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Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and I found the term in Terence Tao's blog and if my memory is correct I believe he touched upon it in a YouTube video. More specifically, I am having trouble understanding the following excerpt from the same blog:

Philosophically, one of the main reasons why it is so hard to control the distribution of the primes is that we do not currently have too many tools with which one can rule out “conspiracies” between the primes, in which the primes (or the von Mangoldt function) decide to correlate with some structured object (and in particular, with a totally multiplicative function) which then visibly distorts the distribution of the primes. For instance, one could imagine a scenario in which the probability that a randomly chosen large integer ${n}$ is prime is not asymptotic to ${\frac{1}{\log n}}$ (as is given by the prime number theorem), but instead to fluctuate depending on the phase of the complex number ${n^{it}}$ for some fixed real number ${t}$, thus for instance the probability might be significantly less than ${1/\log > n}$ when ${t \log n}$ is close to an integer, and significantly more than ${1/\log n}$ when ${t \log n}$ is close to a half-integer. This would contradict the prime number theorem, and so this scenario would have to be somehow eradicated in the course of proving that theorem. In the language of Dirichlet series, this conspiracy is more commonly known as a zero of the Riemann zeta function at ${1+it}$.

Please note that my motivation is chiefly a philosophical interpretation than a number theoretic one, so it would be very helpful if the reasoning is directed in that way.

Thank you.

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Let $N$ be a 100-digit even number. Pick $10^{96}$ odd numbers, uniformly at random, from among the integers up to $N$. The chances that among those $10^{96}$ numbers you can't find two that add up to $N$ are very, very close to zero - so close to zero that if it actually happened you might suspect that the numbers you chose had conspired with each other to avoid adding up to $N$.

Goldbach's conjecture says that every even number exceeding $4$ is a sum of two odd primes. The number of odd primes up to the $100$-digit number $N$ exceeds $10^{96}$, so, speaking informally, we say it would take a conspiracy among the primes for Goldbach's conjecture to fail for that number $N$.

So that's what "conspiracy" means; some situation that is extremely unlikely to happen on any mild assumption about the distribution of the primes could only happen if there is a conspiracy among the primes.

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  • $\begingroup$ The last paragraph illuminates. So basically if 'heads' show up all the time when a coin is tossed, we can say that the event is "conspired", correct? $\endgroup$ – Sniper Clown Mar 6 '12 at 7:21

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