Some problems of analytic number theory for

polynomials over a finite field

Aisenstadt lecture I

CRM, September 2014

Zeev Rudnick, Tel Aviv University

Joint work with J. Keating

Analogy: Fq[t] vs. integers

Qualitative analogues:

integers Z ↔ polynomials Fq[t]

primes p ↔ irreducible polynomial P(t) (“prime”)

positive p>0 ↔ monic polynomial P(t)=td+….

In both cases we have the Fundamental Theorem of Arithmetic – unique

factorization into primes (prime polynomials).

Quantitative analogues

Prime Number Theorem ↔ Prime Polynomial Theorem

Fq=finite field with q elements

Fq[t]= polynomials f(t) = a0+a1t+a2t2+....+adt

d , with coefficients ai in Fq

Notation: f(x) ~ g(x) if limx→∞ f(x)/g(x)=1

The Prime Number Theorem

π(x) = number of primes p ≤ x

Von Mangoldt function

otherwise,0

,log)(

kpnpn

xnxxn

~)(:)(

PNT

2

2 logloglog:)(Li~)(

x

x

x

x

t

dtxx

x

PNT:

Counting prime powers with weights:

The Riemann Hypothesis:

)()(Li)( )1(2/1 oxOxx )()( )1(2/1 oxOxx

The Prime Polynomial Theorem

nPPnq )deg(e,irreducibl monic#:

)2/

(n

nqO

n

nqnq

x

x

t

dtxx

x

log~

log:)(Li~)(

2

Compare with PNT

The Prime Polynomial Theorem

)()(Li)(:RH )1(2/1 oxOxx

#{positive integers<x} = x↔qn =#{monic pols of degree n}

Counting primes in short intervals

| | / 2

( ; ) : ( )n x H

x H n

The Prime Number Theorem implies ψ(x,H)~H for H≈x .

The Riemann Hypothesis implies ψ(x,H)~H as long as H >x1/2+o(1) .

Hoheisel (1930): H>x0.99997 ……… Heath-Brown (1988) H>X7/12-o(1)

Conjecture: ψ(x,H)~H holds for all H> xε .

Maier (1985): False for H=(log X)N !

Selberg (1943): Assuming RH, for almost all x we have ψ(x,H)~H as long as H >>(log X)2

otherwise,0

,log)(

kpnpn

counts prime powers in interval [x-H/2, x+H/2] of length H around x

Question (regularity in the distribution of primes):

When can we find primes (+ asymptotic formula) in short intervals?

Variance of primes in short intervals

2

2

1

1( ; ) ~ (log lo ( log 2 )g )

X

x H H dx H X HX

X H X

Goldston & Montgomery (1987) studied the variance of ψ(x,H). They conjectured that

Goldston & Montgomery showed that their conjecture follows from: RH +

“Strong Pair Correlation conjecture” on the zeros

Montgomery & Soundararajan 2002

Pair correlation of Riemann zeros

No progress has been made towards the Riemann Hypothesis

since the 19-th century. A fundamental discovery was made in

1972 by Montgomery, who found that nearby zeros tend to

“repel” each other :

Montgomery studied the “pair correlation function”, which

measures the repulsion between pairs of zeros ½+iγ, ½ +iγ’

as measured in the scale of their mean spacing.

Montgomery’s pair correlation conjecture:

Dyson: this is the same as that for GUE in Random Matrix Theory !!

repulsion

a

a

dxx

xa

T

T

2sin

1~|'|2

log,':)',(#

~#

1

Prime polynomials in “short intervals”

0

0

|| ||

( ; ) : ( )hf f q

f h f

If deg f0 >h then # {f: ||f-f0 ||≤ qh } =qh+1 =:H

Counting prime polynomials in short interval around f0

A short interval around f0 : {f: ||f-f0 ||≤ qh }= {𝑓: deg(𝑓 − 𝑓0) ≤ ℎ}

Norm of a polynomial: ||f||:=#Fq[t]/(f)=qdeg(f)

( analogy: for 0≠nεZ, |n|=#Z/nZ )

Von Mangoldt function deg , , "prime"

( )0, otherwise

kP f cP Pf

Asymptotics for polynomials in

short intervals

Bank, Bary-Soroker and Rosenzweig (2013): as q→∞, if 2<h<deg(f0)

)()()(:);( 2/11

||

0

0q

HOHqOqfhf hh

qff h

This is a function-field analogue of the conjecture that ψ(x,H)~H for all H> xε .

Statistics of Ψ(f;h)

1

degmonic

1( ; ) h

nf n

f

f h qq

Expected value:

Goal: statistics as we vary f0 over all polynomials of fixed degree n, n>h.

We wish to understand the variance and compare with the Goldston-Montgomery conjecture

Variance of prime polynomials in

short intervals

2

2

log ( l1

( ; ) ~ ( ))g oo g 2l

X

Xx H H d Hx HX

Compare with the Goldston & Montgomery conjecture:

Dictionary: X↔qn , H ↔ qh+1 , log ↔ deg Xε<H<X1-ε

2 2

deg ( 2)

1( ; ) ~ trace n

nf n U n h

f h H H U dUq

2( )H n h

Thm (JP Keating & ZR 2012) : Fix n, 0<h<n-4. Then as q→∞,

0),,min()(trace2

)(

nNndUUNU

nMatrix integral

Variance - overview

dUUqqhfq

hnU

nh

fnf

h

n

)2(

21

monicdeg

21 trace~);(

1)2(1 hnqh

(JP Keating & ZR 2012) : as q→∞, the variance is given by a matrix integral:

The argument consists of two steps:

• Reduction to a problem about zeros of certain L-functions

• Equidistribution of Frobenius matrices

Dirichlet characters & L-functions for Fq[t] For polynomial Q(t) we get congruence relation A=B mod Q if A-B=CQ

A Dirichlet character modulo Q is a function χ : Fq[t] →C× satisfying

• χ(AB)= χ(A) χ(B)

• χ(A+CQ) = χ(A)

• χ(1)=1

• χ is “even” if it is trivial on scalars Fq

The L-function associated to χ :

for Re(s)>1

1

primemonic ||||

)(1

||||

)(:),(

Ps

fs P

P

f

fsL

If χ is “even” then there is a trivial zero at s=0

If χ is nontrivial (“primitive”) then

• L(s, χ) is a polynomial in u:=q-s of degree deg(Q)-1

• functional equation L(s, χ) ↔ L(1-s, χ-1 )

• RH (Weil, 1940’s): All non-trivial zeros lie on Re(s)=1/2

The Frobenius conjugacy class

1

2

.

.

N

i

i

i

e

e

e

Θχ = unitary nxn matrix, n=deg Q-2,

called the “unitarized Frobenius matrix”

)det()1(),( 2/1

qqIqsL ss

If the character χ is even and primitive mod Q, then can write

N. Katz (2012): As χ varies over all “even” characters mod tN, Θχ become equidistributed

in the projective unitary group PU(N-2).

lim𝑞→∞

1

#{𝜒} 𝐹(Θ 𝜒 )

𝜒𝑚𝑜𝑑𝑡𝑁evenprimitive

= 𝐹 𝑈 𝑑𝑈

𝑃𝑈(𝑁−2)

i.e. for any nice function F on PU(N-2)# ,

Variance & Equidistribution

2

1

mod"even"

1 1var ( , ) ~ tr

# n h

n

h

t

hq

Step 1 (spectral interpretation) connect to zeros of L-functions: as q→∞

2 2

mod ( 2)"even"

1lim tr tr 2

# n h

n n

qt PU n h

U dU n h

Step 2: Apply equidistribution

monicdeg

2);(

1:);(var

fnf

nHhf

qh

Computing the variance

QED )2(~);(var hnHh

Primes in long arithmetic progressions

Friedlander – Goldston (1996) conjecture that G(X,Q)~X log Q for X1/2+ε<Q<X,

and that

Hooley (1974) conjectured that (in unspecified ranges)

Variance

Primes in arithmetic progressions:

QAnXn

nAQX

mod

)(:),;(

2

)(),;(:),(

1),(mod

QAQA Q

XAQXQXG

QXQXG log~),(

12/1

|

),(1

log2loglog),( XQXXo

p

pXQXQXG

Qp

“It may well be that these also hold for smaller Q, but below Q=X1/2 we are

somewhat skeptical.”

Prime polynomials in AP’s

Counting prime polynomials in arithmetic progressions:

Variance:

Theorem (Keating & ZR, 2012): Given a sequence of finite fields Fq, and squarefree QεFq[t],

n≥deg Q-1, then as q→∞

Comparison with Hooley, Friedlander-Goldston – hence conjecture should be OK for Q>Xo(1)

QAfnf

fAQn

moddeg

)(:),;(2

)(),;(:);(

1),(mod

QAQA

n

Q

qAQnQnG

dUUq

QnG

QU

n

n )1(deg

2

)trace(~);(

qQqQnG n ),1(deg~);(

QXQXG log~),(

Fiorilli (2013) suggests G(X,Q)~X log Q continues to hold for Q>(loglog X)1+o(1)

Sums of Möbius in short intervals

2||

)(:);(H

xn

nHxM

Randomness of Möbius is manifested by amount of cancellation in sum over short intervals

Good & Churchhouse (1968): For x random and small H, M(x;H)/√H is

asymptotically normal with mean zero and variance 1/ζ(2)

1

22

)(,)2(

~);(1

XXHHXH

dxHxMX

X

X

has a normal distribution )2(/

);(

H

HxM

- expect M(x;H) to behave as though μ was random ±1 on square-free integers

Good & Churchhouse 1968

Don Russell, Robbie, Bob Churchhouse,

ICL Engineers 19.09.67

Good & Churchhouse 1968

Don Russell, Robbie, Bob Churchhouse,

ICL Engineers 19.09.67

Mobius values over “short intervals” for Fq[t]

hqAf

nf

fhAM

||

deg

)(:);(

J. Keating & ZR (2014) Assume h<n-4. As q→∞ the variance of M(∙,h) is

)2(

2

deg

Symtrace~2);(1

hnUnAn

dUUnHhAMq

If deg f0 >h then # {f: ||f-f0 ||≤ qh } =qh+1 =: H

A short interval around 𝐴 ∈ 𝑭𝑞[𝑡] : {f: ||f-A ||≤ qh }= {𝑓: deg(𝑓 − 𝐴) ≤ ℎ}

Norm of a polynomial: ||f||:=# 𝑭𝑞 𝑡 /(𝑓)= qdeg(f)

Comparison with Good-Churchhouse

)2(

2

deg

Symtrace~2);(1

hnUnAn

dUUnHhAMq

The argument consists of two steps:

• Reduction to a problem about zeros of certain L-functions

• Equidistribution of Frobenius matrices associated to “even” Dirichlet

characters of (𝐹 𝑡 /(𝑡𝑁))× (Katz 2013).

J. Keating & ZR (2014): Assume h<n-4. As q→∞ the variance of M(∙,h) is

1

22

)(,)2(

~);(1

XXHHXH

dxHxMX

X

X

“Good-Churchhouse conjecture”

1||f||

1:)2(

monic

[x]F2

qf

q

q

)(

2n 1Symtrace

NU

dUU

)2(~

q

H

Value distribution for Fq[t] Plots of sums Mobius values M(A;h) over “short intervals” for Fq[t]

n=4, h=0 n=6, h=0

hqAf

nf

fhAM

||

deg

)(:);(

Summary

We studied function field analogues of some open problems in analytic number

theory.

One can obtain results which:

• go much further than one can hope to do in the number field setting in the

foreseeable future.

• shed light on several conjectures in number fields which are currently

intractable.

We use tools specific to functional fields; unavailable over the integers.

Thank you for your attention !