)$ - first-order derivative of the simple prime-power counting function Riemann defined the following relationships between $\pi(x)$ and $J(x)$: $$J(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{1}{n}\pi( x^{\frac{1}{n}})$$ $$\pi(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\mu(n)} {n}J(x^{\frac{1}{n}})$$ The relationship between $\pi(x)$ and $K(x)$ is slightly simpler (the $\frac{1}{n}$ term is removed): $$K(x)=\sum_{n=1}^{Floor(Log(2,x))}\pi(x^{\frac{1} {n}})$$ $$\pi(x)=\sum_{n=1}^{Floor(Log(2,x))}\mu(n)K(x^{\f rac{1}{n}})$$ Note the relationship between $\pi(x)$ and $K(x)$ is analogous to the relationship between $\vartheta(x)$ and $\psi(x)$: $$\psi(x)=\sum_{n=1}^{Floor(Log(2,x))}\vartheta(x^ {\frac{1}{n}})$$ $$\vartheta(x)=\sum_{n=1}^{Floor(Log(2,x))}\mu(n)\ psi(x^{\frac{1}{n}})$$ The relationship between $J(x)$ and $K(x)$ is a bit more complex: $$K(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\phi(gsfd (n))}{gsfd(n)}J(x^{\frac{1}{n}})$$ $$J(x)=\sum_{n=1}^{Floor(Log(2,x))}\frac{\mu(gsfd( n))\phi(n)gsfd(n)}{n^2}K(x^{\frac{1}{n}})$$ The $\phi$ function above is Euler's totient function and the $gsfd(n)$ function above is the greatest square-free divisor of $n$ (which is also known as the radical or square-free kernel of $n$). There are relationships between $\zeta(s)$ and $\pi(x)$, $J(x)$, $J'(x)$, $\psi(x)$, and $\psi'(x)$ such as the following: $$\log\zeta(s)=s\int_{0}^{\infty}\frac{\pi(x)}{x(x ^s-1)}dx$$ $$\log\zeta(s)=s\int_{0}^{\infty}J(x)x^{-s-1}dx$$ $$\log\zeta(s)=\int_{0}^{\infty}x^{-s}J'(x)dx$$ $$-\frac{\zeta(s)}{(\zeta(s)}=s\int_{0}^{\infty}\psi( x)x^{-s-1}dx$$ $$-\frac{\zeta'(s)}{\zeta(s)}=\int_{0}^{\infty}x^{-s}\psi'(x)dx$$ My questions are: 1) What is the relationship between $\zeta(s)$ and the simple prime-power counting function $K(x)$? 2) What is the relationship between $\zeta(s)$ and the first order derivative $K'(x)$ of the simple prime-power counting function? I'm looking for integral relationships such as those specified above for $\pi(x)$, $J(x)$, $J'(x)$, $\psi(x)$, and $\psi'(x)$.

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Rahul k Number Theory 1 May 14th, 2015 06:10 AM Jonas Castillo T Number Theory 6 May 9th, 2015 06:26 PM wyK1NG Complex Analysis 12 June 11th, 2014 12:25 AM fastandbulbous Number Theory 5 March 17th, 2014 11:23 AM capea Real Analysis 1 April 2nd, 2012 07:56 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top