# Primes

### Some articles on primes, prime:

Selberg Sieve - Description
... Let A be a set of positive integers ≤ x and let P be a set of primes ... for p dividing d, when d is a product of distinct primes from P ... Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z ...
Turán Sieve - Description
... Let A be a set of positive integers ≤ x and let P be a set of primes ... the Ap for p dividing d, when d is a product of distinct primes from P ... a positive real number and P(z) denote the product of the primes in P which are ≤ z ...
Landau's Problems - Progress Toward Solutions - Twin Prime Conjecture
... showed that the size of the gap between primes could be far smaller than the average size of the prime gap Earlier, they conditionally proved a weaker version of the twin prime conjecture ... is the prime-counting function ... The twin prime conjecture replaces 20 with 2 ...
Maier's Theorem
... In number theory, Maier's theorem (Maier 1985) is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives the wrong answer ... It states that if π is the prime counting function and λ is greater than 1 then does not have a limit as x tends to infinity more precisely the lim ... Cramér's probabilistic model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the Borel–Cantelli lemma) ...
Carmichael's Totient Function Conjecture - Lower Bounds
... counterexample must be divisible by squares of the primes dividing its totient value ... Klee's results imply that 8 and Fermat primes (primes of the form 2k+1) excluding 3 do not divide the smallest counterexample ...