An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeros of the zeta function and leads to a necessary and sufficient condition for the Riemann Hypothesis to hold.

Abstract. We compute the asymptotic behaviour of the number of partitions of large vectors (n1, n2) of Z 2 + in the critical regime n1 √ n2 and in the subcritical regime. This work completes the results established in the fifties by Auluck, Nanda, and Wright.

Given a convex cone C in R d , an integral zonotope T is the sum of segments [0, vi] (i = 1, . . . , m) where each vi ∈ C is a vector with integer coordinates. The endpoint of T is k = m 1 vi. Let T (C, k) be the family of all integral zonotopes in C whose endpoint is k ∈ C. We prove that, for large k, the zonotopes in T (C, k) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in T (C, k) are very close to a fixed convex set. We also establish several combinatorial properties of a typical zonotope in T (C, k).2010 Mathematics Subject Classification. Primary 52B20; Secondary 60C05, 05A17.

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An equivalence is proven between the Riemann hypothesis and the speed of convergence to 6∕ 2 of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0. K E Y W O R D SArithmetic error terms, Euler totient function, Riemann hypothesis M S C ( 2 0 1 0 ) 11N37, 11M26