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In this paper we prove the following theorems in incidence geometry. 1. There is $δ > 0$ such that for any $P_{1}, \dots, P_{4}$ , and $Q_{1}, \dots, Q_{n} \in ℂ^{2}$ , if there are $\leq n^{(1 + δ) / 2}$ many distinct lines between $P_{i}$ and $Q_{j}$ for all $i$ , $j$ , then $P_{1}, \dots, P_{4}$ are collinear. If the number of the distinct lines is $< c n^{1 / 2}$ then the cross ratio of the four points is algebraic. 2. Given $c > 0$ , there is $δ > 0$ such that for any $P_{1}, P_{2}, P_{3} \in ℂ^{2}$ noncollinear, and $Q_{1}, \dots, Q_{n} \in ℂ^{2}$ , if there are $\leq c n^{1 / 2}$ many distinct lines between $P_{i}$ and $Q_{j}$ for all $i$ , $j$ , then for any $P \in ℂ^{2} ∖ {P_{1}, P_{2}, P_{3}}$ , we have $δ n$ distinct lines between $P$ and $Q_{j}$ . 3. Given $c > 0$ , there is...

Sums and differences of additive arithmetic functions in mean square.

P.D.T.A. Elliott (1979)

Journal für die reine und angewandte Mathematik

Sums and differences of power-free numbers

Julia Brandes (2015)

Acta Arithmetica

We employ a generalised version of Heath-Brown's square sieve in order to establish an asymptotic estimate of the number of solutions a, b ∈ ℕ to the equations a + b = n and a - b = n, where a is k-free and b is l-free. This is the first time that this problem has been studied with distinct powers k and l.

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Summation formulae and their relation to Dirichlet's series

Summation identities for representation of certain real numbers.

Summation of singular series corresponding to representations of numbers by some quadratic forms in twelve variables.

Summation of singular series corresponding to the representation of integers by some quadratic forms in fourteen variables.

Summation operators and "explicit formulas"

Summatory function of the Möbius function. I: Experimental upper bounds. (Fonction sommatoire de la fonction de Möbius. I: Majorations expérimentales.)

Summatory function of the Möbius function. II: Elementary asymptotic upper bounds. (Fonction sommatoire de la fonction de Möbius. II: Majorations asymptotiques élémentaires.)

Summen mit der Möbius-Funktion.

Summen von Quadranten in Zahlringen.

Summirung der Gausschen Reihen.

Sum-product estimates applied to Waring's problem mod $p$ .

Sum-product theorems and incidence geometry

Sums and differences of additive arithmetic functions in mean square.

Sums and differences of power-free numbers

Sums and products of distinct sets and distinct elements in $ℂ$ .

Sums for U(2n,q²) and their applications

Sums involving Fibonacci and Pell numbers.

Sums involving moments of reciprocals of binomial coefficients.

Sums involving the greatest integer function and Riemann-Stieltjes integration.

Sums involving the inverses of binomial coefficients.

Currently displaying 881 – 900 of 1526