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 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 $\psi'(x)$. Tags $zetas$, counting, function, mellin, number-theory, prime numbers, primepower, relationship, riemannzeta, simple Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded 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

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