18 research outputs found
Square-free values of the Carmichael function
No full textAbstractWe obtain an asymptotic formula for the number of square-free values among p−1, for primes p⩽x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1⩽n⩽x,L(x)=(κ+o(1))xln1−αx,where α=0.37395… is the Artin constant, and κ=0.80328… is another absolute constant
On the prime number lemma of Selberg
No full textThe key result needed in almost all elementary proofs of the Prime Number Theorem is a prime number lemma proved by Atle Selberg in 1948. Without restricting ourselves to purely elementary techniques we show how the error term in Selberg's fundamental lemma relates to the error term in the Prime Number Theorem. In spite of all the interest in this topic over the last sixty years this particular question seems to have been overlooked in the past
Non-Abelian Generalizations of the Erdős-Kac Theorem
No full textAbstractLet a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:The number of n ≤ x coprime to a satisfyingis asymptotic to as x tends to infinity.</jats:p
Riemann and his zeta function
No full textAn exposition is given, partly historical and partly mathemat-ical, of the Riemann zeta function (s) and the associated Rie-mann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of (s). Rel-evance of these investigations to the theory of the distribution of prime numbers is discussed
