Re: [math-fun] DGF --> OGF?
Does an "arithmetic function" take its values in Z ? If so, there will be lots of Dirichlet series that have non-integral coefficients, the simplest of which might be A(s) = (1/2) 1/2^s = 1/2^(s+1). Note that if a Dirichlet series with real coefficients equals the 0 function, then all its coefficients are 0. So A(s) above won't have any *other* such series than 1/2 * 1/2^s. It seems like a good question to ask which functions A(s) *do* have a Dirichlet series with integral coefficients, but hard. It's interesting that the set D_I of all Dirichlet series with integer coefficients are closed under addition and multiplication, so form a subring of the ring D formed by all such series. Then we can say the "obstruction" to a function A(s)'s Dirichlet expansion having integer coefficients is the image phi(A(s)) in the quotient ring D / D_I by the quotient map phi : D —> D / D_I. —Dan Mike Stay wrote: ----- Can every suitably nice function A(s) be "inverted" to give an arithmetic function a(n) such that A(s) = sum_{n >= 1} a(n)/n^s? (E.g. by Perron's formula?) In other words, given ordinary generating functions A_i(x) = sum_{n >= 0} a^i_n x^n, I can combine them into a Dirichlet generating function A(s) = prod_{i >= 1} A_i(p_i^{-s}). My question is whether given a suitably nice but otherwise arbitrary function of s and think of it as a Dirichlet generating function, can I decompose it into a collection of ordinary generating functions? If so, what constitutes "suitably nice"? Merely convergence for Re(s) > sigma? -----
On Wed, May 8, 2019 at 2:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
Does an "arithmetic function" take its values in Z ?
It seems I've confused arithmetic fn with multiplicative fn. An arithmetic function is any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers, so my "in other words" is not the same as the first question I asked. It looks like I can use the formula here to extract the coefficients: https://mathoverflow.net/questions/329980/extracting-dirichlet-series-coeffi... but there's no guarantee that the resulting a(n) are a multiplicative function, so there's no guarantee that the coefficients can be "unpacked" into a bunch of OGFs. Thanks!
If so, there will be lots of Dirichlet series that have non-integral coefficients, the simplest of which might be A(s) = (1/2) 1/2^s = 1/2^(s+1). Note that if a Dirichlet series with real coefficients equals the 0 function, then all its coefficients are 0. So A(s) above won't have any *other* such series than 1/2 * 1/2^s.
It seems like a good question to ask which functions A(s) *do* have a Dirichlet series with integral coefficients, but hard.
It's interesting that the set D_I of all Dirichlet series with integer coefficients are closed under addition and multiplication, so form a subring of the ring D formed by all such series. Then we can say the "obstruction" to a function A(s)'s Dirichlet expansion having integer coefficients is the image phi(A(s)) in the quotient ring D / D_I by the quotient map
phi : D —> D / D_I.
—Dan
Mike Stay wrote: ----- Can every suitably nice function A(s) be "inverted" to give an arithmetic function a(n) such that A(s) = sum_{n >= 1} a(n)/n^s? (E.g. by Perron's formula?)
In other words, given ordinary generating functions A_i(x) = sum_{n >= 0} a^i_n x^n,
I can combine them into a Dirichlet generating function A(s) = prod_{i >= 1} A_i(p_i^{-s}).
My question is whether given a suitably nice but otherwise arbitrary function of s and think of it as a Dirichlet generating function, can
I decompose it into a collection of ordinary generating functions?
If so, what constitutes "suitably nice"? Merely convergence for Re(s) > sigma? -----
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
On Wed, May 8, 2019 at 3:34 PM Mike Stay <metaweta@gmail.com> wrote:
On Wed, May 8, 2019 at 2:10 PM Dan Asimov <dasimov@earthlink.net> wrote:
Does an "arithmetic function" take its values in Z ?
It seems I've confused arithmetic fn with multiplicative fn. An arithmetic function is any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers, so my "in other words" is not the same as the first question I asked. It looks like I can use the formula here to extract the coefficients: https://mathoverflow.net/questions/329980/extracting-dirichlet-series-coeffi... but there's no guarantee that the resulting a(n) are a multiplicative function, so there's no guarantee that the coefficients can be "unpacked" into a bunch of OGFs.
Thanks!
Are there results about when a Dirichlet series is multiplicative? Like (just guessing here) all its complex zeros are confined to the critical strip? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
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