It is unsolved as of today.
However, some latest researches are:
Edit:
FWIW, do note that the shortest "proof" of RH by Mark Colyvan as mentioned in IEP here using paraconsistent logic:
As the founders of relevant logic,
Anderson and Belnap, urge in their
canonical book Entailment, a ‘proof’
submitted to a mathematics journal in
which the essential steps fail to
provide a reason to believe the
conclusion, e.g. a proof by explosion,
would be rejected out of hand. Mark
Colyvan (2008) illustrates the point
by noting that no one has laid claim
to a startlingly simple proof of the
Riemann hypothesis:
Riemann’s Hypothesis: All the zeros of the zeta function have real part equal to > 1/2.
Proof: Let R stand for the Russell set, the set of all sets that are not
members of themselves. It is
straightforward to show that this set
is both a member of itself and not a
member of itself. Therefore, all the
zeros of Riemann’s zeta function have
real part equal to 1/2.
Needless to say, the Riemann
hypothesis remains an open problem at
time of writing.
The cited 2008 article by Colyvan however does not use this for RH but Fermat's Last Theorem.
Speaking of non-classical approach, Douglas S. Bridges further writes about the self-referential nature in Reality and Virtual Reality in Mathematics (2006):
There is an even more dramatic example of a proof which might cause the
same unease. A famous conjecture of Riemann in the nineteenth century, the
Riemann Hypothesis, remains unsolved today despite the efforts of some of the
greatest mathematicians in the intervening 150 years. Early last century, the
English mathematician J.E. Littlewood produced a theorem whose difficult proof
was split into two cases. In the first case, Littlewood assumed that the Riemann
Hypothesis was true, and in the second that it was false. Writing R to denote the
Riemann Hypothesis, and P to denote the conclusion of Littlewoods theorem,
we can express his proof in the schematic form
$(R\bigvee\neg R)\Rightarrow P.$ (1)
Here I have introduced the standard logical symbols $\bigvee$ (or), $\neg$ (not), and $\Rightarrow$ (implies)
What is the meaning of Littlewoods proof? Since we are unable at this
date to decide whether or not the Riemann Hypothesis is true, we cannot say
which of the two cases of his proof actually applies. If, as most mathematicians
expect, the Riemann Hypothesis turns out to be provable, then that part of
Littlewoods proof that is based on the assumption that the Riemann Hypothesis
is false is worthless and can be thrown away. Moreover, in such a proof, if $P$
is an existential statement one that asserts the existence of a certain object
$x$ with certain properties then the two cases of a proof of $P$ that follows the
Littlewoods schematic form (1) may produce di¤erent objects $x$ with the desired
properties (as in our earlier proof involving $\sqrt{2}^{\sqrt{2}}$
); under such circumstances, we
might be unable to tell which of the two possibilities for x was the desired one
until we could prove the truth or falsity of the Riemann Hypothesis.
The formalist might attempt to remove our unease about Littlewoods proof
as follows. Suppose that the desired conclusion $P$ of Littlewoods theorem is false.
Then Littlewoods arguments, schematised in (1), show that neither the Riemann
Hypothesis nor its negation can hold (since each of these alternatives leads us
to a proof of $P$): In other words, if $P$ is false, then the Riemann Hypothesis is
false and it is false that the Riemann Hypothesis is false! This is plainly absurd.
Hence we conclude that $P$ cannot be false and is therefore true.
