Barrier Approach Visual Guide

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Overview of the “Barrier Approach”
to lower the upper bound of the
de Bruijn-Newman constant.

D.H.J. Polymath
June 2018

High level storyline

 The basic De Bruijn idea leading to the function Ht(x+iy).
 How to effectively bound a good estimate for Ht?
 Some observations on the zeros of Ht.
 How could zeros of Ht be 'blocked' to lower the Λ upper bound.
 The key ideas behind the “Barrier approach”.
 How to ensure no zeros have passed the Barrier?
 How to show that Ht doesn’t vanish from the Barrier to Nb ?
 Numerical results showing Λ < 0.22 (and lower, but conditionally on RH).
 Software used and detailed results available.

Basic idea by De Bruijn

Fourier
transform

New family
introduced
by De Bruijn

First step: develop an effectively bounded estimate.

Estimating and effectively bounding Ht(x+iy)
Error terms

Main estimate
Designed for:

Optionally: a more effective C-term is available

Normalize by B0 and bound effectively

Main estimate lower bound

Error upper bounds

(triangle)
Hence,

(lemma)

Choice of ‘Euler mollifiers’

If Lower bound ≥ Upper bound then

Real example of trajectories of real and complex zeros of Ht(x+iy)
Once a zero becomes real, it stays
real forever and ends up roughly
equally spaced with:
Zeroes get denser as one moves
away from the origin, so there
are more zeros to the right of xn
then to the left, hence their
trajectories “lean” leftwards.

t

The complex parts of zeros
attract each other and the real
parts repel each other. From
isolating the imaginary “force”,
it can be derived that all
complex zeroes will be forced
into the real axis in a finite time
leading to the bound:

0

y

x
trajectory of a complex zero

trajectory of a real zero

The De Bruijn – Newman Λ and a ‘ceiling’ the complex zeroes can’t cross
1. Introduce a ‘ceiling’ and
verify that Ht0 (x+iy) ≠ 0
for = 0 . . ∞, = 0 ..1 (or
the blue hyperbola).

The blue DbN hyperbola
is only valid for t ≥ 0.

2. If so, then the new
upper bound:
Λ ≤ t0 + 0.5 y02
has been established.

x3
x4
Complex zeros are “attracted” to their
conjugates and “fall” to the real line
with a lower bounded speed.

x2
Even this extreme trajectory is
theoretically possible since
there is no upper bound on
the speed by which zeros fall
to the real line.

x1
A “Lehmer pair”.

Possible trajectory of a complex zero (Ht (x+ |

|>0)

= 0)

“ Barrier” approach to assure Ht (x+iy) ≠ 0 for a certain y>y0, t0.
Numerical
verification

0.5

≤

Analytical
proof

t0

+

1

y0

0

t

y

0

-1

-t

0

x
1. Area where the RH has been
verified e.g. 6x1010 certain, 1013
to be confirmed. Or assume
that it has been verified up to X.

X X+1

Na
3. Verify Ht0 (x+iy0) ≠ 0
i.e. Lemma lower bound
> Error upper bound.

∞

Nb

4. Analytical proof
that Ht0 (x+iy0) ≠ 0

2. Verify Ht (x+iy) ≠ 0
in the Barrier area
x=X..X+1, y=y0..1, t=0..t0

Possible trajectories of a complex zeros that should be “blocked”.

“Barrier” approach: how to clear the barrier?
t=t + (minABBeff - 0.5) /| |
t
Known areas where
Ht (x+iy) ≠ 0

0. Pick a promising
combination of t0 ,y0
and an X to lower
Λ ≤ t0 + 0.5 y02

3. Use an adaptive mesh to
establish the optimal next t
and continue with 2. until t=t0.

t0
0.5

Λ

upper bound required
1

y

0
1
Will be detected
by barrier.
Y0

0

X

x

X+1

1. Area cleared since all zeros have been, or are
assumed to be, verified to
be on the critical line.

Can’t happen due
to double barrier.

Location of Barrier can be optimized
by selecting an X and X+1 with a
relatively high value of ABBeff (and
where the ‘mollified’ lower lemma
bound is sufficiently positive).

Fast integral based
approach developed.

2. For a given t, clear
rectangle X..X+1, y=y0..1 (or
a point on the hyperbola)
using the argument principle
and Rouché's theorem.
=

upper bound required

Possible trajectories of a complex zeros that should be “blocked”.

Available software tools:
 Barrier_Location_Optimizer
 Stored_Sums_Generator
 WindingNumber_Calculator

“Barrier” approach: how to verify the area from the barrier up to Nb?
3. Only the lower Lemma bound for the line
y0, t0 needs to be verified to stay above the
error bounds, since the Lemma bound
monotonically increases for y going to 1.
t

0.5

Λ
1

t0

y

y0
0

x

X X+1

Nb

Na
1. Select a ‘mollifier’ that
makes the Lemma bound
sufficiently positive.

2. A fast Approximate Triangle
bound is used to establish the
Nb point after which analytical
proof takes over (currently
‘unmollified’ bound only).

A fast “Sawtooth” mechanism has been
developed, that only calculates the required
incremental Lemma Bound terms and only
requires a full calculation when the incremental
bound passes a user defined threshold.

Possible trajectory of a complex zeros that should be “blocked”.

Available software tools:
 Na_Lemmabound_calculator
 Nb_Location_Finder
 LemmaBound_Sawtooth_calculator

The Barrier model in action: some real numbers (wip)
Selected with
LemmaBound utility

Selected with Barrier
Location optimizer

x

Barrier
offset

RH
verified?

t0

y0

Λ

Winding
number

mollifier
Lemma
# primes bound value

Selected with
Nb Location finder

Na

Triangle
bound value

Nb

6.00E+10

155019

yes

0.20

0.20

0.22

0

4

0.067

69098

0.077

1.7E+06

1.00E+11

78031

yes

0.19

0.20

0.21

0

4

0.067

89206

0.081

6.0E+06

1.00E+12

46880

yes

0.18

0.20

0.20

0

3

0.135

282094

0.089

1.3E+07

5.00E+12

194858

yes 1)

0.17

0.20

0.19

0

3

0.180

630783

0.116

1.5E+07

1.00E+13

9995

not yet

0.16

0.20

0.18

0

3

0.109

892062

0.091

3.0E+07

1.00E+14

2659

not yet

0.15

0.20

0.17

0

3

0.195

2820947

0.076

7.0E+07

1.00E+15

21104

not yet

0.14

0.20

0.16

0

3

0.251

8920620

0.073

2.0E+08

1.00E+16

172302

not yet

0.13

0.20

0.15

0

3

0.278

28209479

0.077

7.0E+08

1.00E+17

31656

not yet

0.12

0.20

0.14

0

3

0.279

89206205

0.080

3.0E+09

1.00E+18

44592

not yet

0.11

0.20

0.13

tbd

2

0.207

282094791

0.103

2.0E+10

1.00E+19

12010

not yet

0.10

0.20

0.12

tbd

2

0.128

892062059

0.097

1.5E+11

1.00E+20

37221

not yet

0.09

0.20

0.11

tbd

3

0.037

2820947918

0.075

1.5E+12

1) Gourdon-Demichel 2004

Software used and useful links
All software was developed in two languages and all results were reconciled:
 Symbolic math programming language - pari/gp (https://pari.math.u-bordeaux.fr )
 Short development time
 Relatively fast
 Arithmetic Balls C-based library - Arb (http://arblib.org)
 Longer development time
 Very fast (up to 20 x pari/gp)

All software and results are free to use (under the LGPL-terms) and can be found here:
https://github.com/km-git-acc/dbn_upper_bound



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