Remainder in the Wiener-Ikehara theorem
$begingroup$
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form
$$ sum_{nleq X} a(n) = cX^a(log X)^m + Obig(X(log X)^{m-1}big),$$
i.e. with an error term better than $o(1)$. I have looked into Tenenbaum's book (Theorem 7.13) but both the condition and the expression for the error term seem overly complicated. I also looked into Korevaar's book but couldn't find it there either.
All I managed to get is this article (in french) where Theorem A.1 in Appendix A gives this type of result under some growth condition. The authors say that this is a well-known theorem but they could not find it in the literature, so I wondered where this might appear.
reference-request tauberian-theory wieners-tauberian-theorem
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
$endgroup$
add a comment |
$begingroup$
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form
$$ sum_{nleq X} a(n) = cX^a(log X)^m + Obig(X(log X)^{m-1}big),$$
i.e. with an error term better than $o(1)$. I have looked into Tenenbaum's book (Theorem 7.13) but both the condition and the expression for the error term seem overly complicated. I also looked into Korevaar's book but couldn't find it there either.
All I managed to get is this article (in french) where Theorem A.1 in Appendix A gives this type of result under some growth condition. The authors say that this is a well-known theorem but they could not find it in the literature, so I wondered where this might appear.
reference-request tauberian-theory wieners-tauberian-theorem
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
$endgroup$
$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
1
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
1
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42
add a comment |
$begingroup$
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form
$$ sum_{nleq X} a(n) = cX^a(log X)^m + Obig(X(log X)^{m-1}big),$$
i.e. with an error term better than $o(1)$. I have looked into Tenenbaum's book (Theorem 7.13) but both the condition and the expression for the error term seem overly complicated. I also looked into Korevaar's book but couldn't find it there either.
All I managed to get is this article (in french) where Theorem A.1 in Appendix A gives this type of result under some growth condition. The authors say that this is a well-known theorem but they could not find it in the literature, so I wondered where this might appear.
reference-request tauberian-theory wieners-tauberian-theorem
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
$endgroup$
I am looking for a reference for a variant of the Wiener-Ikehara theorem (for Dirichlet series) giving result of the form
$$ sum_{nleq X} a(n) = cX^a(log X)^m + Obig(X(log X)^{m-1}big),$$
i.e. with an error term better than $o(1)$. I have looked into Tenenbaum's book (Theorem 7.13) but both the condition and the expression for the error term seem overly complicated. I also looked into Korevaar's book but couldn't find it there either.
All I managed to get is this article (in french) where Theorem A.1 in Appendix A gives this type of result under some growth condition. The authors say that this is a well-known theorem but they could not find it in the literature, so I wondered where this might appear.
reference-request tauberian-theory wieners-tauberian-theorem
reference-request tauberian-theory wieners-tauberian-theorem
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
edited Dec 14 '18 at 9:51
Daniele Tampieri
2,2422722
2,2422722
asked Dec 14 '18 at 9:40
hrthrt
462312
asked Dec 14 '18 at 9:40
hrthrt
462312
asked Dec 14 '18 at 9:40
hrthrt
462312
462312
$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
1
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
1
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42
add a comment |
$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
1
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
1
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42
$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
1
1
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
1
1
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42
add a comment |
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$begingroup$
@reuns My $L$-series are very nice, basically powers of $zeta$ up to some holomorphic factor.
$endgroup$
– hrt
Dec 14 '18 at 10:28
1
$begingroup$
Then what is your precise Dirichlet series ? What do you assume for $sum_{n=1}^infty a(n) n^{-s}$ ?
$endgroup$
– reuns
Dec 14 '18 at 10:30
1
$begingroup$
I have $sum a(n)n^{-s} = H(s)zeta(s)^k$ where $H$ is holomorphic and explicit, e.g. $H(s) = prod_{p|A}(1+p^{-s})$ for some fixed parameter $A$.
$endgroup$
– hrt
Dec 14 '18 at 10:33
$begingroup$
Then $H(s)$ is a finite Dirichlet series so it is enough to look at the partial sums of the coefficients of $zeta(s)^k$ which is Titchmarsh's chapter "(Dirichlet) divisor problem" (the error term will be much more precise than what you wrote)
$endgroup$
– reuns
Dec 14 '18 at 10:42
$begingroup$
Perhaps you could have a look at the following monograph of T. Ganelius (1969): "Tauberian Remainder Theorems*, LNM 232 Springer Verlag. I am not familiar with the book but I stumbled on it while searching references on paricular power series remainder estimates, and it seems relevant for your problem.
$endgroup$
– Daniele Tampieri
Dec 14 '18 at 10:42