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 2009-01-02, 22:46 #1 Dougy     Aug 2004 Melbourne, Australia 23·19 Posts 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?
2009-01-02, 22:59   #2
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by Dougy 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..

 2009-01-05, 05:09 #3 Dougy     Aug 2004 Melbourne, Australia 23·19 Posts 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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