![]() Selberg (1942) showed that a positive proportion of In 1914, Hardy proved that an infinite number of values forĬan be found for which and (Havil 2003, p. 213). Wiener showed that the prime number theorem is literally equivalent to the assertion that the Riemann These results are summarized in the following table, whereįalling in the critical strip lie on the critical This computation verifies that the Riemann hypothesis is true at least Nontrivial zeros of the function lie on the critical (2004) then used a faster method by Odlyzko and Schönhage to verify that the Nontrivial zeros lie on the critical line. S. Wedeniwski used ZetaGrid ( ) to prove that the first The Riemann hypothesis was computationally tested and found to be true for the first Line), does not earn the $1 million award. Hypothesis (e.g., by using a computer to actually find a zero off the critical In 2000, the Clay Mathematics Institute ( ) offered a $1 million prize ( )įor proof of the Riemann hypothesis. Problems and number 1 of Smale's problems. Proof of the Riemann hypothesis is number 8 of Hilbert's Furthermore, Conrey and Li (1998) proveĪ counterexample to de Branges's approach, which essentially means that theory developed Proofs seem to be present in these papers. The generalized Riemann hypothesis (de Branges 2003, 2004 Boutin 2004), but no actual Riemann hypothesis (de Branges 1986, 1992, 1994) and in fact claiming to prove Has written a number of papers discussing a potential approach to the generalized Hypothesis was reported in Time magazine, even after a flaw in the proof hadīeen unearthed by Siegel (Borwein and Bailey 2003, p. 97 Conrey 2003). The late 1940s, H. Rademacher's erroneous proof of the falsehood of Riemann's ![]() Mertens conjecture has been proven false, completely invalidating this claim. However, the proof itself was never published, nor was it found in Stieltjes papersįollowing his death (Derbyshire 2004, pp. 160-161 and 250). Stieltjes (1885) published a note claiming to have proved the MertensĪ result stronger than the Riemann hypothesis and from which it would have followed. The Riemann hypothesis has thus far resisted all attempts to prove it. To several decimal digits (Granville 2002 Borwein and Bailey 2003, p. 68). That Riemann had made detailed numerical calculations of small zeros of the Riemann While it was long believed that Riemann's hypothesis was the result of deep intuition on the part of Riemann, an examination of his papers by C. L. Siegel showed ![]() Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix). Has a zero with real part larger than 1/2. The " critical line" (where denotes the real part of ).Ī more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann First published in Riemann's groundbreaking 1859 paper (Riemann 1859), the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the
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