Counting sums of two squares: the Meissel-Lehmer method

Computing pi(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method

Let π(x) denote the number of primes ≤ x. Our aim in this paper is to present some refinements of a combinatorial method for computing single values of π(x), initiated by the German astronomer Meissel in 1870, extended and simplified by Lehmer in 1959, and improved in 1985 by Lagarias, Miller and Odlyzko. We show that it is possible to compute π(x) in O( x 2/3 log2 x ) time and O(x1/3 log x log...

Sums of Two Squares

n = 1: 1 = 0 + 1; n = 2 (prime): 2 = 1 + 1; n = 3 (prime) is not a sum of two squares. n = 4: 4 = 2 + 0. n = 5 (prime): 5 = 2 + 1. n = 6 is not a sum of two squares. n = 7 (prime) is not a sum of two squares. n = 8: 8 = 2 + 2. n = 9: 9 = 3 + 0. n = 10: 10 = 3 + 1. n = 11 (prime) is not a sum of two squares. n = 12 is not a sum of two squares. n = 13 (prime): 13 = 3 + 2. n = 14 is not a sum of t...

Sums of Two and Four Squares

This document gives the formal proofs of the following results about the sums of two and four squares: 1. Any prime number p ≡ 1 mod 4 can be written as the sum of two squares. 2. (Lagrange) Any natural number can be written as the sum of four

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < eγn log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler-Mascheroni constant. We exhibit a broad class of subsets S of the natural numbers such that the Robin inequality holds for all but finitely many n ∈ S. As a special case, we determin...

Sums of distinct squares