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PostPosted: Thu, 19 Jan 2012 20:10:27 UTC 
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Say I have a magical function, f:\mathbb{Z}\to\mathbb{R} satisfying

f(n)=0\iff n=0
f(n)\ge 0, \forall n
f(n)=f(-n)
f(xy)=f(x)f(y)
f(x+y)\le f(x)+f(y)
f|_{_\mathbb{N}}\not\equiv 1

Then if f(n)\le 1 \forall n\in\mathbb{Z}, show that \exists !p, prime, such that f(p)<1.

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PostPosted: Fri, 20 Jan 2012 06:22:09 UTC 
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Shadow wrote:
Say I have a magical function, f:\mathbb{Z}\to\mathbb{R} satisfying

f(n)=0\iff n=0
f(n)\ge 0, \forall n
f(n)=f(-n)
f(xy)=f(x)f(y)
f(x+y)\le f(x)+f(y)

Then if f(n)\le 1 \forall n\in\mathbb{Z}, show that \exists !p, prime, such that f(p)<1.


I think you want a condition to rule out f(n)=1\forall n\neq 0.

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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PostPosted: Fri, 20 Jan 2012 08:23:04 UTC 
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outermeasure wrote:
Shadow wrote:
Say I have a magical function, f:\mathbb{Z}\to\mathbb{R} satisfying

f(n)=0\iff n=0
f(n)\ge 0, \forall n
f(n)=f(-n)
f(xy)=f(x)f(y)
f(x+y)\le f(x)+f(y)

Then if f(n)\le 1 \forall n\in\mathbb{Z}, show that \exists !p, prime, such that f(p)<1.


I think you want a condition to rule out f(n)=1\forall n\neq 0.


Ah yes, you're right. Thanks for noticing that outermeasure. :)

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PostPosted: Sat, 21 Jan 2012 06:37:17 UTC 
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Hint:
Spoiler:
What can you say about q^{(p-1)r}?

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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