Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasion of his retirement Andrew Odlyzko School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko December 2, 2011 Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 1 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 2 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 3 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 4 / 15
Riemann explicit connection between primes and zeros: Li ( x ρ ) + O ( x 1 / 2 log x ) � π ( x ) = Li ( x ) − ρ where ρ runs over the nontrivial zeros of ζ ( s ), and similar formulas for other functions, such as M ( x ) = � n ≤ x µ ( n ) Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 5 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 6 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 7 / 15
Blocks of zeros at large heights: 10 23 O. 10 24 Gourdon 10 32 Bober and Hiary Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 8 / 15
Algorithms for verifying RH for first n zeros: n 2+ o (1) Euler–Maclaurin 3 2 + o (1) Riemann–Siegel n n 1+ o (1) O.–Sch¨ onhage Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 9 / 15
Algorithms for single values of zeta at height t : t 1+ o (1) Euler–Maclaurin t 1 / 2+ o (1) Riemann–Siegel t 3 / 8+ o (1) Sch¨ onhage t 1 / 3+ o (1) Heath-Brown t 4 / 13+ o (1) Hiary Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 10 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 11 / 15
Critical strip: Nearest neighbor spacings: Empirical minus expected 0.010 0.005 density difference 0.0 -0.005 -0.010 0.0 0.5 1.0 1.5 2.0 2.5 3.0 normalized spacing Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 12 / 15
Beware the law of small numbers (especially in number theory): N ( t ) = 1 + 1 π θ ( t ) + S ( t ) where θ ( t ) is a smooth function, and S ( t ) is small: | S ( t ) | = O (log t ) � t 1 S ( u ) 2 du ∼ c log log t t 10 | S ( t ) | < 1 for t < 280 | S ( t ) | < 2 for t < 6 . 8 × 10 6 largest observed value of | S ( t ) | only a bit over 3 Andrew Odlyzko ( School of Mathematics University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 13 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 14 / 15
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 University of Minnesota odlyzko@umn.edu http://www.dtc.umn.edu/ ∼ odlyzko ) Computation and the Riemann Hypothesis: Dedicated to Herman te Riele on the occasio December 2, 2011 15 / 15
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