Intereting Posts

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How to find asymptotically (or some reasonable bound, at least $ o(n) $) number of numbers, representable as a sum of squares of 2 numbers? (in case of bound I am interested in both: lower and upper bounds)

I know how to find explicitly the number of ways to represent given number in such a way. (can be found here)

Thank you!

- Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)
- Does dividing by zero ever make sense?
- effective version of Mertens Theorem for the Euler product
- Prime Harmonic Series $\sum\limits_{p\in\mathbb P}\frac1p$
- To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$
- Effective Upper Bound for the Number of Prime Divisors

P.S. For one lower bound you can use this problem, it’ll give you somewhat $ \Omega (n^{\frac{3}{4}}) $.

- What is the value of $\sum_{p\le x} 1/p^2$?
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- Big O/little o true/false
- Comparing average values of an arithmetic function
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- Asymptotic expression for sum of first n prime numbers?
- What exactly *is* the Riemann zeta function?
- What do following asymptotic symbols mean?
- Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
- Computing the product of p/(p - 2) over the odd primes

Let $S_2(x)$ be the number of integers $\leq x$ which are a sum of two squares. Landau, in 1906, showed that

$$S_2(x) \sim K \frac{x}{\sqrt{\log x}}$$

where

$$K = \frac{1}{\sqrt{2}} \prod_{p \equiv 3 \mod 4} \frac{1}{\sqrt{1-p^{-2}}}$$

I can’t find a proof online, but there are references to the statement in many places, such as here.

Posting pages 260-263 from LeVeque. The evident hair on page 262 is not part of the book proper; it evidently fell from my own head onto the scanner, and is one I could ill afford to lose.

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A number can be written as a sum of two squares if and only if it is not divisible an odd number of times by any prime that is $3$ modulo $4$.

In particular, every prime that is $1$ modulo $4$ is a sum of two squares. By the prime number theorem, there are $\Theta(\frac{n}{\log n})$ primes in the interval $[0, n]$. By Dirichlet’s theorem on arithmetic progressions, asymptotically half of these are $1$ modulo $4$.

Therefore, the number of numbers less than $n$ that can be written as a sum of two squares is $\Omega(\frac{n}{\log n})$

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