Birkhoff and Hall's theta function

Dougy · 2009-01-02, 22:46

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?

R.D. Silverman · 2009-01-02, 22:59

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..

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).

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?

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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).