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What is the relationship between simple prime-power counting function and $logzeta(s)$?
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This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $theta(y)$ is the Heaviside step function which takes a unit step at $y=0$.
(1) $quadpi(x)=sumlimits_{p}theta(x-p),qquadquadtext{(fundamental prime counting function)}$
(2) $quadPi(x)=sumlimits_{n=p^k}frac{1}{k},theta(x-n),quadtext{(Riemann's prime-power counting function)}$
(3) $quad k(x)=sumlimits_{n=p^k}theta(x-n),qquadtext{(simple prime-power counting function)}$
The following plot illustrates $pi(x)$, $Pi(x)$, and $k(x)$ in blue, orange, and green respectively. Note that $pi(x)$ takes a step of $1$ at each prime and $k(x)$ takes a step of $1$ at each prime-power. $Pi(x)$ is more complicated in that it takes a step of $frac{1}{k}$ at each prime-power $p^k$.
The $pi(x)$ and $Pi(x)$ functions defined above are related to $logzeta(s)$ as defined below.
(4) $quadlogzeta(s)=sintlimits_0^inftyPi(x),x^{-s-1},dx,,quadRe(s)>1$
(5) $quadlogzeta(s)=sintlimits_0^inftyfrac{pi(x)}{x,left(x^s-1right)},dx,,qquadquadRe(s)>1$
Question: What is the relationship between $k(x)$ and $logzeta(s)$? More specifically, what is the definition of the function $f(x)$ consistent with (6) below?
(6) $quadlogzeta(s)=sintlimits_0^infty k(x),f(x),dx,,qquadquadRe(s)>1$
The following relationships between $pi(x)$, $Pi(x)$, and $k(x)$ may provide some insight where $rad(n)$ is the greatest square-free divisor of $n$ also referred to as the square-free kernel of $n$.
(7) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{1}{n},pi(x^{1/n})$
(8) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(n)}{n},Pi(x^{1/n})$
(9) $quad k(x)=sumlimits_{n=1}^{log_2(x)}pi(x^{1/n})$
(10) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}mu(n),k(x^{1/n})$
(11) $quad k(x)=sumlimits_{n=1}^{log_2(x)}frac{phi(n)}{n},Pi(x^{1/n})$
(12) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(rad(n)),phi(rad(n))}{n},k(x^{1/n})$
number-theory prime-numbers riemann-zeta mellin-transform
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
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add a comment |
$begingroup$
This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $theta(y)$ is the Heaviside step function which takes a unit step at $y=0$.
(1) $quadpi(x)=sumlimits_{p}theta(x-p),qquadquadtext{(fundamental prime counting function)}$
(2) $quadPi(x)=sumlimits_{n=p^k}frac{1}{k},theta(x-n),quadtext{(Riemann's prime-power counting function)}$
(3) $quad k(x)=sumlimits_{n=p^k}theta(x-n),qquadtext{(simple prime-power counting function)}$
The following plot illustrates $pi(x)$, $Pi(x)$, and $k(x)$ in blue, orange, and green respectively. Note that $pi(x)$ takes a step of $1$ at each prime and $k(x)$ takes a step of $1$ at each prime-power. $Pi(x)$ is more complicated in that it takes a step of $frac{1}{k}$ at each prime-power $p^k$.
The $pi(x)$ and $Pi(x)$ functions defined above are related to $logzeta(s)$ as defined below.
(4) $quadlogzeta(s)=sintlimits_0^inftyPi(x),x^{-s-1},dx,,quadRe(s)>1$
(5) $quadlogzeta(s)=sintlimits_0^inftyfrac{pi(x)}{x,left(x^s-1right)},dx,,qquadquadRe(s)>1$
Question: What is the relationship between $k(x)$ and $logzeta(s)$? More specifically, what is the definition of the function $f(x)$ consistent with (6) below?
(6) $quadlogzeta(s)=sintlimits_0^infty k(x),f(x),dx,,qquadquadRe(s)>1$
The following relationships between $pi(x)$, $Pi(x)$, and $k(x)$ may provide some insight where $rad(n)$ is the greatest square-free divisor of $n$ also referred to as the square-free kernel of $n$.
(7) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{1}{n},pi(x^{1/n})$
(8) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(n)}{n},Pi(x^{1/n})$
(9) $quad k(x)=sumlimits_{n=1}^{log_2(x)}pi(x^{1/n})$
(10) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}mu(n),k(x^{1/n})$
(11) $quad k(x)=sumlimits_{n=1}^{log_2(x)}frac{phi(n)}{n},Pi(x^{1/n})$
(12) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(rad(n)),phi(rad(n))}{n},k(x^{1/n})$
number-theory prime-numbers riemann-zeta mellin-transform
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
$endgroup$
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– Aloizio Macedo♦
Jan 14 at 19:07
add a comment |
$begingroup$
This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $theta(y)$ is the Heaviside step function which takes a unit step at $y=0$.
(1) $quadpi(x)=sumlimits_{p}theta(x-p),qquadquadtext{(fundamental prime counting function)}$
(2) $quadPi(x)=sumlimits_{n=p^k}frac{1}{k},theta(x-n),quadtext{(Riemann's prime-power counting function)}$
(3) $quad k(x)=sumlimits_{n=p^k}theta(x-n),qquadtext{(simple prime-power counting function)}$
The following plot illustrates $pi(x)$, $Pi(x)$, and $k(x)$ in blue, orange, and green respectively. Note that $pi(x)$ takes a step of $1$ at each prime and $k(x)$ takes a step of $1$ at each prime-power. $Pi(x)$ is more complicated in that it takes a step of $frac{1}{k}$ at each prime-power $p^k$.
The $pi(x)$ and $Pi(x)$ functions defined above are related to $logzeta(s)$ as defined below.
(4) $quadlogzeta(s)=sintlimits_0^inftyPi(x),x^{-s-1},dx,,quadRe(s)>1$
(5) $quadlogzeta(s)=sintlimits_0^inftyfrac{pi(x)}{x,left(x^s-1right)},dx,,qquadquadRe(s)>1$
Question: What is the relationship between $k(x)$ and $logzeta(s)$? More specifically, what is the definition of the function $f(x)$ consistent with (6) below?
(6) $quadlogzeta(s)=sintlimits_0^infty k(x),f(x),dx,,qquadquadRe(s)>1$
The following relationships between $pi(x)$, $Pi(x)$, and $k(x)$ may provide some insight where $rad(n)$ is the greatest square-free divisor of $n$ also referred to as the square-free kernel of $n$.
(7) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{1}{n},pi(x^{1/n})$
(8) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(n)}{n},Pi(x^{1/n})$
(9) $quad k(x)=sumlimits_{n=1}^{log_2(x)}pi(x^{1/n})$
(10) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}mu(n),k(x^{1/n})$
(11) $quad k(x)=sumlimits_{n=1}^{log_2(x)}frac{phi(n)}{n},Pi(x^{1/n})$
(12) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(rad(n)),phi(rad(n))}{n},k(x^{1/n})$
number-theory prime-numbers riemann-zeta mellin-transform
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
$endgroup$
This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $theta(y)$ is the Heaviside step function which takes a unit step at $y=0$.
(1) $quadpi(x)=sumlimits_{p}theta(x-p),qquadquadtext{(fundamental prime counting function)}$
(2) $quadPi(x)=sumlimits_{n=p^k}frac{1}{k},theta(x-n),quadtext{(Riemann's prime-power counting function)}$
(3) $quad k(x)=sumlimits_{n=p^k}theta(x-n),qquadtext{(simple prime-power counting function)}$
The following plot illustrates $pi(x)$, $Pi(x)$, and $k(x)$ in blue, orange, and green respectively. Note that $pi(x)$ takes a step of $1$ at each prime and $k(x)$ takes a step of $1$ at each prime-power. $Pi(x)$ is more complicated in that it takes a step of $frac{1}{k}$ at each prime-power $p^k$.
The $pi(x)$ and $Pi(x)$ functions defined above are related to $logzeta(s)$ as defined below.
(4) $quadlogzeta(s)=sintlimits_0^inftyPi(x),x^{-s-1},dx,,quadRe(s)>1$
(5) $quadlogzeta(s)=sintlimits_0^inftyfrac{pi(x)}{x,left(x^s-1right)},dx,,qquadquadRe(s)>1$
Question: What is the relationship between $k(x)$ and $logzeta(s)$? More specifically, what is the definition of the function $f(x)$ consistent with (6) below?
(6) $quadlogzeta(s)=sintlimits_0^infty k(x),f(x),dx,,qquadquadRe(s)>1$
The following relationships between $pi(x)$, $Pi(x)$, and $k(x)$ may provide some insight where $rad(n)$ is the greatest square-free divisor of $n$ also referred to as the square-free kernel of $n$.
(7) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{1}{n},pi(x^{1/n})$
(8) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(n)}{n},Pi(x^{1/n})$
(9) $quad k(x)=sumlimits_{n=1}^{log_2(x)}pi(x^{1/n})$
(10) $quadpi(x)=sumlimits_{n=1}^{log_2(x)}mu(n),k(x^{1/n})$
(11) $quad k(x)=sumlimits_{n=1}^{log_2(x)}frac{phi(n)}{n},Pi(x^{1/n})$
(12) $quadPi(x)=sumlimits_{n=1}^{log_2(x)}frac{mu(rad(n)),phi(rad(n))}{n},k(x^{1/n})$
number-theory prime-numbers riemann-zeta mellin-transform
number-theory prime-numbers riemann-zeta mellin-transform
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
edited Jan 10 at 23:11
Steven Clark
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
asked Jan 10 at 17:19
Steven ClarkSteven Clark
7091413
7091413
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Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– Aloizio Macedo♦
Jan 14 at 19:07
add a comment |
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– Aloizio Macedo♦
Jan 14 at 19:07
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– Aloizio Macedo♦
Jan 14 at 19:07
$begingroup$
Comments are not for extended discussion; this conversation has been moved to chat.
$endgroup$
– Aloizio Macedo♦
Jan 14 at 19:07
add a comment |
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