history-page15

# history-page15 - H n . • It is the other half of...

This preview shows page 1. Sign up to view the full content.

The Riemann Hypothesis For s = a + bi with a > 1, the Riemann zeta function is given by the formula ζ(s) = s n = 1 1 n s . This function can be extended in a natural way to be de±ned for all s except s = 1. The extended function is equal to zero at all negative even integers. These are called the trivial zeros . The Hypothesis: If ζ(s) = 0, and s is not a trivial zero, then a = 1 / 2. Elementary number-theoretic characterization due to Je²rey Lagarias: Let H n = 1 + 1 / 2 + ···+ 1 /n . Then RH is equivalent to: For every number n the sum of the divisors of n is less than or equal to H n + e H n ln
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: H n . • It is the other half of Hilbert’s 8th Problem. • Solver will get million dollar prize from Clay Mathematics Institute. Primality Testing • The obvious brute force way to check whether a number n is prime takes √ n divisions. Thus, if n has k digits there will be about 10 k/ 2 divisions. • In 2002, 3 Indian mathematicians discovered the Agrawal-Kayal-Saxena primality test or AKS primality test that requires no more than Ck 12 + ε operations to check for primality....
View Full Document

## This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

Ask a homework question - tutors are online