Andrew Odlyzko

Mathematical computation, mathematical insight, and mathematical truth

Andrew Odlyzko

School of Mathematics University of Minnesota [email protected] http://www.dtc.umn.edu/∼odlyzko

July 31, 2014

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 1 / 22 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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 2 / 22 Computing in mathematics:

long history

growing role

“You waste that which is plentiful” (George Gilder)

needed to cope with greater complexity and sophistication

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 3 / 22 Computing in mathematics:

Gaining insight and intuition, or just knowledge. Discovering new facts, patterns, and relationships. Graphing to expose mathematical facts, structures, or principles. Rigorously testing and especially falsifying conjectures. Exploring a possible result to see if it merits formal proof. Suggesting approaches for formal proof. Computing replacing lengthy hand derivations. Conﬁrming analytically derived results. Borwein and Bailey, Mathematics by Experiment Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 4 / 22 1975 MIT PhD thesis:

bounds for discriminants of number ﬁelds

key element of published formal proof: nonnegativity of rational function

160 98 u2 − 53 1536u2 − 2048 + ··· + − 3(u2 + 4) 25 u4 − 90u2 + 2809 (u2 + 4)3

linear programming and symbolic algebra packages in the background, everything veriﬁable with paper and pencil

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 5 / 22 Empirical results based on heuristics and theory:

sparse linear systems arising in integer factorization and discrete logs: n2 or n3 ?

Monte Carlo simulations (1983–84) leading to “intelligent gaussian elimination” (later “structured gaussian elimination” and now “ﬁltering”)

stimulated (1984): conjugate gradient for finite fields Lanczos for finite fields Wiedemann algorithm

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 6 / 22 Mertens Conjecture disproof (with te Riele):

ρ x x M(x)= n µ(n)= + ··· =1 ρζ′(ρ) P Xρ

Ingham innovation:

weighted average of M(x) = more tractable sum over ρ

need to ﬁnd x that makes sum large (L3 algorithm, but irrelevant how it is done) and prove it works (rigor !!!)

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 7 / 22 Critical strip:

complex s−plane

critical strip

critical line

0 1/2 1

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 8 / 22 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)

unknown until Siegel discovery in Riemann’s private papers: extensive hand computations of zeros with a very eﬃcient algorithm

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 9 / 22 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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 10 / 22 Zeta zeros and random matrices:

Nearest neighbor spacings among first 1,000,000 zeros density 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

normalized spacing Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 11 / 22 Zeta zeros and random matrices:

Nearest neighbor spacings among 1,041,600 zeros near the 2*10^20-th zero density 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

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

T 1 1 2k Ik (T ) = |ζ( + it)| dt T 2 Z0 Known: I1(T ) ∼ log T as T →∞

I T 1 T 4 T 2( ) ∼ 2π2 (log ) as →∞ Conjectures: for all k ≥ 1,

k2 (i) Ik (T ) ∼ ck (log T ) as T →∞

G 2(k+1) (ii) ck = a(k) G(2k+1)

∞ 2 2 Γ(m + k) − a(k) = (1 − 1/p)k p m (m!)2Γ(k)2 p m=0 ! Y X

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 14 / 22 Moment computations:

Computations:

T +H 1 1 2k Jk (T , H) = |ζ( + it)| dt H 2 ZT 23 6 Values of Jk (T , H) for T near 10 -rd zero: H interval of length approx. 10 zeros

k =1 k =2 k =3 50.11 34.89 × 104 18.74 × 109 49.94 38.02 × 104 27.78 × 109 50.07 38.47 × 104 37.69 × 109 49.64 27.86 × 104 8.47 × 109 50.27 53.03 × 104 75.77 × 109 50.11 41.01 × 104 46.18 × 109 50.23 36.59 × 104 19.43 × 109 50.52 38.74 × 104 24.57 × 109 Consecutive blocks of approx. 106 zeros.

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 15 / 22 Numerical evidence, “genericity,” and truth:

empirically, primes and zeros exhibit random behavior

random models produce nice results, and are often testable

BUT primes show deviations from random predictions at small scale (Maier’s results) and on very large scale (on log(x) scale, harmonic series with frequencies determined by zeta zeros)

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 16 / 22 Zeros of the zeta function:

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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 17 / 22 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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 18 / 22 Beware the law of small numbers (especially in number theory):

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

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

1 t S(u)2du ∼ c log log t t Z10

|S(t)| < 1 for t < 280

|S(t)| < 2 for t < 6.8 × 106 Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 19 / 22 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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 20 / 22 Beware the law of small numbers (cont’d):

we have not seen “typical” behavior of S(t)

we are not likely to ever see “typical” behavior of S(t) by direct computation

counterexamples to the Riemann Hypothesis (if they exist) are likely to be far beyond reach of any direct computational methods we can even imagine

Andrew Odlyzko ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 21 / 22 Sign in program counseling oﬃce, 1970s:

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 ( SchoolMathematical of Mathematics University of Minnesota computation,[email protected] http://www.dtc.umn.edu/ 31, 2014 mathematical 22 / 22