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Computation and the Riemann Hypothesis: Dedicated to Herman Te Riele on the Occasion of His Retirement

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Computation and the Riemann Hypothesis: Dedicated to Herman Te Riele on the Occasion of His Retirement Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasion of his retirement Andrew Odlyzko School of Mathematics University of Minnesota [email protected] http://www.dtc.umn.edu/∼odlyzko December 2, 2011 Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 1/15 occasion 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 UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 2/15 occasion Herman te Riele’s contributions towards elucidating some of the mysteries of primes: numerical verifications of Riemann Hypothesis (RH) improved bounds in counterexamples to the π(x) < Li(x) conjecture disproof and improved lower bounds for Mertens conjecture improved bounds for de Bruijn – Newman constant ... Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 3/15 occasion Riemann, 1859: almost all nontrivial zeros of the zeta function are on the critical line (positive assertion, no hint of proof) it is likely that all such zeros are on the critical line (now called the Riemann Hypothesis, RH) (ambiguous: cites computations of Gauss and others, not clear how strongly he believed in it) π(x) < Li(x) Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 4/15 occasion Riemann explicit connection between primes and zeros: π(x) = Li(x) − Li(x ρ) + O(x 1/2 log x) Xρ where ρ runs over the nontrivial zeros of ζ(s), and similar formulas for other functions, such as M x n ( ) = n≤x µ( ) P Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 5/15 occasion Turing and te Riele: Amusing coincidences: this lecture in the Turing Room 2012 is the Turing centenary, as well as te Riele’s 65-th birthday and retirement Turing’s and te Riele’s researches on the zeta function revolved around numerical verifications of RH and use of zeta zeros to study functions such as π(x) − Li(x) Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 6/15 occasion Numerical verifications of RH for first n zeros: Riemann 1859 ? ... Hutchinson 1925 138 Titchmarsh et al. 1935/6 1,041 Turing 1952 1,054 ... te Riele et al. 1986 1,500,000,000 ... Gourdon 2004 10,000,000,000,000 Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 7/15 occasion Blocks of zeros at large heights: O. 1023 Gourdon 1024 Bober and Hiary 1032 Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 8/15 occasion Algorithms for verifying RH for first n zeros: Euler–Maclaurin n2+o(1) 3 +o(1) Riemann–Siegel n 2 O.–Sch¨onhage n1+o(1) Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 9/15 occasion Algorithms for single values of zeta at height t: Euler–Maclaurin t1+o(1) Riemann–Siegel t1/2+o(1) Sch¨onhage t3/8+o(1) Heath-Brown t1/3+o(1) Hiary t4/13+o(1) Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 10/15 occasion The validity of RH: Turing’s skepticism grew with time many other famous number theorists were disbelievers (e.g., Littlewood) skepticism appears to have diminished, because of computations and various heuristics (many assisted by computations) but ... Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 11/15 occasion Critical strip: Nearest neighbor spacings: Empirical minus expected density difference -0.0100.0 -0.005 0.5 0.0 0.005 1.0 0.010 1.5 2.0 2.5 3.0 normalized spacing Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 12/15 occasion Beware the law of small numbers (especially in number theory): N t 1 t S t ( ) = 1 + π θ( ) + ( ) where θ(t) is a smooth function, and S(t) is small: |S(t)| = O(log t) t 1 S u 2du c t t ( ) ∼ log log Z10 |S(t)| < 1 for t < 280 |S(t)| < 2 for t < 6.8 × 106 largest observed value of |S(t)| only a bit over 3 Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 13/15 occasion Conclusion (from a sign in a computing support office): We are sorry that we have not been able to solve all of your problems, and we realize that you are about as confused now as when you came to us for help. However, we hope that you are now confused on a higher level of understanding than before. Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 14/15 occasion More information, papers and data sets: http://www.dtc.umn.edu/∼odlyzko/ in particular, recent paper with Dennis Hejhal, “Alan Turing and the Riemann zeta function” Andrew Odlyzko ( School of Mathematics UniversityComputation of Minnesota and [email protected] Riemann Hypothesis: http://www.dtc.umn.edu/ Dedicated toDecember2,2011 Herman te∼ Rieodlyzkole on) the 15/15 occasion.
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