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In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers and therefore is not(More)
1 Voltaire said that Archimedes had more imagination than Homer. Unfortunately, as far as we know, Archimedes's only use of it, outside of mathematics, was to condemn goldsmiths and to kill people. More peaceful uses of mathematicians' imagination, to the outside of mathematics, in decreasing order of impact, were provided by Multi-Millionaire Richard(More)
I. Let Q(N; q; a) denote the number of squares in the arithmetic progression qn+a; n = 1; 2; ; N; and let Q(N) be the maximum of Q(N; q; a) over all non-trivial arithmetic progressions qn + a. It seems to be remarkably diicult to obtain non-trivial upper bounds for Q(N). There are currently two proofs known of the weak bound Q(N) = o(N) (which is an old(More)
The Riemann zeta function is the function of the complex variable s, defined in the half-plane 1 (s) > 1 by the absolutely convergent series ζ(s) := ∞ n=1 1 n s , and in the whole complex plane C by analytic continuation. As shown by Riemann, ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1, and satisfies the(More)