What makes proving the Riemann Hypothesis so difficult?

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!!

Solution 1:

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.