We give estimates for exponential sums of the form n N Λ(n) exp(2πiag n /m), where m is a positive integer, a and g are integers relatively prime to m, and Λ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form p N exp(2πiaM p /m), where M p is the Mersenne number; M p = 2 p − 1 for any prime p. We also estimate some closely related sums, including n N µ(n) exp(2πiag n /m) and n N µ 2 (n) exp(2πiag n /m), where µ is the Möbius function.

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We present combinatorial Gray codes and explicit designs of efficient algorithms for lexicographical combinatorial generation of the sets of nonnesting and sparse nonnesting set partitions of length n.

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Abstract. We present an explicit bijection between noncrossing and nonnesting partitions of Coxeter systems of type D which preserves openers, closers and transients. OverviewThe lattice of set partitions of a set of n elements can be interpreted as the intersection lattice for the hyperplane arragement corresponding to a root system of type A n−1 , i.e. the symmetric group of n objects, S n . In particular, two of its subposets are very wellbehaved and widely studied, i.e. the lattice of, respectively, noncrossing and nonnesting partitions, which have a lot of interesting combinatorial properties, see e.g. [4,7,8,9,12,14,15,18,21,22,23,24,25] and the references therein.Recently, Victor Reiner [19] and Alexander Postnikov, see Christos Athanasiadis' paper [2, §6, Remark 2] generalized the notion of, respectively, noncrossing and nonnesting partitions to all classical reflection groups, and a new very active research area sprung up, namely generalizing previous known results held for noncrossing and nonnesting partitions of type A to their type B and D analogue, see e.g. [1,3,13,26,27] and the references therein.In particular, it is well known that in type A the number of noncrossing partition equals the number of nonnesting partition, and several bijections have been established. Recently one of us in [16,17] solved the corresponding problem for the type B (and type C) presenting an explicit bijection which falls back to one already known when restricted to the type A.The goal of this paper is to present an explicit bijection for the type D, therefore solving the problem for all classical Weyl groups, and to show that this bijection has interesting additional combinatorial properties. For instance, for any classical reflection group and any partition it is possible to define three different maps, called openers, closers, and transients, going from the partition itself to subsets of the involved reflection group, and we show that our bijection preserves all these functions. Furthermore, our map remains a bijection if restricted to the two sets of respectively, noncrossing and nonnesting partitions which are both of type B and type D, and therefore it is also a bijection between the partitions which are noncrossing in type D but which are actually crossing in type B, and the partition which are nonnesting in type D but which are actually nesting in type B.Very recently, two other bijections have been presented for solving the type D problem, see [10,20], but both of them have different designs and settings. Moreover in [10] openers, closers and transients are not preserved.2000 Mathematics Subject Classification. 05A18; 05E15.