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Why could Mertens not prove the prime number theorem?
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Is there a probability density function satisfying the following conditions?
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Aug
15 |
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Is there a probability density function satisfying the following conditions?
@Fedja: No I still don't understand. Can you exemplify with one working example of a solution that works for all $y > 0$. |
Aug
15 |
comment |
Is there a probability density function satisfying the following conditions?
@Nik, I had the same formulation that you have given and I can draw several distributions through $(0,1)$ and $(1,e^{-y})$. However satisfying condition 5 is the problem. For example $p(x,y) = e^{-yx^a}$ satisfies the first four conditions. Now if conditions 5 is also going to be true the we find that $y$ must be exactly equal to $\Gamma(1+1/a)^a$. But the minimum value of $\Gamma(1+1/a)^a$ is $e^{-\gamma}$. So this solutions is good for $y > e^{-\gamma}$ but it does not hold for $0 < y < e^{-\gamma}$. Likewise I have not been able to find a function $p(x,y)$ which satisfies 5 for all $y > 0$. |
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Aug
14 |
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Is there a probability density function satisfying the following conditions?
@NikWeaver: Yes that is correct |
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14 |
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Is there a probability density function satisfying the following conditions?
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14 |
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Aug
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Asymptotic bounds on $\pi^{-1}(x)$ (inverse prime counting function)
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