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Old 2009-01-02, 22:46   #1
Dougy
 
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Aug 2004
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Default 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 \theta(n) is defined by \theta(p^a)=(p^a-1)(p^a-p)...(p^a-p^{a-1}) for prime p, and theta is multiplicative, that is \theta(dt)=\theta(d)\theta(t) if \gcd(d,t)=1. (Sloane's A061350)
I have shown that for prime q, the largest b such that q^b dividing \theta(n) is O(\log^2(n)).
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?
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Old 2009-01-02, 22:59   #2
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by Dougy View Post
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 \theta(n) is defined by \theta(p^a)=(p^a-1)(p^a-p)...(p^a-p^{a-1}) for prime p, and theta is multiplicative, that is \theta(dt)=\theta(d)\theta(t) if \gcd(d,t)=1. (Sloane's A061350)
I have shown that for prime q, the largest b such that q^b dividing \theta(n) is O(\log^2(n)).
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?
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..
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Old 2009-01-05, 05:09   #3
Dougy
 
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Aug 2004
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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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