Additive problems with almost prime squares
19 Jun 2023
TL;DR: In this article , it was shown that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a smooth number, and that the number of such representations is of the expected order of magnitude.
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Abstract: Abstract We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes $$p-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.
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Citations
On an ErdŐs–Kac-Type Conjecture of Elliott
Ofir Gorodetsky,Lasse Grimmelt +1 more
TL;DR: The Erdős–Kac-Type Conjecture of Elliott holds true, and it is proved using the Bombieri–Vinogradov theorem and related results.
On a conjecture of Elliott concerning a probabilistic variant of Titchmarsh's divisor problem
Ofir Gorodetsky,Lasse Grimmelt +1 more
- 01 Jan 2023
TL;DR: The conjecture of Elliott concerning a probabilistic variant of Titchmarsh's divisor problem is proved using the Bombieri--Vinogradov theorem and related results involving Poisson--Dirichlet distribution.
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Henryk Iwaniec
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Opera De Cribro
John B. Friedlander,Henryk Iwaniec +1 more
- 22 Jun 2010
TL;DR: A wide range of applications are included, both to traditional questions such as those concerning primes, and to areas previously unexplored by sieve methods, such as elliptic curves, points on cubic surfaces and quantum ergodicity.
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Primes in arithmetic progressions to large moduli
TL;DR: In this article, the authors present a set of notations for the use of the word "cascade" in the form of a sequence of n-grams, where each node corresponds to a node in a tree.