Turing and the Riemann zeta function Andrew Odlyzko School of Mathematics University of Minnesota
[email protected] http://www.dtc.umn.edu/∼odlyzko May 11, 2012 Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the
[email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 1/23 The mysteries of prime numbers: Mathematicians have tried in vain to discover some order in the sequence of prime numbers but we have every reason to believe that there are some mysteries which the human mind will never penetrate. To convince ourselves, we have only to cast a glance at tables of primes (which some have constructed to values beyond 100,000) and we should perceive that there reigns neither order nor rule. Euler, 1751 Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the
[email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 2/23 Riemann, 1850s: ∞ s 1 s C s ζ( ) = ns , ∈ , Re( ) > 1 . Xn=1 Showed ζ(s) can be continued analytically to C \{1} and has a first order pole at s = 1 with residue 1. If s ξ(s) = π− /2Γ(s/2)ζ(s) , then (functional equation) ξ(s) = ξ(1 − s) . Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the
[email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 3/23 Critical strip: complex s−plane critical strip critical line 0 1/2 1 Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the
[email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 4/23 Riemann explicit