Two remarks in prime number theory

Some Remarks on Number Theory

This note contains some disconnected minor remarks on number theory . 1 . Let (1) Iz j I=1, 1<j<co be an infinite sequence of numbers on the unit circle . Put n s(k, n) _ z~, Ak = Jim sup I s(k, n) j=1 k=oo and denote by B k the upper bound of the numbers I s(k,n)j . If z j = e 2nij' a =A 0 then all the Ak 's are finite and if the continued fraction development of a has bounded denominators the...

Some Topics in Prime Number Theory

(ii) p has no divisor other than 1 and itself We denote by P the set of all primes and we write the elements of P in the increasing order as p 1 < p 2 < p 3 < · · · Now consider = p 1 · · · p n + 1. Observe that is an integer > 1. Therefore there exists q ∈ P with q |. Then q = p i for 1 ≤ i ≤ n since q || − 1. Thus q > p n. We started with primes p 1 , · · · , p n and we found a prime q > p n....

Three Topics in Additive Prime Number Theory

We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectu...

New results in comparative prime number theory

Two classic theorems from number theory: The Prime Number Theorem and Dirichlet’s Theorem