Problems of the Millennium : The Riemann Hypothesis ( 2004
نویسنده
چکیده
The Riemann Hypothesis (RH) is one of the seven millennium prize problems put forth by the Clay Mathematical Institute in 2000. Bombieri’s statement [Bo1] written for that occasion is excellent. My plan here is to expand on some of his comments as well as to discuss some recent developments. RH has attracted further attention of late in view of the fact that three popular books were written about it in 2003 and a fourth one is forthcoming in 2005. Unlike the case of Fermat’s Last Theorem, where the work of Frey, Serre and Ribet redefined the outlook on the problem by connecting it to a mainstream conjecture in the theory of modular forms (the “Shimura-Taniyama-Weil Conjecture”), in the case of RH we cannot point to any such dramatic advance. However, that doesn’t mean that things have stood still.
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Million Dollar Mathematics
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with s = 12 + it , and shows that ξ(t) is an even entire function of t whose zeros have imaginary part between −i/2 and i/2. He further states, sketching the proof, that in the range between 0 and T the function ξ(t) has about (T/2π) log(T/2π)− T/2π zeros. Riemann then continues: “Mann finden nun in der That etwa so wiel reelle Nullstellen innerhalb dieser Grenzen, und es ist sehr wahrscheinlic...
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