Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = a0 + a1 2z + a2 3z + . . . + an nz + . . .
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = a0 + a1 2z + a2 3z + . . . + an nz + . . .
Usually z is replaced by −z.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = a0 + a1 2−z + a2 3−z + . . . + an n−z + . . .
Usually z is replaced by −z.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = a0 + a1 2−z + a2 3−z + . . . + an n−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
ζ(z) = 1 + 2−z + 3−z + . . . + n−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
50 summands
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
100 summands
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
300 summands
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Unfortunately the series
converges only for Re z > 1
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Unfortunately the series
converges only for Re z > 1
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Unfortunately the series
converges only for Re z > 1
and convergence is so slow
that this image is useless.
Dirichlet series and the Riemann zeta function
Dirichlet series are sums of exponential functions
f (z) = 1 + 2−z + 3−z + 4−z + 5−z + 6−z + . . .
The simplest Dirichlet series with all coefficients 1 defines the Riemann Zeta
function.
1000 summands
Unfortunately the series
converges only for Re z > 1
and convergence is so slow
that this image is useless.
Riemann’s explicit formula
In his groundbreaking paper “Über die
Anzahl der Primzahlen unter einer gegeben
Größe” of 1859, Bernhard Riemann derives
an explicit formula for analytic continuation
the Zeta function,
I
(−x)z−1
dx,
2 sin(π)Γ(z)ζ(z) = i
x
C e −1
where the curve C starts at +∞, runs once
around the origin in positive direction and
returns to +∞.
E. Wegert ()
2 / 12
Riemann’s explicit formula
In his groundbreaking paper “Über die
Anzahl der Primzahlen unter einer gegeben
Größe” of 1859, Bernhard Riemann derives
an explicit formula for analytic continuation
the Zeta function,
I
(−x)z−1
dx,
2 sin(π)Γ(z)ζ(z) = i
x
C e −1
where the curve C starts at +∞, runs once
around the origin in positive direction and
returns to +∞. This formula implies that
Zeta has a simple pole at 1 and “trivial”
zeros at −2, −4, −6, . . ., but there are
others, nicely aligned along Re z = 1/2.
E. Wegert ()
2 / 12
Non-trivial zeros of the Zeta function
Berhard Riemann was aware that all “non-trivial” zeros are located in the
“critical strip”
S := {z ∈ C : 0 ≤ Re z ≤ 1}.
E. Wegert ()
3 / 12
Non-trivial zeros of the Zeta function
Berhard Riemann was aware that all “non-trivial” zeros are located in the
“critical strip”
S := {z ∈ C : 0 ≤ Re z ≤ 1}.
Moreover, he heuristically estimated the number N(T ) of non-trivial zeros
which satisfy 0 < Im z < T by
T
T
T
N(T ) ≈
log
−
2π
2π 2π
and claimed:
E. Wegert ()
3 / 12
The Riemann Hypothesis
By saying that the (nontrivial) zeros “are real”, Riemann claims that they
lie exactly in the middle of the critical strip, i.e., their real part equals 1/2.
This innocent statement is the famous Riemann Hypothesis.
E. Wegert ()
4 / 12
The Riemann Hypothesis
By saying that the (nontrivial) zeros “are real”, Riemann claims that they
lie exactly in the middle of the critical strip, i.e., their real part equals 1/2.
This innocent statement is the famous Riemann Hypothesis.
In the years after Riemann’s death the difficulty of the problem was not
recognized. This changed when David Hilbert included the conjecture as
number 8 in his famous list of the 23 most important unsolved problem in
mathematics.
E. Wegert ()
4 / 12
The Riemann Hypothesis
By saying that the (nontrivial) zeros “are real”, Riemann claims that they
lie exactly in the middle of the critical strip, i.e., their real part equals 1/2.
This innocent statement is the famous Riemann Hypothesis.
In the years after Riemann’s death the difficulty of the problem was not
recognized. This changed when David Hilbert included the conjecture as
number 8 in his famous list of the 23 most important unsolved problem in
mathematics.
More than 100 years later basically nothing has changed. In 2000, the Clay
Mathematics Institute has pronounced the Riemann Hypothesis as one of
the seven Millennium Problems and donated a 1 Million US dollar price for
its verification.
E. Wegert ()
4 / 12
The Riemann Hypothesis
By saying that the (nontrivial) zeros “are real”, Riemann claims that they
lie exactly in the middle of the critical strip, i.e., their real part equals 1/2.
This innocent statement is the famous Riemann Hypothesis.
In the years after Riemann’s death the difficulty of the problem was not
recognized. This changed when David Hilbert included the conjecture as
number 8 in his famous list of the 23 most important unsolved problem in
mathematics.
More than 100 years later basically nothing has changed. In 2000, the Clay
Mathematics Institute has pronounced the Riemann Hypothesis as one of
the seven Millennium Problems and donated a 1 Million US dollar price for
its verification.
Though it is known today that more than 10 000 000 000 000 non-trivial
zeros indeed lie on the critical line, the problem still withstands all attacks
and seems far from being solved.
E. Wegert ()
4 / 12
Non-trivial zeros of the Zeta function
E. Wegert ()
5 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
Here we are in the critical strip at height
Im z = 121 415.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
Here we are in the critical strip at height
Im z = 121 415.
The white line is the critical line.
In particular the right side is remarkably
colorful.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
Here we are in the critical strip at height
Im z = 121 415.
The white line is the critical line.
In particular the right side is remarkably
colorful.
E. Wegert ()
6 / 12
The Zeta function in the critical strip
The phase portrait of the Zeta function is
surprisingly rich.
The most interesting region is the critical
strip.
Here we are in the critical strip at height
Im z = 121 415.
The white line is the critical line.
In particular the right side is remarkably
colorful.
In order to explore this region further we
send out scouts.
E. Wegert ()
6 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2.
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2.
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
When one travels around a
string, the corresponding color
point moves along the color
wheel. The number of
complete revolutions in
(counter-clockwise) direction
made by this point is the
chromatic number chrom S of
the string.
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
When one travels around a
string, the corresponding color
point moves along the color
wheel. The number of
complete revolutions in
(counter-clockwise) direction
made by this point is the
chromatic number chrom S of
the string.
chrom S = 0
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
When one travels around a
string, the corresponding color
point moves along the color
wheel. The number of
complete revolutions in
(counter-clockwise) direction
made by this point is the
chromatic number chrom S of
the string.
chrom S = 1
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
When one travels around a
string, the corresponding color
point moves along the color
wheel. The number of
complete revolutions in
(counter-clockwise) direction
made by this point is the
chromatic number chrom S of
the string.
chrom S = −1
E. Wegert ()
7 / 12
Strings and their chromatic numbers chrom S
A string is a simple closed curve (Jordan curve)
which fits into an open strip of width 1/2. Strings
are colored continuously by colors of the hsv–color
wheel.
When one travels around a
string, the corresponding color
point moves along the color
wheel. The number of
complete revolutions in
(counter-clockwise) direction
made by this point is the
chromatic number chrom S of
the string.
chrom S = 0
E. Wegert ()
7 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
Strings live in the right half of the critical strip.
They are allowed to move in vertical direction.
Strings try to find a place where they become
invisible.
We say that a string can hide itself if it can find a
position where its colors differ from the background
by an amount which can be made arbitrarily small.
Which strings can hide themselves in the phase
portrait of the Riemann Zeta function ?
E. Wegert ()
8 / 12
Universality of the Zeta function
The following result illustrates the unbelievable
richness of the phase portrait of the Riemann Zeta
function. It is the translation of the
Karatsuba-Voronin universality theorem into the
language of phase plots.
E. Wegert ()
8 / 12
Universality of the Zeta function
The following result illustrates the unbelievable
richness of the phase portrait of the Riemann Zeta
function. It is the translation of the
Karatsuba-Voronin universality theorem into the
language of phase plots.
Theorem
Every string with chromatic number zero can hide
itself in the phase plot of the Riemann Zeta
function.
E. Wegert ()
8 / 12
Universality of the Zeta function
The following result illustrates the unbelievable
richness of the phase portrait of the Riemann Zeta
function. It is the translation of the
Karatsuba-Voronin universality theorem into the
language of phase plots.
Theorem
Every string with chromatic number zero can hide
itself in the phase plot of the Riemann Zeta
function.
Note that the shape as well as the color of the
string can be chosen arbitrarily, provided its
chromatic number is zero !
E. Wegert ()
8 / 12
Universality of the Zeta function
The following result illustrates the unbelievable
richness of the phase portrait of the Riemann Zeta
function. It is the translation of the
Karatsuba-Voronin universality theorem into the
language of phase plots.
Theorem
Every string with chromatic number zero can hide
itself in the phase plot of the Riemann Zeta
function.
Note that the shape as well as the color of the
string can be chosen arbitrarily, provided its
chromatic number is zero !
The necessity of the condition chrom S = 0 is
equivalent to the Riemann Hypothesis.
E. Wegert ()
8 / 12
The Zeta Function in the critical strip
These images are phase portraits of Zeta in the critical strip. Every picture
covers about 20 units in direction of the imaginary axis.
E. Wegert ()
9 / 12
The Zeta Function in the critical strip
These images are phase portraits of Zeta in the critical strip. Every picture
covers about 20 units in direction of the imaginary axis.
The yellow diagonal stripes indicate kind of “stochastic” periodicity.
E. Wegert ()
9 / 12
Mean values of the Zeta function
This observation has inspired Jörn Steuding and myself to investigate
mean values of the Zeta function on “vertical arithmetic sequences”.
Theorem (J.Steuding, E.W.)
Let s ∈ C \ {1} such that 0 < σ := Re s < 1 and Im s ≥ 0. If
d = 2π/log m with m ∈ Z and m ≥ 2, then
1
N
X
ζ(s + in d) =
0≤n<N
1
+ O(N −σ log N) as
1 − m−s
N → +∞.
The proof is elementary (but not simple), Experimental Mathematics
2012, No.3.
E. Wegert ()
10 / 12
Mean values of the Zeta function: experiments
The pictures show the results of some numerical simulations where we
compared the mean values with the asymptotic formulas for
d = 2π/log 2 ≈ 9.0647.
This case corresponds to the observed “periodic” behavior in the phase
portraits and gives at least a first hint to an explanation.
E. Wegert ()
11 / 12