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.121221222... is not quadratic.

Florian Luca (2005)

Revista Matemática Complutense

In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑n≥0 (an / bf(n)), where an ∈ Z and 1 ≤ |an| ≤ K for all n ≥ 0, is neither rational nor quadratic.

A higher rank Selberg sieve and applications

Akshaa Vatwani (2018)

Czechoslovak Mathematical Journal

We develop an axiomatic formulation of the higher rank version of the classical Selberg sieve. This allows us to derive a simplified proof of the Zhang and Maynard-Tao result on bounded gaps between primes. We also apply the sieve to other subsequences of the primes and obtain bounded gaps in various settings.

A larger GL 2 large sieve in the level aspect

Goran Djanković (2012)

Open Mathematics

In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL 2 cusp forms modular with respect to the congruence subgroups Γ1(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q 2−δ for any δ > 0, where N is the length of linear forms in the large sieve.

Additive properties of dense subsets of sifted sequences

Olivier Ramaré, Imre Z. Ruzsa (2001)

Journal de théorie des nombres de Bordeaux

We examine additive properties of dense subsets of sifted sequences, and in particular prove under very general assumptions that such a sequence is an additive asymptotic basis whose order is very well controlled.

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