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Binary Quadratic Forms with Large Discriminants and Sums of Two Squareful Numbers II
Author(s) -
Blomer Valentin
Publication year - 2005
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610704006040
Subject(s) - mathematics , combinatorics , binary number , binary quadratic form , quadratic equation , quadratic form (statistics) , upper and lower bounds , discrete mathematics , quadratic function , arithmetic , mathematical analysis , geometry
Let F = ( F 1 , …, F m ) be an m ‐tuple of primitive positive binary quadratic forms and let U F ( x ) be the number of integers not exceeding x that can be represented simultaneously by all the forms F j , j = 1, …, m . Sharp upper and lower bounds for U F ( x ) are given uniformly in the discriminants of the quadratic forms. As an application, a problem of Erdős is considered. Let V ( x ) be the number of integers not exceeding x that are representable as a sum of two squareful numbers. Then V ( x ) = x (log x ) −α+ o (1) with α = 1 − 2 −1/3 = 0.206….