Riemann hypothesis

For excellent insight into Riemann and his mathematics (primarily focusing on the Riemann zeta function and the Riemann hypothesis ), see J. Derbyshire. Prime Obsession Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Penguin, New York, 2004).]... 

If one wants a deterministically polynomial-time algorithm that only outputs prime numbers, and where the corresponding factoring assumption follows from that made above, one has to rely on Cramer s conjecture (see above) and search for each prime from some random number upwards in steps of two, and test each number with the pure Miller primality test [Mill76], which relies on the extended Riemann hypothesis. 

Weil s construction in vi) above was the basis of his epoch-making proof of the Riemann Hypothesis for curves over finite fields, which really put characteristic p algebraic geometry on its feet. 

The Jacobian is introduced in the standard books referred to in our notes to Lecture I esp. Serre (op. cit.) shows that all rational differentials to on curves C are pull-backs of translation-invariant differentials rj on algebraic groups J via some rational map (f> C - J. Moreover Gunning (op. cit., esp. Math. Notes 12, subtitled Jacobi Varieties ) treats in detail many of the topics of this and the next Lecture. For Weil s original algebraic construction of the Jacobian and its application to the Riemann hypothesis, see ... 

Montgomery [6] showed, assuming the Riemann hypothesis, that the pair correlation function of the unfolded zeros for j has a form. 

In this article we have addressed a general expression for the multiparticle distribution functions of ideal Fermi gas in the ground state. The expression is of a determinant form for any spatial dimension, and has a mathematical structure similar to that for the correlation function of the eigenvalues of random unitary matrices, especially in the one-dimensional case. Noting intimate relationships between the random matrix theory and the Riemann hypothesis, the present... 

Mill76 Gary L. Miller Riemann s Hypothesis and Tests for Primality Journal of Computer and System Sciences 13 (1983) 300-317. 

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