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Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number t ≥ 2, the number of solutions to the equation (n / m) = t for natural numbers 1 ≤ m < n is bounded. In this paper we establish this result in the interior region exp(log^(2/3+ε)n) ≤m ≤n −exp(log^(2/3+ε)n) for any fixed ε > 0. Indeed, when t is sufficiently large depending on ε, we show that there are at most four solutions (or at most two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation (n)m=t, where (n)_m: = n(n−1)…(n−m+1) denotes the falling factorial.

Let λ denote the Liouville function. We show that, as X → ∞, ∫^(2X)_X sup_(P(Y)∈ℝ[Y]deg(P)≤k) ∣ ∑_(x≤n≤x+H) λ(n)e(−P(n))∣ dx = o(XH) for all fixed k and X^θ ≤ H ≤ X with 0 < θ < 1 fixed but arbitrarily small. Previously this was only established for k ≤ 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases e(−P(n)) by degree k nilsequences F⎯(g(n)Γ). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result ∫^(2X)_X ‖λ‖_(U^(k+1)([x,x+H])) dx = o(X) in the same range of H. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of λ over short polynomial progressions (n + P₁(m),…,n + P_k(m)), which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range H ≥ exp ((log X)^(5/8 + ε)), thus strengthening also previous work on the Fourier uniformity of the Liouville function.

We introduce a simple approach to study partial sums of multiplicative functions which are close to their mean value. As a first application, we show that a completely multiplicative function f : N→C satisfies ∑ f(n)_ (n≤x) = cx+O(1) with c ≠ 0 if and only if f(p) = 1 for all but finitely many primes and |f(p)| < 1 for the remaining primes. This answers a question of Imre Ruzsa. For the case c = 0, we show, under the additional hypothesis ∑_(p:|f(p)|<1)1/p < ∞, that f has bounded partial sums if and only if f(p) = χ(p)p^(it) for some non-principal Dirichlet character χ modulo q and t ∈ ℝ except on a finite set of primes that contains the primes dividing q, wherein |f(p)| < 1. This essentially resolves another problem of Ruzsa and generalizes previous work of the first and the second author on Chudakov's conjecture. We also consider some other applications, which include a proof of a recent conjecture of Aymone concerning the discrepancy of square-free supported multiplicative functions.