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I am a Biology major (please don't shame me!) but I really enjoy mathematics. Recently I have been reading about this conjecture and its importance in understanding the distribution of primes.

After being studied for so long, how come all attempts have failed? Why is this such a difficult thing to study and work on?

Would finding a closed form of $$\sum_{n=1}^{\infty} \frac {1}{n^s}$$ help the issue? Have any attempts been made at this at all?

Thank you very much for your time educating me and your help!!

Osama
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Apple
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    I think one of the reasons is that many talented mathematicians have tried so hard in many years, but to no avail in proving or disproving this hypothesis. – Akira Sep 14 '18 at 04:30
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    By the way, Euler famously discovered that $\sum 1/n^2 = \pi^2/6$, using a brilliant argument. You might like to read about that if you haven't already. – littleO Sep 14 '18 at 04:39
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    I think the most important thing is lake of an enbloc formula to specific $\zeta(s)$ relating it to other familiar functions like $\Gamma (s)$. – Nosrati Sep 14 '18 at 06:10
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    @LeAnhDung This is not a "reason", we can only conlude from that fact that a proof must be extremely difficult. But this argument does not explain where the difficulty lies. – Peter Sep 14 '18 at 07:28
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    Although most mathematicians believe that the Riemann hypothesis is true, it is well possible that it is false. In this case, it would not be a wonder that noone can prove it. In fact there are arguments against the truth of the Riemann hypothesis which are apparently not taken serious, but nevertheless doubts are left, and noone can actually claim that the conjecture is "almost surely true". In the case of Goldbach's conjecture, the situation is different. – Peter Sep 14 '18 at 07:31
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    A closed form of $\zeta(s)$ would not necessarily solve the problem. Apparently, the behaviour of the zeta-function is rather complicated in the complex numbers. Probably, a useful structure would be necessary for the proof of the Riemann hypothesis. – Peter Sep 14 '18 at 07:36
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    An argument first put forward by Littlewood: "A long-open conjecture in alaysis generally turns out to be false. A long-open conjecture in algebra generally turns out to be true." – Saša Sep 14 '18 at 09:30
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    @Nosrati ?? 1. Tons of results can be proven about objects with no explicit formula. 2. There are well known links between $\zeta$ and $\Gamma$ (see the WP page for a start). – Did Sep 14 '18 at 10:13
  • https://medium.com/@SereneBiologist/how-close-are-we-to-solving-the-riemann-hypothesis-6dbb631fc0f9 This link gives some interesting observations about the hypothesis along with the statement "if it is true, then it is only barely true". This has to do with a constant (De Brujin-Newman constant). The Riemann hypothesis is true if this constant is non-positive. It was recently proved it was non-megative... so the hypothesis is true if the constant is zero. – QC_QAOA May 16 '21 at 04:52

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This is not an answer but it's too long for a comment: the reason why it is so difficult to prove the Reimann Hypothesis could be that you cannot prove something that is not true.

Here is an interesting quote from a wonderful book written on the subject, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, by John Derbyshire:

You can decompose the zeta function into different parts, each of which tell you something about different zeta's behavior. One of these parts is the so-called $S$ function. For the entire range for which zeta has so far been studied - which is to say, for arguments on the critical line up to a height of around $10^{23}$ - $S$ mainly hovers between -1 and +1. The largest value known is around 3.2. There are strong reasons to think that if $S$ were ever to get up to around 100, then RH might be in trouble. The operative word there is "might"; $S$ attaining a value near 100 is a necessary condition for the RH to be in trouble but not a sufficient one.

Could values of the $S$ function ever get that big? Why, yes! As a matter of fact, Atle Selberg proved in 1946 that $S$ is unbounded; that is to say, it will eventually, if you go high enough up the critical line, exceed any number you name! The rate of growth of $S$ is so creepingly slow that the heights involved are beyond imagining; but certainly $S$ will eventually get up to 100. Just how far would we have to explore up the critical line for $S$ to be that big? Probably around $10^{10^{10000}}$. Way beyond the range of our current computational abilities.

Saša
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  • I followed Atle Selberg's lectures in Oslo in the late 1980's, and remember one of the professors asking about the possibility that RH could be false. Selberg was very dismissive: "Of course it it true! We know that. We just don't know why." – Per Erik Manne Nov 19 '21 at 09:58