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There are infinitely many Carmichael numbers
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Defect zero p-blocks for finite simple groups
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whoseExpand
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Large character sums: Pretentious characters and the Pólya-Vinogradov theorem
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides aExpand
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The Distribution of Values of L(1; )
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ABC allows us to count squarefrees
We show several consequences of the abc-conjecture for questions in analytic number theory which were of interest to Paul Erd} os: For any given polynomial f(x) 2 Zx], we deduce, from theExpand
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er. We shall see how their legacy has inuenced research for most of the rest of the century, particularly through the 'schools' of Selberg, and of Erdos, and with the \large sieve" in the sixties.Expand
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On the Equations zm = F(x, y) and Axp + Byq = Czr
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Extreme values of $|ζ(1+it)|$
Improving on a result of J.E. Littlewood, N. Levinson [3] showed that there are arbitrarily large t for which |ζ(1 + it)| ≥ e log2 t + O(1). (Throughout ζ(s) is the Riemann-zeta function, and logjExpand
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We give a relatively easy proof of the Erd ˝ os-Kac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results inExpand
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