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

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 ...

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)

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

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)

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

O. 1023

Gourdon 1024

Bober and Hiary 1032

Euler–Maclaurin n2+o(1)

3 +o(1) Riemann–Siegel n 2

O.–Sch¨onhage n1+o(1)

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)

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 ...

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.

http://www.dtc.umn.edu/∼odlyzko/

in particular, recent paper with Dennis Hejhal, “Alan Turing and the Riemann zeta function”