Abstract. A theorem of Leibman [19] asserts that a polynomial orbit (g(n)Γ)n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed sub-nilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N] in a nilmanifold. More specifically we show that there is a factorization g = εg ′ γ, where ε(n) is “smooth”, (γ(n)Γ)n∈Z is periodic and “rational”, and (g ′ (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G ′ /Γ ′ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a subsequent paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper [12]. 1.

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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining