Numerology or Number Theory?

Numerology or Number Theory?

Nathan Ryan

September 27th, 2012

Nathan Ryan Numerology or Number Theory?

If you want the number you can find it everywhere. 216 stepsfrom your street corner to your front door, 216 seconds youspend in the elevator, whatever. When your mind becomesobsessed you filter everything else out and find that thingeverywhere. You’ve chosen 216 and you’ll find it everywhere innature. But, Max, as soon as you discard scientific rigour, youare no longer a mathematician, you’re a numerologist.


Numerology has unquestionably stimulated investigations innumber theory and bequeathed to us some most difficultproblems Ore, Number Theory and its History

I Perfect numbersI Amicable numbersI Sociable numbers


Six is a perfect number in itself, and not because God createdall things in six days; rather the converse is true God createdall things in six days because the number is perfect.

St. Augustine, ca. 1600 years ago in City of God.

Our ancestor Jacob prepared his present in a wise way. Thisnumber 220 is a hidden secret, being one of a pair of numberssuch that the parts of it are equal to the other one 284, andconversely. And Jacob had this in mind; this has been tried bythe ancients in securing the love of kings and dignitaries.

Abraham Azulai, ca. 500 years ago.


If as many numbers as we please beginning from a unit be setout continuously in double proportion, until the sum of allbecomes a prime, and if the sum multiplied into the last makesome number, the product will be perfect.

Euclid, ca. 2300 years ago:I That is, if 1 + 2 + · · ·+ 2k = 2k+1 − 1 is prime, then

2k (2k+1 − 1) is perfect.


First series of experiments:I Are there odd perfect numbers?I Are there amicable pairs coprime to 6? 30? 210?I Are there sociable numbers of period 3? 4?


By experimental mathematics one typically means1. gaining insight and intuition;2. visualizing math principles;3. discovering new relationships;4. testing and especially falsifying conjectures;5. exploring a possible result to see if it merits formal proof;6. suggesting approaches for formal proof;7. computing replacing lengthy hand derivations;8. confirming analytically derived results.

And by experimental number theory I mean these ideas appliedin number theoretic settings.


What is numerology?I Historically numerology is any study of the purported

divine, mystical or other special relationship between acount or measurement and observed or perceived events.

I To an experimental mathematician its noticing a fewinstances of a (potential) larger pattern and making aninference from that. Then the inference is tested.


The (in)famous Riemann zeta function:I The function

ζ(s) =∞∑

n=1

1ns

converges when Res > 1 and is never 0 in this region.I If we set

ξ(s) = 12π−s/2s(s − 1)Γ(s/2)ζ(s),

then this function satisfies a functional equationξ(s) = ξ(1− s). To cancel out the poles of Γ(s) at thenegative integers, we see ζ(s) has zeros at the negativeeven integers −2,−4, . . . . These are the trivial zeros.

I The Riemann hypothesis says

All the nontrivial zeros of ζ(s) lie on the line Res = 12


Why ζ?I Special values: the probability that as x goes to infinity N

positive integers less than x chosen uniformly at randomwill be relatively prime approaches ζ(N);

I ζ(2) = π2/6, ζ(3) is known to be irrational but is ittrascendental?

Why RH?I No zeros along the Res = 1 line is equivalent to the Prime

Number Theorem. (Wiener, 1951)I The Riemann hypothesis is equivalent to the “best

possible” bound for the error of the prime number theorem.(van Koch, 1901)


Year number of zeros Authors1859ish 3 Riemann

1903 15 Gram1914 79 Backlund1925 138 Hutchinson1935 1041 Titchmarsh1953 1104 Turing1956 15000 Lehmer1956 25000 Lehmer1958 35337 Meller1966 250000 Lehman1968 3500000 Rosser-Yohe-Schoenfeld1977 40000000 Brent1979 81000001 Brent1982 200000001 Brent-van de Lune-te Riele-Winter1983 300000001 van de Lune-te Riele1986 1500000001 van de Lune-te Riele-Winter2004 900000000000 Wedeniwski2004 10000000000000 Gourdon-Demichel


Why bother?I Mallory: “Because it’s there”I Polya: “Intuition comes to us much earlier and with much

less outside influence than formal arguments.”I Galileo: “All truths are easy to understand once they are

discovered; the point is to discover them.”


What are elliptic curves?I An elliptic curve is a smooth, projective algebraic curve of

genus one, on which there is a specified point O.

Theorem (Mordell)The group of rational points E(Q) on an elliptic curve is afinitely generated abelian group.


Conjecture (Birch and Swinnerton-Dyer)Let r be the rank of E(Q) and let Np = E(Fp). Then∏

p≤x

Np

p≈ C log(x)r as x →∞.

They only went up to p ≤ 1000!!!


Elliptic curve L-functions:I Define ap = p + 1− Np and set for almost all p,

Lp(T ) = 1− apT + pT 2 (for exposition we omit those p forwhich the curve E has not-good reduction).

I Then we define the series

L(E , s) =∏

p

1Lp(p−s)

.

I The coefficients of this L-series are multiplicative and hasan analytic continuation to all of C. If we define

Λ(E , s) = (NE/Q)s/2(2π)−sΓ(s)L(E , s)

thenΛ(E , s) = ωΛ(E , 2− s).


Conjecture (BSD (t-shirt version))Let r be the rank of the group E(Q) and let

ran := ords=1 L(E , s).

Then r = ran.Evidence:

I (Cremona 2011) There are 614308 isogeny class of ellipticcurves with conductor NE/Q ≤ 140000. All have ran ≤ 3and in every case ran = r .


Conjecture (BSD (awesome version))Let everything you see be an important invariant of an ellipticcurve E:

L(r)(E , 1)

r !=

#Sha(E)ΩERE∏

p|n cp

(#Etor)2 .

I Cremona has also checked this for all his curves (modulothe huge fact that almost nothing is known about Sha).


I Numerology has lead to a lot of interesting number theorybut this is still not experimental number theory

I Why not?I a matter of scaleI a matter of rigorI a matter of the tools being usedI a matter of its “generalizability”


Three examples from my work:I Bocherer’s ConjectureI Harder’s ConjectureI Rankin-Convolution of two Siegel modular forms


Classical Modular FormsI Let H be the complex upper halfplane and k ∈ Z. Let

γ =(

a bc d

)∈ SL(2, Z) = Γ act on z ∈ H via fractional linear

transformations: γ(z) = az+bcz+d . Then a modular form of

weight k is a function f : H → C that is holomorphic on H,and holomorphic at ∞ so that

f (γ(z)) = (cz + d)k f (z), for all γ ∈ Γ.

I Note that f (z) = f (z + 1) and thus f has a Fourierexpansion (here q = exp(2πiz))

f (z) =∑n≥0

af (n)qn.

I Note that MFs have: (1) a group, (2) a halfplane, (3)holomorphy conditions, and (4) a functional equation interms of the action of the group on the halfplane.


Siegel Modular FormsI Let Hg = Z = X + iY ∈ M(g, C) : Z t = Z , Y > 0 and

k ∈ Z. Let γ =(

A BC D

)∈ Sp(2g, Z) = Γ act on Z ∈ Hg via

fractional linear transformations:γ(Z ) = (AZ + B)(CZ + D)−1. Then a Siegel modular formof weight k and genus g is a function F : Hg → C that isholomorphic on Hg , and holomorphic at ∞ so that

F (γ(Z )) = det(CZ + D)kF (Z ), for all γ ∈ Γ.

I Note that F (Z ) = F (Z + Ig) and thus F has a Fourierexpansion (here q is suitably defined)

F (Z ) =∑T≥0

aF (T )qT .

where the T are positive semidefinite quadratic forms.


Examples of Siegel modular forms:I classical modular forms are Siegel of genus 1;I given integers g and ` and a positive definite

integer-valued quadratic form Q in ` variables,

θ(g)Q (Z ) =

∑s∈M(g×`,Z)

e2πiTr(Qst Zs);

is a Siegel modular form of genus g and weight `/2;I given a classical modular form of weight 2k − 2 one can

(Saito-Kurokawa) “lift” it to an arithmeticallyindistinguishable Siegel modular form of weight k .


Why do modular forms belong to arithmetic?I theta series like the aboveI spaces of modular forms of a fixed weight are finite

dimensional and there is a commuting family of normaloperators T (1), T (2), T (3), . . . acting on these spaces. Byeasy linear algebra there is a basis of eigenforms that areeigenforms for all of the operators T (n). We write

f |T (n) = λf (n)f

when f is an eigenform.I These are the Hecke eigenforms and their coefficients are

of arithmetic interest.


Let F be a Hecke eigenform of genus 2 and weight k . Then wecan define

L(s, F ) =∏

p

1Lp(p−s)

=∞∑

n=1

bnn−s

where

Lp(T ) = 1−λF (p)T+(λF (p)2−λF (p2)−p2k−4)T 2−λF (p)p2k−3T 3+p4k−6T 4.

and bn is the result of carrying out the multiplication.I This is an L-function in that it has an analytic continuation,

a functional equation (though I typically normalize so thatthe functional equation is s 7→ 1− s), has multiplicativecoefficients bn, etc.

I Consider a quadratic Dirichlet character χD =(

D·

). Then

one can “twist” the original L-function to get

L(s, F ⊗ χD) =∞∑

n=1

bnχD(n)n−s.


L functions and central valuesI vanishings are of particular interest (BSD says a vanishing

implies E(Q) is infinite); vanishings of families of twistsespecially (Random matrix theorem predictions);

I want to do this with SMFs but how do we knowL(s, 1/2, F ⊗ χD) is 0? It’s a floating point number on acomputer.


I We need a discretization: Bocherer’s Conjecture (asinvestigated by myself and Gonzalo Tornarıa) relates thecentral values L(1/2, F ⊗ χD) to an average of Fouriercoefficients of F indexed by quadratic forms of discriminantD.

I In particular,

B`,F (D)2 = α`D kF L(1/2, F ⊗ χ`) L(F , 1/2, F ⊗ χD) |D`|k−1

for some positive constant kF independent of ` and D.I made use of some “magical” numerical integration

techniques and convergence accelerators.


Consider the classical modular form

∆(q) = q − 24q2 + 252q3 − 1472q4 + · · · =∞∑

n=1

τ(n)qn.

I Ramanujan discovered a number of congruences involvingthe coefficients of ∆: e.g.,

τ(p) ≡ p11 + 1 (mod 691) for all primes p.

I This is part of a more general phenomenon: if a “large”prime ` ≥ k − 1 divides the numerator of the zeta-valueζ(−k + 1), then the constant term of the Eisenstein seriesEk is zero modulo `. This can be interpreted to say thatthere is a congruence mod ` between this Eisensteinseries and some cuspidal eigenform of weight k .


Harder’s Conjecture:I Let f be a classical Hecke eigenform of weight r and let

t ∈ Z. If a “large” prime ` divides the algebraic part of thespecial value of L(f , t), then there exists a Siegel modularform F of a weight determined by t and r so that there is acongruence involved the Hecke eigenvalues of F and fmodulo `.

I In work with Alex Ghitza and David Sulon we verified thisconjecture in a dozen or so cases.

I In order to find the algebraic part of an a special value wemade heavy use of the “magical” LLL algorithm.


Consider two Siegel Hecke eigenforms F , G of genus 2 andweight k . One can write their Fourier expansions as

F (Z ) =∞∑

N=1

FNqN , and G(Z ) =∞∑

N=1

GNqN .

I one can form the infinite sum

DF ,G(s) := ζ(2s − 2k + 4)∑N≥1

〈FN , GN〉N−s

where 〈·, ·〉 is the so-called Petersson inner-product.I when F is a Saito-Kurokawa lift the series DF ,F is just the

L-function of F we defined above.


What about DF ,F (s) when F is not a lift?I Skoruppa, Stromberg and I prove numerically that in this

case F is not even an L-function since the coefficients ofDF ,F (s) aren’t multiplicative;

I We used the “magic” of interval arithmetic to be provablysure that our computations were correct.


To conclude I’d like to say that much of my research follows aTheory developing algorithms, knowing what questions to

askComputation implementing the algorithms, carrying out large

scale computations efficientlyTheory understanding what we see in our computations,

using the data to rework our conjectures andhypotheses, apply the data to some largerframework, ask more questions

Cycle rinse and repeat


Numerology or Number Theory?

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