How much of the Riemann Hypothesis has been solved?
Solution 1:
The Riemann zeta function has two kinds of zeros, trivial zeroes (at the negative even integers, $-2$, $-4$, et c.) and the non-trivial zeroes. There are infinitely many non-trivial zeroes and all of them are known to lie in the strip having real parts between $0$ and $1$ (in detail, the strip $\{x + \mathrm{i}y \in \mathbb{C} \mid 0<x<1\}$). (That there are none on the line $x = 1$ (or further to the right) is the prime number theorem. That there are then none on the line $x=0$ follows from the functional equation for the zeta function.) The conjecture is that all the non-trivial zeroes lie on the critical line ($x = \frac{1}{2}$).
[F] showed that in fact, the strip can be narrowed slightly. For $|y| \geq 3$, any roots must have $x \in \left( \frac{1}{57.54 (\ln |y|)^{2/3} (\ln \ln |y|)^{1/3}}, 1-\frac{1}{57.54 (\ln |y|)^{2/3} (\ln \ln |y|)^{1/3}} \right)$.
[H] and [HL] showed that there are infinitely many zeroes on the critical line. [L] showed that one-third of the non-trivial zeroes are on the critical line. [C] improved this to two-fifths. [BL] showed that, [*] for any particular distance from the critical line, the proportion of zeroes that far away or farther decreases to zero as we allow an upper bound on $y$ to increase. (This is the general shape of these results. For some $Y>0$, we count roots in the finite strip $\{x+\mathrm{i}y \mid 0 < x < 1 \text{ and } -Y < y < Y\}$. Then look at what fraction of those are on the critical line or are far enough away from the critical line as we let $Y$ increase.)
Around 1859, Riemann calculated the location of the first few zeroes, determining that they were on the critical line. This work was not published, but used the Riemann-Siegel formula found on scratch paper in his collected works and published in [S] (1932). In 1986, van de Lune, te Riele, and Winter somewhat famously found the 1.5 billion zeroes of least positive imaginary part were all on the critical line. This has been extended by [GD] (2004) to the first 10 trillion such zeroes.
No non-trivial zero is known in closed form. (It seems rather unlikely that any zero will have a closed form.)
[*] The first version of this line read "for any particular distance from the critical line, only finitely many zeroes are that far away or farther from the line." This isn't quite correct. There could still be infinitely many such zeroes, for instance the $2^\text{nd}$, $(2^2)^\text{th}$, $(2^3)^\text{th}$, and so on roots are an infinite set of roots whose proportion of the total number of roots decreases to zero.
[BL] Bohr, H.; Landau, E., "Ein Satz über Dirichletsche Reihen mit Anwendung auf die ζ-Funktion und die L-Funktionen", Rendiconti del Circolo Matematico di Palermo, 37 (1): 269–272, 1914.
[C] Conrey, J. B., "More than two fifths of the zeros of the Riemann zeta function are on the critical line", J. Reine angew. Math., 399: 1–16, 1989.
[F] Ford, K., "Vinogradov's integral and bounds for the Riemann zeta function". Proc. London Math. Soc. 85 (3): 565–633, 2002.
[GD] Gourdon, Xavier, "The $10^{13}$ first zeros of the Riemann Zeta function, and zeros computation at very large height", self-published, 2004. See also Computation of zeros of the Zeta function
[H] Hardy, G. H., "Sur les Zéros de la Fonction ζ(s) de Riemann", C. R. Acad. Sci. Paris, 158: 1012–1014, 1914.
[HL] Hardy, G. H.; Littlewood, J. E., "The zeros of Riemann's zeta-function on the critical line", Math. Z., 10 (3–4): 283–317, 1921.
[L] Levinson, N., "More than one-third of the zeros of Riemann's zeta function are on σ = 1/2", Adv. Math., 13 (4): 383–436, 1974.
[S] Siegel, C. L., "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B: Studien 2: 45–80, 1932.