1. No category

History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
How to Not Count Primes
An Analytic Prime Counting Function
D.J. Platt
Department of Mathematics
University of Bristol
28 February 2011 / TCC Number Theory Day
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Outline
1
A Brief History
2
Riemann and π(x)
3
Lagarias, Odlyzko and Galway
4
Our Approach
5
So What Do We Need?
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Sieve of Eratostenes
Enumerate and Count
for(i=2;i<x;i++) prime[i]=true;
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Sieve of Eratostenes
Enumerate and Count
for(i=2;i<x;i++) prime[i]=true;
for(i=2;i<sqrt(x);i++) if(prime[i]) for(j=i*i;j<x;j+=i)
prime[i]=false;
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Sieve of Eratostenes
Enumerate and Count
for(i=2;i<x;i++) prime[i]=true;
for(i=2;i<sqrt(x);i++) if(prime[i]) for(j=i*i;j<x;j+=i)
prime[i]=false;
O(x log(x) log(log(x))) in time
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Sieve of Eratostenes
Enumerate and Count
for(i=2;i<x;i++) prime[i]=true;
for(i=2;i<sqrt(x);i++) if(prime[i]) for(j=i*i;j<x;j+=i)
prime[i]=false;
O(x log(x) log(log(x))) in time
O(x) in space
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Other Sieves?
“Better” sieves are known
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Other Sieves?
“Better” sieves are known
BUT
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Other Sieves?
“Better” sieves are known
BUT
By the Prime Number Theorem....
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Other Sieves?
“Better” sieves are known
BUT
By the Prime Number Theorem....
x
All sieves must be at least O log(x)
in time
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric Methods
√
π(x) = x − 1 + π( x) . . .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric Methods
√
π(x) = x − 1 + π( x) . . .
− x2 . . .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric Methods
√
π(x) = x − 1 + π( x) . . .
− x2 . . .
− x3 + x6 . . .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric Methods
√
π(x) = x − 1 + π( x) . . .
− x2 . . .
− x3 + x6 . . .
x x x
− x5 + 10
+ 15 − 30 . . .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric/Sieve Methods
Meissel/Lehmer/Lagarias/Miller/Odlyzko/Deléglise/Rivat
Use various sieve “tricks” to reduce search tree
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric/Sieve Methods
Meissel/Lehmer/Lagarias/Miller/Odlyzko/Deléglise/Rivat
Use varioussieve “tricks”
to reduce search tree
2
Is about O
x3
log2 (x)
in time
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric/Sieve Methods
Meissel/Lehmer/Lagarias/Miller/Odlyzko/Deléglise/Rivat
Use varioussieve “tricks”
to reduce search tree
2
Is about O
x3
log2 (x)
in time
Has been used to compute π 1023
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Combinatoric/Sieve Methods
Meissel/Lehmer/Lagarias/Miller/Odlyzko/Deléglise/Rivat
Use varioussieve “tricks”
to reduce search tree
2
Is about O
x3
log2 (x)
in time
Has been used to compute π 1023
BUT...
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
The End Of The Road?
Parallelised version of M/L/L/M/O/D/R method by Gourdon
ran for 360 days
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
The End Of The Road?
Parallelised version of M/L/L/M/O/D/R method by Gourdon
ran for 360 days
Was ±1 out in global checks
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
The End Of The Road?
Parallelised version of M/L/L/M/O/D/R method by Gourdon
ran for 360 days
Was ±1 out in global checks
Project was abandoned
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
The End Of The Road?
Parallelised version of M/L/L/M/O/D/R method by Gourdon
ran for 360 days
Was ±1 out in global checks
Project was abandoned
6 years later in 2007 Oliveira e Silva recomputed it
correctly.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
The End Of The Road?
Parallelised version of M/L/L/M/O/D/R method by Gourdon
ran for 360 days
Was ±1 out in global checks
Project was abandoned
6 years later in 2007 Oliveira e Silva recomputed it
correctly.
Have such methods reached their limit?
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
π ∗ (x)
"
π ∗ (x)
:=
1
2
#
P
pn <x
1
n
+
P
pn ≤x
D.J. Platt
1
n
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
π ∗ (x)
"
π ∗ (x)
:=
1
2
#
P
pn <x
1
n
+
P
pn ≤x
1
n
π(x) is cheap to recover from π ∗ (x) (Möbius inversion)
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
π ∗ (x)
"
π ∗ (x)
:=
1
2
#
P
pn <x
1
n
+
P
pn ≤x
1
n
π(x) is cheap to recover from π ∗ (x) (Möbius inversion)
R∞
Riemann showed given <s > 1, log sζ(s) = π ∗ (x)x −s dx
x
0
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
π ∗ (x)
"
π ∗ (x)
:=
1
2
#
P
pn <x
1
n
+
P
pn ≤x
1
n
π(x) is cheap to recover from π ∗ (x) (Möbius inversion)
R∞
Riemann showed given <s > 1, log sζ(s) = π ∗ (x)x −s dx
x
0
By Mellin inversion π ∗ (x) =
D.J. Platt
1
2πi
σ+i∞
R
σ−i∞
π(x)
log ζ(s)x s ds
s (σ > 1)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
π ∗ (x)
"
π ∗ (x)
:=
1
2
#
P
pn <x
1
n
+
P
pn ≤x
1
n
π(x) is cheap to recover from π ∗ (x) (Möbius inversion)
R∞
Riemann showed given <s > 1, log sζ(s) = π ∗ (x)x −s dx
x
0
By Mellin inversion π ∗ (x) =
1
2πi
(after some analysis) = Li(x) −
σ+i∞
R
σ−i∞
P
ρ
D.J. Platt
π(x)
log ζ(s)x s ds
s (σ > 1)
Li (x ρ ) − log 2 + O x −1
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Forcing Convergence
L.-O. suggested introducing a suitable Mellin transform pair
ˆ
φ(t) and φ(s)
so that
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Forcing Convergence
L.-O. suggested introducing a suitable Mellin transform pair
ˆ
φ(t) and φ(s)
so that
σ+i∞
R
1
ˆ log ζ(s)ds + P 1 [χx (pm ) − φ (pm )]
π ∗ (x) = 2πi
φ(s)
m
pm
σ−i∞
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Forcing Convergence
L.-O. suggested introducing a suitable Mellin transform pair
ˆ
φ(t) and φ(s)
so that
σ+i∞
R
1
ˆ log ζ(s)ds + P 1 [χx (pm ) − φ (pm )]
π ∗ (x) = 2πi
φ(s)
m
pm
σ−i∞

t <x
 1
1/2 t = x
where χx (t) :=

0
t >x
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Forcing Convergence
L.-O. suggested introducing a suitable Mellin transform pair
ˆ
φ(t) and φ(s)
so that
σ+i∞
R
1
ˆ log ζ(s)ds + P 1 [χx (pm ) − φ (pm )]
π ∗ (x) = 2πi
φ(s)
m
pm
σ−i∞

t <x
 1
1/2 t = x
where χx (t) :=

0
t >x
1
This algorithm is potentially O x 2 in time.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Forcing Convergence
L.-O. suggested introducing a suitable Mellin transform pair
ˆ
φ(t) and φ(s)
so that
σ+i∞
R
1
ˆ log ζ(s)ds + P 1 [χx (pm ) − φ (pm )]
π ∗ (x) = 2πi
φ(s)
m
pm
σ−i∞

t <x
 1
1/2 t = x
where χx (t) :=

0
t >x
1
This algorithm is potentially O x 2 in time.
ˆ
Note φ(s)
=
Riemann.
xs
s
gives φ(t) = χx (t) and we are back to
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
J. Buethe, J. Franke, A. Jost, T. Kleinjung
They have developed a version of this algorithm, but based
on Weil’s explict formula.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
J. Buethe, J. Franke, A. Jost, T. Kleinjung
They have developed a version of this algorithm, but based
on Weil’s explict formula.
It is contingent on R.H. (and no I don’t know why).
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
J. Buethe, J. Franke, A. Jost, T. Kleinjung
They have developed a version of this algorithm, but based
on Weil’s explict formula.
It is contingent on R.H. (and no I don’t know why).
They have computed a value for π 1024 .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Galway
ˆ
G. suggested φ(s)
= exp
D.J. Platt
λ2 s2
2
π(x)
xs
s ,
φ(t) = 12 erfc
log( xt )
√
(2)λ
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Galway
ˆ
G. suggested φ(s)
= exp
λ2 s2
2
xs
s ,
φ(t) = 12 erfc
log( xt )
√
Self-duality of Gaussian implies these are “optimal”
D.J. Platt
π(x)
(2)λ
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Galway
ˆ
G. suggested φ(s)
= exp
λ2 s2
2
xs
s ,
φ(t) = 12 erfc
log( xt )
√
(2)λ
Self-duality of Gaussian implies these are “optimal”
Now everything converges absolutely so just compute
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Galway
ˆ
G. suggested φ(s)
= exp
λ2 s2
2
xs
s ,
φ(t) = 12 erfc
log( xt )
√
(2)λ
Self-duality of Gaussian implies these are “optimal”
Now everything converges absolutely so just compute
σ+i∞
R
P1
1
m
m
ˆ log ζ(s)ds +
π ∗ (x) = 2πi
φ(s)
m [χx (p ) − φ (p )]
pm
σ−i∞
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Sum
The sum is a prime sieve, centred on x.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Sum
The sum is a prime sieve, centred on x.
At (e.g.) x = 1024 it needs to be 1016 or so wide.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Sum
The sum is a prime sieve, centred on x.
At (e.g.) x = 1024 it needs to be 1016 or so wide.
This is (just) achievable.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Integral
Numerical integration is problematic.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Integral
Numerical integration is problematic.
So let G(s) be a primitive of φˆ such that
G(2 + i∞) = −G(2 − i∞)
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Integral
Numerical integration is problematic.
So let G(s) be a primitive of φˆ such that
G(2 + i∞) = −G(2 − i∞)
Then the
Pintegral becomes
G(1) − G(ρ) − log2 + O x −1
ρ
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing The Integral
Numerical integration is problematic.
So let G(s) be a primitive of φˆ such that
G(2 + i∞) = −G(2 − i∞)
Then the
Pintegral becomes
G(1) − G(ρ) − log2 + O x −1
ρ
(Note similarity to Riemann’s explicit formula.)
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing G
Start with
ˆ 0 + ih) = φ(s
ˆ 0 ) exp(ih(s0 λ2 + log(x)))
φ(s
exp
−λ2 h2
2
1+ sih
0
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing G
Start with
ˆ 0 + ih) = φ(s
ˆ 0 ) exp(ih(s0 λ2 + log(x)))
φ(s
exp
−λ2 h2
2
1+ sih
0
Write everything as a Taylor series
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Computing G
Start with
ˆ 0 + ih) = φ(s
ˆ 0 ) exp(ih(s0 λ2 + log(x)))
φ(s
exp
−λ2 h2
2
1+ sih
0
Write everything as a Taylor series
Integrate term by term
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
We Need...
A way of computing approximations to real numbers
rigorously.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
We Need...
A way of computing approximations to real numbers
rigorously.
An efficient prime sieve
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
We Need...
A way of computing approximations to real numbers
rigorously.
An efficient prime sieve
Accurate and precise zeros of ζ (maybe 1011 of them)
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Rigorous Computations
Computers are finite, the real number system isn’t.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Rigorous Computations
Computers are finite, the real number system isn’t.
Rounding and truncation must be managed.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Rigorous Computations
Computers are finite, the real number system isn’t.
Rounding and truncation must be managed.
We use multiple precision interval arithmetic (MPFI by
Revol and Rouillier)
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Rigorous Computations
Computers are finite, the real number system isn’t.
Rounding and truncation must be managed.
We use multiple precision interval arithmetic (MPFI by
Revol and Rouillier)
This is one or two orders of magnitude slower than
hardware.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
1, t and t 2 form a basis for this 3 term approximation
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
1, t and t 2 form a basis for this 3 term approximation
We sieve intervals I about 232 wide centred on t0 and
compute the integers
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
1, t and t 2 form a basis for this 3 term approximation
We sieve intervals I about 232 wide centred on t0 and
compute the integers
P
1
p∈I
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
1, t and t 2 form a basis for this 3 term approximation
We sieve intervals I about 232 wide centred on t0 and
compute the integers
P
p∈I
P
1
(p − t0 )
p∈I
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
An Efficient Prime Sieve
Our sieve contains too many primes to compute φ(p)
rigorously each time, so
We compute a quadratic approximation to φ around some
t0 .
1, t and t 2 form a basis for this 3 term approximation
We sieve intervals I about 232 wide centred on t0 and
compute the integers
P
p∈I
P
1
(p − t0 )
p∈I
P
(p − t0 )2
p∈I
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Zeros of ζ
We have developed a rigorous, FFT based algorithm for
computing ζ on the 12 line.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Zeros of ζ
We have developed a rigorous, FFT based algorithm for
computing ζ on the 12 line.
It computes many evenly spaced values of ζ in average
time O (t ).
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Zeros of ζ
We have developed a rigorous, FFT based algorithm for
computing ζ on the 12 line.
It computes many evenly spaced values of ζ in average
time O (t ).
We can interpolate using these values to locate the zeros
of ζ.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Zeros of ζ
We have developed a rigorous, FFT based algorithm for
computing ζ on the 12 line.
It computes many evenly spaced values of ζ in average
time O (t ).
We can interpolate using these values to locate the zeros
of ζ.
We use Turing’s method to confirm that no zeros have
been missed.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Results
We have tested the basic algorithm using Odlyzko’s first
100, 000 zeros to compute π 1011 .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Results
We have tested the basic algorithm using Odlyzko’s first
100, 000 zeros to compute π 1011 .
Using UoB’s Bluecrystal
cluster, we are computing enough
22
zeros to reach π 10 .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Summary
We have a working version of the Lagarias and Odlyzko
analytic π(x) algorithm.
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Summary
We have a working version of the Lagarias and Odlyzko
analytic π(x) algorithm.
It might be able to reach π 1024 .
D.J. Platt
π(x)
History
Riemann and π(x)
Lagarias, Odlyzko and Galway
Our Approach
So What Do We Need?
Summary
Summary
We have a working version of the Lagarias and Odlyzko
analytic π(x) algorithm.
It might be able to reach π 1024 .
As a spin off, we will have lots of accurate and precise
zeros of ζ to give away.
D.J. Platt
π(x)