# Why is proving the Riemann Hypothesis so hard?

The Riemann Hypothesis is considered by many to be the most important unsolved problem in pure mathematics.

Several attempts have been made in the last 150 years (here some of them are reported).
RH is the only problem that has been listed in both Hilbert's 23 Problems and the Millennium Problems by the Clay Institute; and yet it remains unsolved, seemingly resisting all attacks, and quickly becoming a piece of mathematical folklore as an "impossible" problem.
For instance, it is reported that Hilbert himself declared:

If I were to awaken after having slept for a thousand years, my first question would be:
"Has the Riemann hypothesis been proven?"

Implications and applications of this possible result have already been addressed many times on this website (reporting here some of them):
What does proving the Riemann Hypothesis accomplish?
What is so interesting about the zeroes of the Riemann $$\zeta$$ function?
Why do mathematicians care so much about zeta functions?
What is the link between Primes and zeroes of Riemann zeta function?

What I'm asking is: why is a conjecture on the zeroes of a specific complex function so hard to prove (or disprove)?
What are the main obstacles and obstructions to this problem's solution?

Edit: although the question has already been addressed before on Math.SE (here and, in some way, here), no answers have admittedly been given, and I personally find that the comments this question received in the last hours (for which I am grateful) addressed the problem much more clearly than the comments in the questions above.
I believe there's room for improvement, but I apologize if the question is too broad or violates the guidelines in some other way.

• The Riemann zeta function far from a simple object. On half of the complex plane, it's an infinite series. On the other half it is an analytic continuation of an infinite series, an even hairier object. We don't even half a general formula for the roots of degree 5 polynomials. Clearly the roots of an infinite series are a very elusive beast. Dec 7, 2020 at 1:24
• there are similar functions to RZ which have zeroes in the critical strip but not on the line showing that analytic properties like the functional equation are unlikely to suffice to prove RH if it's true; also RZ is a highly transcendental function (it is universal in a well-defined sense as one can approximate any non zero analytic function locally by values of RZ in the critical strip) and for example it sends any vertical line $1/2<\Re \sigma <1$ into a dense set in the plane etc Dec 7, 2020 at 4:57
• I'm voting to reopen since the linked question barely got any relevant answers. Dec 7, 2020 at 12:45
• @lisyarus Isn't it likely the linked question did not receive relevant answers because the question is opinion based and possibly unanswerable? Mar 6, 2021 at 9:23
• It's a survivor bias. We solve the easy problems pretty quickly. What's left? The hard ones. Mar 6, 2021 at 11:06

This is a bit an opinion based question and answer.

The RH is about $$\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$$, not $$\zeta(s)$$.

On the $$\zeta(s)$$ side we can easily exploit that it is the Dirichlet series of the integers.

Surprisingly (or not?) on the $$\log\zeta(s),\frac1{\zeta(s)},\frac{\zeta'(s)}{\zeta(s)}$$ side we can't, and complicated structures appear, eg. the primes.

The same kind of structures appear for many other Dirichlet series (the Dirichlet L-functions, more generally the Selberg class) and the RH is (more or less) assumed to hold for all of them.

This set of Dirichlet series with a RH is discrete/isolated: you can't change slightly the coefficients without loosing one of the key properties (analytic continuation, functional equation, Euler product, growth of the coefficients).

So we need a setting where all those key properties are present: an arithmetical-analytical-algebraic setting. It is hard, by definition.

In practice most elementary approaches to the RH fail because they apply the same way to $$(G(\chi_5)/5^{1/2})^{-1/2}L(s,\chi_5)+\overline{(G(\chi_5)/5^{1/2})}^{-1/2}L(s,\overline{\chi_5})$$ lacking only the Euler product, and having a bunch of zeros off the critical line.