The Riemann Hypothesis
نویسنده
چکیده
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 a 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 “Man findet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind,” which can be translated as “Indeed, one finds between those limits about that many real zeros, and it is very likely that all zeros are real.” The statement that all zeros of the function ξ(t) are real is the Riemann hypothesis. The function ζ(s) has zeros at the negative even integers −2,−4, . . . and one refers to them as the trivial zeros. The other zeros are the complex numbers 12 + iα, where α is a zero of ξ(t). Thus, in terms of the function ζ(s), we can state the
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