Questions related to Moebius Transform of Characteristic Function of the Primes
$begingroup$
Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$. Note that A143519(n) is not multiplicative.
(1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$
https://oeis.org/A143519
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below.
(2) $quad${1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109}
The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below.
(3) $quad${14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682}
Question (1): Does $f(x)$ have a finite number of integer zeros, and if so what is the largest integer zero of $f(x)$?
Question (2): Does $f(x)$ have an finite number of zero crossings, and if so what is the largest zero crossing of $f(x)$?
Question (3): What is the asymptotic for the long term growth of $f(x)$? What are the associated error bounds predicted by the Prime Number Theorem and the Riemann Hypothesis?
The Dirichlet transform of $A143519(n)$ is $frac{P(s)}{zeta(s)}$ where $P(s)$ is the prime zeta function.
(4) $quadfrac{P(s)}{zeta(s)}=sumlimits_{n=1}^inftyfrac{A143519(n)}{n^s},quadRe(s)>1?$
The following figure illustrates the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above in orange where formula (4) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$.
Figure (2): Illustration of formula (4) for $frac{P(s)}{zeta(s)}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (4) for $frac{P(s)}{zeta(s)}$ evaluated along the line $Re(s)=1$ in orange where formula (4) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$. The red discrete portions of the plots illustrate the evaluation of formula (4) for $frac{P(1+i,t)}{zeta(1+i,t)}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (4) for $left|frac{P(1+i,t)}{zeta(1+i,t)}right|$
Figure (4): Illustration of formula (4) for $Releft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (5): Illustration of formula (4) for $Imleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (6): Illustration of formula (4) for $Argleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Question (4): What is the range of convergence of the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Note $frac{P(s)}{zeta(s)}$ has a pole at each non-trivial zeta zero.
Question (5): Are there explicit formulas for $f(x)$ and $frac{P(s)}{zeta(s)}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series riemann-hypothesis
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
$endgroup$
add a comment |
$begingroup$
Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$. Note that A143519(n) is not multiplicative.
(1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$
https://oeis.org/A143519
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below.
(2) $quad${1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109}
The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below.
(3) $quad${14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682}
Question (1): Does $f(x)$ have a finite number of integer zeros, and if so what is the largest integer zero of $f(x)$?
Question (2): Does $f(x)$ have an finite number of zero crossings, and if so what is the largest zero crossing of $f(x)$?
Question (3): What is the asymptotic for the long term growth of $f(x)$? What are the associated error bounds predicted by the Prime Number Theorem and the Riemann Hypothesis?
The Dirichlet transform of $A143519(n)$ is $frac{P(s)}{zeta(s)}$ where $P(s)$ is the prime zeta function.
(4) $quadfrac{P(s)}{zeta(s)}=sumlimits_{n=1}^inftyfrac{A143519(n)}{n^s},quadRe(s)>1?$
The following figure illustrates the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above in orange where formula (4) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$.
Figure (2): Illustration of formula (4) for $frac{P(s)}{zeta(s)}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (4) for $frac{P(s)}{zeta(s)}$ evaluated along the line $Re(s)=1$ in orange where formula (4) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$. The red discrete portions of the plots illustrate the evaluation of formula (4) for $frac{P(1+i,t)}{zeta(1+i,t)}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (4) for $left|frac{P(1+i,t)}{zeta(1+i,t)}right|$
Figure (4): Illustration of formula (4) for $Releft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (5): Illustration of formula (4) for $Imleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (6): Illustration of formula (4) for $Argleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Question (4): What is the range of convergence of the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Note $frac{P(s)}{zeta(s)}$ has a pole at each non-trivial zeta zero.
Question (5): Are there explicit formulas for $f(x)$ and $frac{P(s)}{zeta(s)}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series riemann-hypothesis
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
$endgroup$
add a comment |
$begingroup$
Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$. Note that A143519(n) is not multiplicative.
(1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$
https://oeis.org/A143519
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below.
(2) $quad${1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109}
The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below.
(3) $quad${14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682}
Question (1): Does $f(x)$ have a finite number of integer zeros, and if so what is the largest integer zero of $f(x)$?
Question (2): Does $f(x)$ have an finite number of zero crossings, and if so what is the largest zero crossing of $f(x)$?
Question (3): What is the asymptotic for the long term growth of $f(x)$? What are the associated error bounds predicted by the Prime Number Theorem and the Riemann Hypothesis?
The Dirichlet transform of $A143519(n)$ is $frac{P(s)}{zeta(s)}$ where $P(s)$ is the prime zeta function.
(4) $quadfrac{P(s)}{zeta(s)}=sumlimits_{n=1}^inftyfrac{A143519(n)}{n^s},quadRe(s)>1?$
The following figure illustrates the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above in orange where formula (4) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$.
Figure (2): Illustration of formula (4) for $frac{P(s)}{zeta(s)}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (4) for $frac{P(s)}{zeta(s)}$ evaluated along the line $Re(s)=1$ in orange where formula (4) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$. The red discrete portions of the plots illustrate the evaluation of formula (4) for $frac{P(1+i,t)}{zeta(1+i,t)}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (4) for $left|frac{P(1+i,t)}{zeta(1+i,t)}right|$
Figure (4): Illustration of formula (4) for $Releft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (5): Illustration of formula (4) for $Imleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (6): Illustration of formula (4) for $Argleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Question (4): What is the range of convergence of the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Note $frac{P(s)}{zeta(s)}$ has a pole at each non-trivial zeta zero.
Question (5): Are there explicit formulas for $f(x)$ and $frac{P(s)}{zeta(s)}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series riemann-hypothesis
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
$endgroup$
Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$. Note that A143519(n) is not multiplicative.
(1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$
https://oeis.org/A143519
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below.
(2) $quad${1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109}
The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below.
(3) $quad${14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682}
Question (1): Does $f(x)$ have a finite number of integer zeros, and if so what is the largest integer zero of $f(x)$?
Question (2): Does $f(x)$ have an finite number of zero crossings, and if so what is the largest zero crossing of $f(x)$?
Question (3): What is the asymptotic for the long term growth of $f(x)$? What are the associated error bounds predicted by the Prime Number Theorem and the Riemann Hypothesis?
The Dirichlet transform of $A143519(n)$ is $frac{P(s)}{zeta(s)}$ where $P(s)$ is the prime zeta function.
(4) $quadfrac{P(s)}{zeta(s)}=sumlimits_{n=1}^inftyfrac{A143519(n)}{n^s},quadRe(s)>1?$
The following figure illustrates the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above in orange where formula (4) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$.
Figure (2): Illustration of formula (4) for $frac{P(s)}{zeta(s)}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (4) for $frac{P(s)}{zeta(s)}$ evaluated along the line $Re(s)=1$ in orange where formula (4) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{P(s)}{zeta(s)}$. The red discrete portions of the plots illustrate the evaluation of formula (4) for $frac{P(1+i,t)}{zeta(1+i,t)}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (4) for $left|frac{P(1+i,t)}{zeta(1+i,t)}right|$
Figure (4): Illustration of formula (4) for $Releft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (5): Illustration of formula (4) for $Imleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Figure (6): Illustration of formula (4) for $Argleft(frac{P(1+i,t)}{zeta(1+i,t)}right)$
Question (4): What is the range of convergence of the Dirichlet series for $frac{P(s)}{zeta(s)}$ defined in (4) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Note $frac{P(s)}{zeta(s)}$ has a pole at each non-trivial zeta zero.
Question (5): Are there explicit formulas for $f(x)$ and $frac{P(s)}{zeta(s)}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series riemann-hypothesis
number-theory prime-numbers riemann-zeta dirichlet-series riemann-hypothesis
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
edited Jan 14 at 20:01
Steven Clark
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
asked Oct 1 '18 at 22:38
Steven ClarkSteven Clark
7311413
7311413
add a comment |
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