Results 1 
3 of
3
Different Approaches to the Distribution of Primes
 MILAN JOURNAL OF MATHEMATICS
, 2009
"... In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this lecture celebrating the 150th anniversary of the seminal paper of Riemann, we discuss various approaches to interesting questions concerning the distribution of primes, including several that do not involve the Riemann zetafunction.
Sieve Methods
"... called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of squarefree numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of squarefree numbers can be deduced using a squarefree sieve 1 . We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler
An Overview of Sieve Method and its History
, 2005
"... Trying to decompose an integer into a product of integers, we feel irritation. There should dwell the reason why any prime appears like a real gem that one can touch and hold. We thus muse ever and again how and when ancient people discovered the way of sifting out primes and began appreciating them ..."
Abstract
 Add to MetaCart
(Show Context)
Trying to decompose an integer into a product of integers, we feel irritation. There should dwell the reason why any prime appears like a real gem that one can touch and hold. We thus muse ever and again how and when ancient people discovered the way of sifting out primes and began appreciating them. Perhaps those who conceived the divisibility had already some sieves in their minds. Indeed, a wealth of evidences have been excavated supporting our view. The story to be told below must have originated more than five millennia ago 1) , while the primordial intellectual irritation has remained fresh and fundamental till today. The history of Sieve Method is rich and fascinating; we would need a volume to exhaust the story. In the present article we shall instead concentrate onto several pivotal ideas that made the progress possible; so the scope is inevitably limited. Nevertheless, you will encounter instances of precious mathematical achievements that people in the future will certainly continue to relate. Notes are to be read as essential parts, although they are in the style of personal memoranda. Mathematical symbols and definitions are introduced where they are needed for the first time, and will continue to be effective until otherwise stated. Theorems are given somewhat