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Results tagged with sums-of-squares
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user 9232
5
votes
1
answer
309
views
How often is the value of a quadratic polynomial equal to a sum of two integer squares?
Let $b:\mathbb N\to \{0,1\}$ be the indicator function of integers that are a sum of two non-zero integer squares. Let $f(t)\in \mathbb Z[t]$ be an irreducible polynomial of degree $2$ with positive l …
- 3,062
4
votes
Accepted
Sieve bound for the sum of two squares
One can do a bit better. For simpler presentation assume that we instead consider the function $b'$ that is the indicator function of integers all of whose prime divisors are $1 \mod 4$. We have $b'\l …
- 3,062
4
votes
Divisor sums over values of binary forms of primes
An answer regarding the use of large sieve suggested by Lucia (too long for a comment).
I guess her/his thought was along the following lines:
The divisors $d$ of $p^2+q^2$ are of order $x^2$ and …
- 3,062
7
votes
2
answers
424
views
Divisor sums over values of binary forms of primes
Let $\tau$ be the divisor function, that is
$$
\tau(n)=\sharp\{d \in \mathbb{N}, d|n\}.
$$
I was wondering if anyone has ever proved an asymptotic estimate
for the sum
$$S(x):=\sum_{p,q\leq x}\tau(p …
- 3,062
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number …
- 3,062