The Riemann hypothesis has thus far resisted all attempts to prove it. Stieltjes (1885) published a note claiming to have proved the Mertens conjecture with , a result stronger than the Riemann hypothesis and from which it would have followed. However, the proof itself was never published, nor was it found in Stieltjes papers following his death, so it is strongly believed his claim to have possessed such a proof was erroneous (Derbyshire 2004, pp. 160-161 and 250). In the late 1940s, H. Rademacher's erroneous proof of the falsehood of Riemann's hypothesis was reported in Time magazine, even after a flaw in the proof had been unearthed by Siegel (Borwein and Bailey 2003, p. 97; Conrey 2003). de Branges has written a number of papers discussing a potential approach to the generalized Riemann hypothesis (de Branges 1986, 1992, 1994) and in fact claiming to prove the generalized Riemann hypothesis (de Branges 2003, 2004; Boutin 2004), but no actual proofs seem to be present in these papers. Furthermore, Conrey and Li (1998) prove a counterexample to de Branges's approach, which essentially means that theory developed by de Branges is not viable.
