How prime numbers are related to special functions?












5












$begingroup$


We know that the Riemann zeta function is defined as



$$zeta(s)=sum_{n=1}^inftyfrac{1}{n^s},$$



for all $Re(s)>1$.



Because of Euler product formula we also know that



$$zeta(s) = sum_{n=1}^inftyfrac{1}{n^s} = prod_{p text{ prime}} frac{1}{1-p^{-s}},$$
for all $Re(s)>1.$



There are a lot of functions related to Riemann zeta function. For example




  • $zeta(s)=zeta(s,1)$ where $zeta(s,q)$ is the Hurwitz zeta function.

  • $zeta(s)=operatorname{Li}_s(1)$, where $operatorname{Li}_s(z)$ is the polylogarithm.

  • $zeta(s)=(1-2^{-s})^{-1}chi_s(1)$, where $chi_s(z)$ Legendre chi function

  • $zeta(s)=Phi (1,s,1)$, where $Phi(z, s, alpha)$ is the Lerch zeta function

  • $ zeta(s) = (1-2^{1-s})^{-1}eta(s)$, where $eta(s)$ is the Dirichlet eta function

  • $dots$ and there are lot of other related functions such as multiple zeta function, Barnes zeta function, the Clausen function, etc.


Question. Are there Euler product formula type statements to other special functions?






number-theory prime-numbers special-functions riemann-zeta polylogarithm







share|cite|improve this question











$endgroup$












  • $begingroup$
    You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:02










  • $begingroup$
    @GerryMyerson Or they are just not that well-known to be there.
    $endgroup$
    – user153012
    Oct 19 '14 at 9:04










  • $begingroup$
    Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:19










  • $begingroup$
    @GerryMyerson You could find some related thing here.
    $endgroup$
    – user153012
    Oct 20 '14 at 2:01












  • $begingroup$
    I see a lot of products there, but none over primes.
    $endgroup$
    – Gerry Myerson
    Oct 20 '14 at 4:52
















5












$begingroup$


We know that the Riemann zeta function is defined as



$$zeta(s)=sum_{n=1}^inftyfrac{1}{n^s},$$



for all $Re(s)>1$.



Because of Euler product formula we also know that



$$zeta(s) = sum_{n=1}^inftyfrac{1}{n^s} = prod_{p text{ prime}} frac{1}{1-p^{-s}},$$
for all $Re(s)>1.$



There are a lot of functions related to Riemann zeta function. For example




  • $zeta(s)=zeta(s,1)$ where $zeta(s,q)$ is the Hurwitz zeta function.

  • $zeta(s)=operatorname{Li}_s(1)$, where $operatorname{Li}_s(z)$ is the polylogarithm.

  • $zeta(s)=(1-2^{-s})^{-1}chi_s(1)$, where $chi_s(z)$ Legendre chi function

  • $zeta(s)=Phi (1,s,1)$, where $Phi(z, s, alpha)$ is the Lerch zeta function

  • $ zeta(s) = (1-2^{1-s})^{-1}eta(s)$, where $eta(s)$ is the Dirichlet eta function

  • $dots$ and there are lot of other related functions such as multiple zeta function, Barnes zeta function, the Clausen function, etc.


Question. Are there Euler product formula type statements to other special functions?






number-theory prime-numbers special-functions riemann-zeta polylogarithm







share|cite|improve this question











$endgroup$












  • $begingroup$
    You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:02










  • $begingroup$
    @GerryMyerson Or they are just not that well-known to be there.
    $endgroup$
    – user153012
    Oct 19 '14 at 9:04










  • $begingroup$
    Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:19










  • $begingroup$
    @GerryMyerson You could find some related thing here.
    $endgroup$
    – user153012
    Oct 20 '14 at 2:01












  • $begingroup$
    I see a lot of products there, but none over primes.
    $endgroup$
    – Gerry Myerson
    Oct 20 '14 at 4:52














5












5








5


3



$begingroup$


We know that the Riemann zeta function is defined as



$$zeta(s)=sum_{n=1}^inftyfrac{1}{n^s},$$



for all $Re(s)>1$.



Because of Euler product formula we also know that



$$zeta(s) = sum_{n=1}^inftyfrac{1}{n^s} = prod_{p text{ prime}} frac{1}{1-p^{-s}},$$
for all $Re(s)>1.$



There are a lot of functions related to Riemann zeta function. For example




  • $zeta(s)=zeta(s,1)$ where $zeta(s,q)$ is the Hurwitz zeta function.

  • $zeta(s)=operatorname{Li}_s(1)$, where $operatorname{Li}_s(z)$ is the polylogarithm.

  • $zeta(s)=(1-2^{-s})^{-1}chi_s(1)$, where $chi_s(z)$ Legendre chi function

  • $zeta(s)=Phi (1,s,1)$, where $Phi(z, s, alpha)$ is the Lerch zeta function

  • $ zeta(s) = (1-2^{1-s})^{-1}eta(s)$, where $eta(s)$ is the Dirichlet eta function

  • $dots$ and there are lot of other related functions such as multiple zeta function, Barnes zeta function, the Clausen function, etc.


Question. Are there Euler product formula type statements to other special functions?






number-theory prime-numbers special-functions riemann-zeta polylogarithm







share|cite|improve this question











$endgroup$




We know that the Riemann zeta function is defined as



$$zeta(s)=sum_{n=1}^inftyfrac{1}{n^s},$$



for all $Re(s)>1$.



Because of Euler product formula we also know that



$$zeta(s) = sum_{n=1}^inftyfrac{1}{n^s} = prod_{p text{ prime}} frac{1}{1-p^{-s}},$$
for all $Re(s)>1.$



There are a lot of functions related to Riemann zeta function. For example




  • $zeta(s)=zeta(s,1)$ where $zeta(s,q)$ is the Hurwitz zeta function.

  • $zeta(s)=operatorname{Li}_s(1)$, where $operatorname{Li}_s(z)$ is the polylogarithm.

  • $zeta(s)=(1-2^{-s})^{-1}chi_s(1)$, where $chi_s(z)$ Legendre chi function

  • $zeta(s)=Phi (1,s,1)$, where $Phi(z, s, alpha)$ is the Lerch zeta function

  • $ zeta(s) = (1-2^{1-s})^{-1}eta(s)$, where $eta(s)$ is the Dirichlet eta function

  • $dots$ and there are lot of other related functions such as multiple zeta function, Barnes zeta function, the Clausen function, etc.


Question. Are there Euler product formula type statements to other special functions?






number-theory prime-numbers special-functions riemann-zeta polylogarithm




number-theory prime-numbers special-functions riemann-zeta polylogarithm






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Oct 17 '14 at 16:32







user153012

















asked Oct 17 '14 at 10:47









user153012user153012

6,39622279




6,39622279












  • $begingroup$
    You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:02










  • $begingroup$
    @GerryMyerson Or they are just not that well-known to be there.
    $endgroup$
    – user153012
    Oct 19 '14 at 9:04










  • $begingroup$
    Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:19










  • $begingroup$
    @GerryMyerson You could find some related thing here.
    $endgroup$
    – user153012
    Oct 20 '14 at 2:01












  • $begingroup$
    I see a lot of products there, but none over primes.
    $endgroup$
    – Gerry Myerson
    Oct 20 '14 at 4:52


















  • $begingroup$
    You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:02










  • $begingroup$
    @GerryMyerson Or they are just not that well-known to be there.
    $endgroup$
    – user153012
    Oct 19 '14 at 9:04










  • $begingroup$
    Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
    $endgroup$
    – Gerry Myerson
    Oct 19 '14 at 9:19










  • $begingroup$
    @GerryMyerson You could find some related thing here.
    $endgroup$
    – user153012
    Oct 20 '14 at 2:01












  • $begingroup$
    I see a lot of products there, but none over primes.
    $endgroup$
    – Gerry Myerson
    Oct 20 '14 at 4:52
















$begingroup$
You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
$endgroup$
– Gerry Myerson
Oct 19 '14 at 9:02




$begingroup$
You have linked to the Wikipedia pages on those functions. Were any Euler products given on those pages? If not, then that's very strong evidence that there are no Euler products.
$endgroup$
– Gerry Myerson
Oct 19 '14 at 9:02












$begingroup$
@GerryMyerson Or they are just not that well-known to be there.
$endgroup$
– user153012
Oct 19 '14 at 9:04




$begingroup$
@GerryMyerson Or they are just not that well-known to be there.
$endgroup$
– user153012
Oct 19 '14 at 9:04












$begingroup$
Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
$endgroup$
– Gerry Myerson
Oct 19 '14 at 9:19




$begingroup$
Well, the functions are well-known, and well-studied, so I figure if they have Euler products then it's well-known that they do.
$endgroup$
– Gerry Myerson
Oct 19 '14 at 9:19












$begingroup$
@GerryMyerson You could find some related thing here.
$endgroup$
– user153012
Oct 20 '14 at 2:01






$begingroup$
@GerryMyerson You could find some related thing here.
$endgroup$
– user153012
Oct 20 '14 at 2:01














$begingroup$
I see a lot of products there, but none over primes.
$endgroup$
– Gerry Myerson
Oct 20 '14 at 4:52




$begingroup$
I see a lot of products there, but none over primes.
$endgroup$
– Gerry Myerson
Oct 20 '14 at 4:52










1 Answer
1






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oldest

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1












$begingroup$

The online L-functions and modular forms database (lmfdb.org) provides detailed information on a very large number of special functions and provides a functional equation and Euler product form for each of them.






share|cite|improve this answer









$endgroup$













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    active

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    1












    $begingroup$

    The online L-functions and modular forms database (lmfdb.org) provides detailed information on a very large number of special functions and provides a functional equation and Euler product form for each of them.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      The online L-functions and modular forms database (lmfdb.org) provides detailed information on a very large number of special functions and provides a functional equation and Euler product form for each of them.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        The online L-functions and modular forms database (lmfdb.org) provides detailed information on a very large number of special functions and provides a functional equation and Euler product form for each of them.






        share|cite|improve this answer









        $endgroup$



        The online L-functions and modular forms database (lmfdb.org) provides detailed information on a very large number of special functions and provides a functional equation and Euler product form for each of them.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 17 '18 at 10:14









        KlangenKlangen

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