Modelling large values of L-functions
How big can things get?
Christopher Hughes
Exeter, 20th January 2016
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
1 / 31
How big can the Riemann zeta function get?
ζ
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
2 / 31
How big can the Riemann zeta function get?
ζ
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
2 / 31
How big can the Riemann zeta function get?
ζ
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
2 / 31
Extreme values of the Riemann zeta function
15
10
5
2000
4000
6000
8000
The running maxima of zeta
for 0 ≤ t ≤ 103
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
3 / 31
Large values of the Riemann zeta function
30
30
25
25
20
20
15
15
10
10
5
5
0
1
2
3
4
5
6
30
30
25
25
20
20
15
15
10
10
5
5
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
The largest value of zeta over an interval of length 2π
for t = 1010 , 1010 + 100, 1010 + 200, 1010 + 300
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
4 / 31
Extreme values of zeta
(Growth of maxima up to height T )
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
5 / 31
Extreme values of zeta
Conjecture (Farmer, Gonek, Hughes)
max
t∈[0,T ]
|ζ( 21
p
1
√
+ o(1)
+ it)| = exp
log T log log T
2
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
6 / 31
Bounds on extreme values of zeta
Theorem (Littlewood; Ramachandra and Sankaranarayanan;
Soundararajan; Chandee and Soundararajan)
Under RH, there exists a C such that
log T
1
max |ζ( 2 + it)| = O exp C
log log T
t∈[0,T ]
Theorem (Bondarenko-Seip)
√
For all c < 1/ 2
s
max
t∈[0,T ]
|ζ( 12
Christopher Hughes (University of York)
+ it)| > exp c
log T log log log T
log log T
Modelling large values of L-functions
!
Exeter, 20th January 2016
7 / 31
An Euler-Hadamard hybrid
Theorem (Gonek, Hughes, Keating)
A simplified form of our theorem is:
ζ( 12 + it) = P(t; X )Z (t; X ) + errors
where
P(t; X ) =
Y
p≤X
1−
1
!−1
1
p 2 +it
and
!
Z (t; X ) = exp
X
Ci(|t − γn | log X )
γn
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
8 / 31
An Euler-Hadamard hybrid: Primes only
Graph of |P(t + t0 ; X )|, with t0 = γ1012 +40 ,
with X = log t0 ≈ 26 (red) and X = 1000 (green).
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
9 / 31
An Euler-Hadamard hybrid: Zeros only
Graph of |Z (t + t0 ; X )|, with t0 = γ1012 +40 ,
with X = log t0 ≈ 26 (red) and X = 1000 (green).
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
10 / 31
An Euler-Hadamard hybrid: Primes and zeros
Graph of |ζ( 21 + i(t + t0 ))| (black) and |P(t + t0 ; X )Z (t + t0 ; X )|,
with t0 = γ1012 +40 , with X = log t0 ≈ 26 (red) and X = 1000 (green).
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
11 / 31
Extreme values of zeta: RMT & zeros
Keating and Snaith modelled the Riemann zeta function with
ZUN (θ) := det(IN − UN e−iθ )
=
N
Y
(1 − ei(θn −θ) )
n=1
where UN is an N × N unitary matrix chosen with Haar measure.
The matrix size N is connected to the height up the critical line T via
N = log
Christopher Hughes (University of York)
T
2π
Modelling large values of L-functions
Exeter, 20th January 2016
12 / 31
Extreme values of zeta: RMT & zeros
0.25
0.20
0.15
0.10
0.05
-10
-5
5
log |ζ( 12
Graph of the value distribution of
+ it)| around the 1020 th zero (red),
against the probability density of log |ZUN (0)| with N = 42 (green).
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
13 / 31
Extreme values of zeta: RMT & zeros
Simply taking the largest value of a characteristic polynomial doesn’t
work.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
14 / 31
Extreme values of zeta: RMT & zeros
Simply taking the largest value of a characteristic polynomial doesn’t
work.
Split the interval [0, T ] up into
M=
T log T
N
blocks, each containing approximately N zeros.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
14 / 31
Extreme values of zeta: RMT & zeros
Simply taking the largest value of a characteristic polynomial doesn’t
work.
Split the interval [0, T ] up into
M=
T log T
N
blocks, each containing approximately N zeros.
Model each block with the characteristic polynomial of an N × N
random unitary matrix.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
14 / 31
Extreme values of zeta: RMT & zeros
Simply taking the largest value of a characteristic polynomial doesn’t
work.
Split the interval [0, T ] up into
M=
T log T
N
blocks, each containing approximately N zeros.
Model each block with the characteristic polynomial of an N × N
random unitary matrix.
Find the smallest K = K (M, N) such that choosing M independent
characteristic polynomials of size N, almost certainly none of them will
be bigger than K .
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
14 / 31
Extreme values of zeta: RMT & zeros
Note that
M
P max max |ZU (j) (θ)| ≤ K = P max |ZUN (θ)| ≤ K
1≤j≤M
θ
θ
N
Theorem
Let 0 < β < 2. If M = exp(N β ), and if
q
1
K = exp
1 − 2 β + ε log M log N
then
P
max max |ZU (j) (θ)| ≤ K
1≤j≤M
θ
→1
N
as N → ∞ for all ε > 0, but for no ε < 0.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
15 / 31
Extreme values of zeta: RMT & zeros
Recall
ζ( 12 + it) = P(t; X )Z (t; X ) + errors
We showed that Z (t; X ) can be modelled by characteristic polynomials
of size
log T
N= γ
e log X
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
16 / 31
Extreme values of zeta: RMT & zeros
Recall
ζ( 12 + it) = P(t; X )Z (t; X ) + errors
We showed that Z (t; X ) can be modelled by characteristic polynomials
of size
log T
N= γ
e log X
Therefore the previous theorem suggests
Conjecture
If X = log T , then
max |Z (t; X )| = exp
t∈[0,T ]
Christopher Hughes (University of York)
p
1
√ + o(1) log T log log T
2
Modelling large values of L-functions
Exeter, 20th January 2016
16 / 31
Extreme values of zeta: RMT & zeros
Theorem
By the PNT, if X = log T then for any t ∈ [0, T ],
p
P(t; X ) = O
log T
exp C
log log T
!!
Thus one is led to the max values conjecture
Conjecture
max |ζ( 21 + it)| = exp
t∈[0,T ]
Christopher Hughes (University of York)
p
1
√ + o(1)
log T log log T
2
Modelling large values of L-functions
Exeter, 20th January 2016
17 / 31
Extreme values of zeta: Primes
First note that

P(t; X ) = exp 

X
p≤X
Christopher Hughes (University of York)
1
p1/2+it
 × O(log X )
Modelling large values of L-functions
Exeter, 20th January 2016
18 / 31
Extreme values of zeta: Primes
First note that

P(t; X ) = exp 

1
X
p≤X
p1/2+it
 × O(log X )
Treat p−it as independent random variables, Up , distributed uniformly
on the unit circle.
The distribution of
X Up
Re
√
p
p≤X
tends to Gaussian with mean 0 and variance
Christopher Hughes (University of York)
Modelling large values of L-functions
1
2
log log X as X → ∞.
Exeter, 20th January 2016
18 / 31
Extreme values of zeta: Primes
p
We let X = exp( log T ) and model the maximum of P(t; X ) by finding
the maximum of the Gaussian random variable sampled T (log T )1/2
times. This suggests
p
1
max |P(t; X )| = O exp ( √ + ε) log T log log T
t∈[0,T ]
2
for all ε > 0 and no ε < 0.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
19 / 31
Extreme values of zeta: Primes
p
We let X = exp( log T ) and model the maximum of P(t; X ) by finding
the maximum of the Gaussian random variable sampled T (log T )1/2
times. This suggests
p
1
max |P(t; X )| = O exp ( √ + ε) log T log log T
t∈[0,T ]
2
for all ε > 0 and no ε < 0.
For such a large X , random matrix theory suggests that
p
max |Z (t; X )| = O exp
log T
.
t∈[0,T ]
This gives another justification of the large values conjecture.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
19 / 31
Large values of zeta
(Distribution of maxima over short intervals)
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
20 / 31
Distribution of the max of characteristic polynomials
In 2012 Fyodorov, Hiary and Keating studied the distribution of the
maximum of a characteristic polynomial of a random unitary matrix via
freezing transitions in certain disordered landscapes with logarithmic
correlations. This mixture of rigorous and heuristic calculation led to:
Conjecture (Fyodorov, Hiary and Keating)
For large N,
log max |ZUN (θ)| ∼ log N −
θ
3
log log N + Y
4
where the random variable Y has the density 4e−2y K0 (2e−y )
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
21 / 31
Distribution of the max of characteristic polynomials
0.4
0.3
0.2
0.1
-2
2
4
The probability density of Y
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
22 / 31
Distribution of the max of characteristic polynomials
Note that
P {Y ≥ K } ≈ 2Ke−2K
for large K .
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
23 / 31
Distribution of the max of characteristic polynomials
Note that
P {Y ≥ K } ≈ 2Ke−2K
for large K .
However, one can show that if K / log N → ∞ but K N then
K2
(1 + o(1))
P max log |ZUN (θ)| ≥ K = exp −
θ
log N
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
23 / 31
Distribution of the max of characteristic polynomials
Note that
P {Y ≥ K } ≈ 2Ke−2K
for large K .
However, one can show that if K / log N → ∞ but K N then
K2
(1 + o(1))
P max log |ZUN (θ)| ≥ K = exp −
θ
log N
Thus there must be a critical K (of the order log N) where the
probability that maxθ |ZU (θ)| ≈ K changes from looking like linear
exponential decay to quadratic exponential decay.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
23 / 31
Large values of zeta
Their work with characteristic polynomials led Fyodorov and Keating to
conjecture that
T
3
T
1
max |ζ( 2 + it)| ∼ exp log log
− log log log
+Y
2π
4
2π
T ≤t≤T +2π
with Y having (approximately) the same distribution as before.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
24 / 31
Large values of zeta
Distribution of −2 log maxt∈[T ,T +2π] |ζ( 12 + it)| (after rescaling to get the
empirical variance to agree) based on 2.5 × 108 zeros near T = 1028 . Graph
by Ghaith Hiary, taken from Fyodorov-Keating.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
25 / 31
Large values of zeta
The conjecture that for almost all T
max log |ζ( 21 + i(T + h))| = log log T −
0≤h≤1
3
log log log T + O(1)
4
was backed up by a different argument of Harper, and later by Arguin,
Belius and Harper.
They considered random Euler products
X Re(Up p−ih )
√
p
0≤h≤1
max
p≤T
where Up are uniform iid on the unit circle.
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
26 / 31
Large values of zeta
Split the sum over primes up
Yk (h) =
X
2k −1 <log p≤2k
Re(Up p−ih )
√
p
These are well-correlated if |h − h0 | < 2−k and nearly uncorrelated for
h and h0 further apart.
Translating this as a branching random walk they were able to show
that
X Re(Up p−ih )
3
= log log T − log log log T + oP (log log log T )
√
p
4
0≤h≤1
max
p≤T
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
27 / 31
Moments at the local maxima
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
28 / 31
Moments of the local maxima
Theorem (Conrey and Ghosh)
As T → ∞
2 e2 − 5
1 X 1
ζ( 2 + itn ) ∼
log T
N(T )
2
tn ≤T
where tn are the points of local maxima of |ζ( 12 + it)|.
This should be compared with Hardy and Littlewood’s result
1
T
Christopher Hughes (University of York)
Z
0
T
|ζ( 12 + it)|2 dt ∼ log T
Modelling large values of L-functions
Exeter, 20th January 2016
29 / 31
Moments of the local maxima
In 2012 Winn succeeding in proving a random matrix version of this
result (in disguised form)
Theorem (Winn)
As N → ∞
#
N
h
2k
2k i
1 X E
ZUN (φn )
∼ C(k ) E ZUN (0)
N
"
n=1
where φn are the points of local maxima of ZUN (θ), and where C(k )
can be given explicitly as a combinatorial sum involving Pochhammer
symbols on partitions.
In particular,
#
N
2
1 X e2 − 5
E
ZUN (φn ) ∼
N
N
2
"
n=1
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
30 / 31
Bibliography
Growth of zeta function D. Farmer, S. Gonek and C. Hughes “The maximum size of L-functions” Journal für die reine und
angewandte Mathematik (2007) 609 215–236
V. Chandee and K. Soundararajan “Bounding |ζ(1/2 + it)| on the Riemann hypothesis” Bull. London
Math. Soc. (2011) 43 243–250
A. Bondarenko and K. Seip “Large GCD sums and extreme values of the Riemann zeta function” (2015)
arXiv:1507.05840
Euler-Hadamard product formula S. Gonek, C. Hughes and J. Keating, “A hybrid Euler-Hadamard product for the Riemann zeta
function” Duke Math. J. (2007) 136 507–549
Modelling zeta using characteristic polynomials J. Keating and N. Snaith “Random matrix theory and ζ(1/2 + it)” Comm. Math.
Phys. (2000) 214 57–89
Distribution of max values Y. Fyodorov, G. Hiary and J. Keating, “Freezing transition, characteristic polynomials of random
matrices, and the Riemann zeta-function” Phys. Rev. Lett. (2012) 108 170601
Y. Fyodorov and J. Keating, “Freezing transitions and extreme values: random matrix theory, ζ(1/2 + it),
and disordered landscapes” (2012) arXiv:1211.6063
A. Harper, “A note on the maximum of the Riemann zeta function, and log-correlated random variables”
(2013) arXiv:1304.0677
L.-P. Arguin, D. Belius and A. Harper, “Maxima of a randomized Riemann Zeta function, and branching
random walks” (2015) arXiv:1506.00629
Moments at the local maxima J. Conrey and A. Ghosh, “A mean value theorem for the Riemann zeta-function at its relative
extrema on the critical line” J. Lond. Math. Soc. (1985) 32 193–202
B. Winn, “Derivative moments for characteristic polynomials from the CUE” Commun. Math. Phys. (2012)
315 531–563
Christopher Hughes (University of York)
Modelling large values of L-functions
Exeter, 20th January 2016
31 / 31