Interestingly after a lifetime of hearing other people discuss Riemann's hypothesis and not quite understanding myself what the heck zeroes of the zeta function could have to do with primes, I read Riemann's own paper, and found it much cklearer and more elementary than any of the later explanations.
As I tried to explain above, Riemann made it clear throughout his own work that to understand a complex function, one should consider it as intimately associated with its maximum domain of definition (via analytic continuation) but that it is completely determined in any open set. Moreover it is also best characterized by its full set of zeroes and poles in that maximum domain.
Hence consider the following reasoning: by euler's product formula, which contains only the primes, the zeta function is completely determiend by the set of primes, i.e. their distribution. Hence knowing the location of all the primes is equivalent to knowing the zeta function.
On the other hand the zeta function has as its natural domain of definition the full complex plane, in which it has exactly one pole at 1, and an infinite number of zeroes (apparently). Hence the zeta function is also essentially determined by knowing exactly where all these zeroes are.
So we have a function, the zeta function, and it is essentially determined in two different ways: 1) by knowing where all the primes are.
2) by knowing where all its zeroes are (the one pole is at z=1).
Hence the two pieces of data: zeroes of zeta, and distribution of primes, are merely two sides of the same coin, and that coin is the zeta function.
hence these two data must contain almost the same information.
More precisely, Riemann shows that Gauss's approximation Li(x) to the number of primes less than or equal to x, overestimates the actual number. I.e. he shows it estimates instead the number of primes plus the numbers of squares of primes plus the number of cubes of primes, ... He states this very clearly.
Then he writes out this statement as an equation and "solves it" for a better formula estimating the actual number of primes, by "mobius inversion".
Throughout the process he makes various statements about how many terms can be ignored in his estimates, and these claims are still unproved to some extent. That is the famous hypothesis.
I did not know any of what I have just explained until I read Riemann's own version of it.
The later versions attempt to prove his claims and fill in the gaps in his sketchy arguemnts. That makes them longer to read and more technical. Riemann himself is very clear in many ways, and very intuitive, much more so than his followers.
So although at one time I spent up to a week per page reading his abelian functions paper, (on which I thought I was an "expert"! hah!), on the other hand the number theory paper was relatively easy to at least peruse for its main points.
I repeat: shake off your reluctance to read riemann: his short paper is much clearer even than edwards' excellent book about it. Edwards himself says this! that is why edwards provides a translation of riemann in his appendix.
An example from my own field was the "Brill Noether" estimate of just how large the genus of a curve needs to be before we can expect all curves of that genus to admit a "d to 1" branched cover of the projective line.
I almost never understood clearly why this number was what it was until I read Riemann. He deduced it very quickly just from his proof of the riemann roch theorem, and until i spent that week thinking about it i had never relaized it could be deduced from just that data.
indeed although it is called the "Brill Noether" number, Riemann deduced it several years earlier than they did. current books on the topic, do not credit riemann with this discovery and do not give his own argument, but rather credit brill and noether.
Thus I say riemann understood these matters better than do our contemporaries who have apparently never read him.
the original discoverer of a topic is in my mind always the best source for the deepest understanding of it. This is of course based mainly on my reading of gauss, riemann, euclid, mumford, serre, kempf, artin, grothendieck, galois, wirtinger, weil, andreotti - mayer, weyl, kodaira, Newton, cantor, milnor, eilenberg - maclane, einstein, galileo, van der waerden, and others.
I did not get this impression from my reading of principia mathematica by russell-whitehead, and in my opinion there is a reason for that.
