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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
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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....
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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.

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