Mixing
Questions related to the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$
$begingroup$
This question is related to the following two functions evaluated with the coefficient function $a(n)=mu(n)log(n)$.
(1) $quad f(x)=sumlimits_{n=1}^x a(n)$
(2) $quadfrac{zeta'(s)}{zeta(s)^2}=sumlimits_{n=1}^infty a(n),n^{-s},quadRe(s)>1?$
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Is it true $f(x)$ has an infinite number of zero crossings?
Question (2): What are the limits on $f(x)$ predicted by the Prime Number Theorem and the Riemann Hypothesis?
The following figure illustrates the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above in orange where formula (2) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$.
Figure (2): Illustration of formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ evaluated along the line $Re(s)=1$ in orange where formula (2) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$. The red discrete portions of the plots illustrate the evaluation of formula (2) for $frac{zeta'(1+i,t)}{zeta(1+i,t)^2}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (2) for $left|frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right|$
Figure (4): Illustration of formula (2) for $Releft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (5): Illustration of formula (2) for $Imleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (6): Illustration of formula (2) for $Argleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Question (3): What is the range of convergence of the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Question (4): Are there explicit formulas for $f(x)$ and $frac{zeta'(s)}{zeta(s)^2}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series mellin-transform
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
$endgroup$
add a comment |
$begingroup$
This question is related to the following two functions evaluated with the coefficient function $a(n)=mu(n)log(n)$.
(1) $quad f(x)=sumlimits_{n=1}^x a(n)$
(2) $quadfrac{zeta'(s)}{zeta(s)^2}=sumlimits_{n=1}^infty a(n),n^{-s},quadRe(s)>1?$
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Is it true $f(x)$ has an infinite number of zero crossings?
Question (2): What are the limits on $f(x)$ predicted by the Prime Number Theorem and the Riemann Hypothesis?
The following figure illustrates the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above in orange where formula (2) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$.
Figure (2): Illustration of formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ evaluated along the line $Re(s)=1$ in orange where formula (2) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$. The red discrete portions of the plots illustrate the evaluation of formula (2) for $frac{zeta'(1+i,t)}{zeta(1+i,t)^2}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (2) for $left|frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right|$
Figure (4): Illustration of formula (2) for $Releft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (5): Illustration of formula (2) for $Imleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (6): Illustration of formula (2) for $Argleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Question (3): What is the range of convergence of the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Question (4): Are there explicit formulas for $f(x)$ and $frac{zeta'(s)}{zeta(s)^2}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series mellin-transform
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
$endgroup$
1
$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30
add a comment |
$begingroup$
This question is related to the following two functions evaluated with the coefficient function $a(n)=mu(n)log(n)$.
(1) $quad f(x)=sumlimits_{n=1}^x a(n)$
(2) $quadfrac{zeta'(s)}{zeta(s)^2}=sumlimits_{n=1}^infty a(n),n^{-s},quadRe(s)>1?$
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Is it true $f(x)$ has an infinite number of zero crossings?
Question (2): What are the limits on $f(x)$ predicted by the Prime Number Theorem and the Riemann Hypothesis?
The following figure illustrates the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above in orange where formula (2) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$.
Figure (2): Illustration of formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ evaluated along the line $Re(s)=1$ in orange where formula (2) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$. The red discrete portions of the plots illustrate the evaluation of formula (2) for $frac{zeta'(1+i,t)}{zeta(1+i,t)^2}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (2) for $left|frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right|$
Figure (4): Illustration of formula (2) for $Releft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (5): Illustration of formula (2) for $Imleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (6): Illustration of formula (2) for $Argleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Question (3): What is the range of convergence of the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Question (4): Are there explicit formulas for $f(x)$ and $frac{zeta'(s)}{zeta(s)^2}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series mellin-transform
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
$endgroup$
This question is related to the following two functions evaluated with the coefficient function $a(n)=mu(n)log(n)$.
(1) $quad f(x)=sumlimits_{n=1}^x a(n)$
(2) $quadfrac{zeta'(s)}{zeta(s)^2}=sumlimits_{n=1}^infty a(n),n^{-s},quadRe(s)>1?$
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Is it true $f(x)$ has an infinite number of zero crossings?
Question (2): What are the limits on $f(x)$ predicted by the Prime Number Theorem and the Riemann Hypothesis?
The following figure illustrates the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above in orange where formula (2) is evaluated over the first $10,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$.
Figure (2): Illustration of formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (2) for $frac{zeta'(s)}{zeta(s)^2}$ evaluated along the line $Re(s)=1$ in orange where formula (2) is evaluated over the first $1,000$ terms. The underlying blue reference function is $frac{zeta'(s)}{zeta(s)^2}$. The red discrete portions of the plots illustrate the evaluation of formula (2) for $frac{zeta'(1+i,t)}{zeta(1+i,t)^2}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (2) for $left|frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right|$
Figure (4): Illustration of formula (2) for $Releft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (5): Illustration of formula (2) for $Imleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Figure (6): Illustration of formula (2) for $Argleft(frac{zeta'(1+i,t)}{zeta(1+i,t)^2}right)$
Question (3): What is the range of convergence of the Dirichlet series for $frac{zeta'(s)}{zeta(s)^2}$ defined in (2) above? Does it converge only for $Re(s)>1$, or does it also converge for $Re(s)=1landIm(s)ne 0$?
Question (4): Are there explicit formulas for $f(x)$ and $frac{zeta'(s)}{zeta(s)^2}$ expressed in terms of the non-trivial zeta zeros?
number-theory prime-numbers riemann-zeta dirichlet-series mellin-transform
number-theory prime-numbers riemann-zeta dirichlet-series mellin-transform
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
asked Jan 14 at 21:43
Steven ClarkSteven Clark
7291413
7291413
1
$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30
add a comment |
1
$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30
1
1
$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30
add a comment |
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$begingroup$
See those kind of proof of the PNT. Going from $b(n)=mu(n)$ to $b(n) log n$ is one of the main tools of ANT. The PNT and explicit formula for $sum_{n le x}mu(n)$ are not very different to those for $pi(x), psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $sum_{n le x}mu(n),sum_{n le x}mu(n)log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh).
$endgroup$
– reuns
Jan 16 at 1:04
$begingroup$
The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time...
$endgroup$
– daniel
Jan 16 at 8:30