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-   -   Birkhoff and Hall's theta function (https://www.mersenneforum.org/showthread.php?t=11265)

 Dougy 2009-01-02 22:46

Birkhoff and Hall's theta function

Hi guys,
Long time no see. I've been busy doing a PhD (in combinatorial number theory), getting married and having a baby girl. (: So very little time to write. Anyway...
Birkhoff and Hall's theta function [TEX]\theta(n)[/TEX] is defined by [TEX]\theta(p^a)=(p^a-1)(p^a-p)...(p^a-p^{a-1})[/TEX] for prime [TEX]p[/TEX], and theta is multiplicative, that is [TEX]\theta(dt)=\theta(d)\theta(t)[/TEX] if [TEX]\gcd(d,t)=1[/TEX]. (Sloane's A061350)
I have shown that for prime [TEX]q[/TEX], the largest [TEX]b[/TEX] such that [TEX]q^b[/TEX] dividing [TEX]\theta(n)[/TEX] is [TEX]O(\log^2(n))[/TEX].
I suspect that this has already been found by someone else. So this is my "homework help." Does anyone know where I can find this?

 R.D. Silverman 2009-01-02 22:59

[QUOTE=Dougy;156517]Hi guys,
Long time no see. I've been busy doing a PhD (in combinatorial number theory), getting married and having a baby girl. (: So very little time to write. Anyway...
Birkhoff and Hall's theta function [TEX]\theta(n)[/TEX] is defined by [TEX]\theta(p^a)=(p^a-1)(p^a-p)...(p^a-p^{a-1})[/TEX] for prime [TEX]p[/TEX], and theta is multiplicative, that is [TEX]\theta(dt)=\theta(d)\theta(t)[/TEX] if [TEX]\gcd(d,t)=1[/TEX]. (Sloane's A061350)
I have shown that for prime [TEX]q[/TEX], the largest [TEX]b[/TEX] such that [TEX]q^b[/TEX] dividing [TEX]\theta(n)[/TEX] is [TEX]O(\log^2(n))[/TEX].
I suspect that this has already been found by someone else. So this is my "homework help." Does anyone know where I can find this?[/QUOTE]

I expect that this might be found in Ramanujan's work on Theta
functions. You might also want to check Bruce Berndt's papers...

Note: I am not very knowledgable about special functions..

 Dougy 2009-01-05 05:09

Thanks for that! I haven't found exactly what I was after but I think I'm getting a bit closer. I'm not 100% sure it has been published before, but I think there's a high chance (it's not a particularly surprising result).

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