(Huh, this post was definitely written while still in ephiphanic mode. Have fun parsing it. I added strikethroughs where appropriate to make it hopefully a little more sensible. I've since explored some of these ideas more coherently elsewhere.)
Primes cancel out everything following which is multiplied by them. For example, 2 cancels out all subsequent even numbers, which ends up being a huge amount of all possible integers, including half of infinity. Then three cancels out an enormous quantity of numbers, although not as many as 2. Then 5 comes along, still striking out an enormous range, albeit not as many as 3. So on with 7, 11, 13, and upward toward infinity. Thus each new prime removes a large but steadily shrinking quantity of future possible primes from being considered as primes. It is the obscurity of this future-changing quality of primes which confuses people into thinking things like 'primes look nearly random, so maybe they are random' (Strangely a common misconception.)(Later edit; I realized one day recently that people probably think primes are random because of Freeman Dyson's Correlations between eigenvalues of a random matrix insight back in the 1970s.)
Here is what confuses some people about primes, since it makes a prime 2 things at once: A prime is BOTH the latest in a series of numbers which cannot be divided AND ALSO the earliest of a huge swath of numbers which have just been removed from possible future primes. People tend to only think about the first definition; the second one seems barely relevant, an easily-ignored artifact rather than an important part of the definition.
I believe the prevailing mystery about primes being random is a consequence of relying too strongly on excluded-middle logic, which tends to use the first definition of a thing as, well, definitive -- to the exclusion of other possible definitions. Especially with math. There is probably an evolutionary-grade efficiency involved in this dynamic, or maybe sociologists or neurologists have studied it intimately and I just don't know what to call it when I'm using search engines, but I'm sure I'm not the only one to understand it.
With binary logic, it appears "a thing can only be one thing at a time," but with ternary logic, a thing can be multiple things at the same time (think "superposition")*.
Intuition tells me the 1/2 of Riemann's zeta function may be a reference to the halving of infinity that occurs each time a new prime is added, as half of the remaining infinity gets canceled out, over and over. Infinity promptly responds by doubling itself again, lest anyone should find the end of mathematical endlessness (he said, casually anthropomorphising infinity), because, well, infinity plays by different rules... like... ternary logic where everything is superimposed? (Hmmm....) People have a hard time thinking of infinity as a moving target (since all other numbers are very stable, never 'moving,' we tend to think of infinity as "a thing" rather than as "a dynamic vastness which can be doubled or halved instantly with the same effort as incrementing a normal number by one"). Think about it. Hm, this insight is more dense than it first appears. 9:00 - 11:57 a.m. Monday Oct 2, 2017.
(Note: I've been circling around the mysterious role of 1/2 for more than a decade, often seeing it in glimpses but rarely as a whole, and it finally came together as just described, shortly after being delighted with the early discussion in this conversation on "Mind and the Wave Function Collapse" which I eagerly want to finish because Henry Stapp's way of saying things is so... succinct and elegant. The insight above leaped to my mind when I saw this Quora question in my email, the current answer for which is missing the magic of the question. I've seen the peculiar importance of 1/2 from the zeta function for a long time, but this is the first time I'm getting a glimpse of how it actually works.) (Update, now it's May 2021 and I just updated that section again, with more emphasis on superposition and ternary logic... it's slowly coming together.)
In summary, as often happens with mathy insights, I presently have a fair confidence that I just exposed an the correct approach to proving the Riemann hypothesis. I'm publishing it here in my blog instead of keeping it hidden because, well, that confidence, and also I don't have time and maybe not the skills to make it more formal any time soon. So, if anyone uses this approach, well, here's this blog entry showing where it originally came from. This blog is pretty obscure and I doubt anyone else will get the binary/ternary part of the insight, wherein a simple explanation for "how primes work" has been hidden for a couple thousand years because of over-reliance on excluded-middle thinking. There is of course a tangentially related insight into the structure of set theory buried within this particular logic point, which I've written about elsewhere, though I suppose it would take a long book to explore in appropriate detail. Anyway, long story short, I get mathy insights like this often, and regularly overrate them -- who doesn't -- so let's see what time tells as to its actual value.
(Speaking of which, it should be easy to determine whether primes were a mystery while native, err, ternary, logic was still more prevalent... who first discovered primes? (Brief Internet search) Yep, primes don't go back further than Greece, when excluded middle logic was first being explored systematically, so it may be possible to establish that prime numbers come into being as a consequence of excluded-middle thinking.) Ok, I'm starting to edit the original thought above, so I'll quit, leaving it in its rough form for now.
*Superposition is the correct way to understand the most important feature of ternary logic. Superposition is literally incomprehensible from within the excluded-middle mindset (which is embedded in language so we learn it as we learn language). From the binary perspective, people tend to think of everything as separated from everything else, down to a particle level. We prefer to see small "particles" instead of interwoven "waves" and get confused when there is a transition between these two forms of vibration. From binary logic, the "third pole" of ternary logic is seen as a weaker, indefinite "maybe" existing between the very certain "Yes" and "No" poles. This is a binary view of ternary logic. In fact, when seen more objectively, the third pole is superposed, and can contain any parts or all of infinity simultaneously. It is not weaker, but in fact has equal certainty with the two flanking extremes of "yes" and "no." It's like describing colors to someone who only understands black and white -- paradoxes abound. Quantum mechanics has brought this particular middle realm into our awareness but the number of people who understand that superposition destroys the very foundations of binary logic can be counted on a single hand. The YouTube video above does a good job of introducing the importance of the third pole as being that which can contain anything up to and including the entire universe -- simultaneously, with our minds being that which determines what part of the universe is there. (May 2021 update: this paragraph was confusing, so I tidied it up and it makes much more sense, so I removed the strikethrough I had put on it during the previous edit. Due to more recent insights, it's all starting to make more coherent sense now.)
ps just found this, it might be useful: https://phys.org/news/2010-05-sum-digits-prime-evenly.html