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 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)

Hence size of π(x) − Li(x) depends on location of zeros

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 5/23 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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 6/23 Turing, Cambridge in the 1930s, and zeta function:

Hardy, Littlewood, Ingham, ..., and Skewes

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 7/23 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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 8/23 Turing zeta machine design, 1939:

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 9/23 Turing and the zeta function:

persistent interest starting as undergraduate design of special zeta function machine, 1939 first computation of zeta zeros on electronic digital computer, 1950 new method for computing the zeta function studies of π(x) − Li(x) (with Skewes) “Turing method” for easy numerical verification of RH skepticism about validity of RH

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 10/23 Numerical verifications of RH for first n zeros:

Riemann 1859 ? ... Hutchinson 1925 138 Titchmarsh et al. 1935/6 1,041 Turing 1950 (published 1953) 1,054 ... te Riele et al. 1986 1,500,000,000 ... Gourdon 2004 10,000,000,000,000

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 11/23 Interest in zeros of zeta function:

numerical verification of RH

π(x) − Li(x) and related functions

(more recently) distribution questions related to hypothetical random matrix connections

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 12/23 Riemann and Ingham:

Riemann:

π(x) = Li(x) − Li(x ρ) + O(x 1/2 log x) Xρ

Ingham:

certain averages of (π(x) − Li(x)) = nice sums over ρ

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 13/23 Riemann and Ingham:

Littlewood (1914): Riemann “conjecture” that π(x) < Li(x) false

Skewes (1933, assuming RH): first counterexample 1034 < 1010

Ingham approach (with extensive computations): < 10317

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 14/23 “Turing method” for numerical verification of RH:

ξ(s) = ξ(1 − s) means ξ(s) takes real values on the critical line hence sign changes of ξ(s) correspond to zeros of ζ(s) that are right on the critical line need to prove all zeros have been found traditionally done using the principle of the argument “Turing method” provides neat approach that uses just values of ξ(s) on the critical line

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 15/23 “Turing method:”

N(t) = number of zeros ρ with 0 < Im(ρ) < t

1 N(t) = 1 + θ(t) + S(t) π where θ(t) is a smooth function, and

|S(t)| = O(log t)

t1 S(u)du = O(log t ) Z 1 t0

Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 16/23 Typical behavior of S(t):

S(t) around zero number 10^23 S(t) −1.0 −0.5 0.0 0.5 1.0

1 2 3 4

Gram point scale Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 17/23 Turing:

Turing method S(t) −4 −2 0 2

5 10 15 20

Gram point scale Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 18/23 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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 19/23 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 log log t t 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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 20/23 extreme among 106 zeros near zero 1023:

extreme S(t) around zero number 10^23 S(t) −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0

5 10 15 20 25

Gram point scale Andrew Odlyzko ( School of Mathematics UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 21/23 Zeta zeros and random matrices:

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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 22/23 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 UniversityTuring of Minnesotaand the [email protected] zeta function http://www.dtc.umn.edu/May11,2012∼odlyzko ) 23/23