Hamburgers, Numbers
and infinite Series
JSPS Science Dialogue

Henrik Bachmann
Nagoya University

Plan of this TalkHamburgers, Germany and MeNumbers
Real numbers, Algebraic numbers, Rational numbers
and Transcendental numbers

Infinite Series
Finite & Infinite Series
Geometric series and Riemann zeta values
Multiple zeta values

Mathematics & Research
What I like about mathematics

What is this?
これは何ですか？

… and this?

Das ist ein Hamburger

Das ist ein Hamburger

This is a hamburger

This is someone fromHamburg

About me• In 1985 I was born in Hamburg, Germany.
• I have an older sister who studied Japanese.
姉妹• From 2006 to 2016 I did my Bachelor, Master and Doctor inMathematics at Hamburg University.
• Since April 2016 I am a JSPS Post-doctoral Fellow in Nagoya.
• I am interested in Number theory.
数論

Where is Germany?

Where is Hamburg?
Hamburg
Old name: Hammaburg
Hamma = Something at a river
限界

Burg = Castle

Special phrase fromHamburg forこんにちわ

What does this have to do with?

• A lot of people moved from Germany to the USA.

• There the people fromHamburg sold a north
German food: A pieceof meat inside a smallbread.

• Later the American people called these breads “Hamburger”.

What do you know about Germany?

Germany & Japan

82.175.684
357.375 km²

1. Berlin (3.520.031)
2. Hamburg (1.787.408)
3. München (1.450.381)
4. Köln (1.060.582)…

127.110.047
377.835 km²

1. Tokyo (9.375.104 )2. Yokohama (3.732.616)
3. Osaka (2.705.262)
4. Nagoya (2.302.696)…

Mathematics overview

What is number theory (数論) ?
• Number theory is a branch of pure mathematics devoted
primarily to the study of the integers (numbers).
• It is sometimes called "The Queen of Mathematics“.

• There are several different areas inside of number theory.
• Classical number theory (Prime numbers, divisors,…)• Analytic number theory (Use functions to study numbers)• Algebraic number theory (Use algebra to study numbers)• Diophantine geometry (Use geometry to study numbers)• ….

Numbers (数)What types of numbers do you know?
Where/When do they appear?
1. Counting:x = number of people in this room

2. Ratios:

x = remaining part of this pizza:

3. Finding the root of a polynomial:
根多項式

x4. Circumference of a circle:
円周率1

Numbers (数)What types of numbers do you know?
Where/When do they appear?
1. Counting:x = number of people in this room

2. Ratios:

x = remaining part of this pizza:

3. Finding the root of a polynomial:
根多項式

x4. Circumference of a circle:
円周率1

Overview of the real numbers (実数)Real numbers

Real numbers: All numbers with adecimal expansion.

Algebraic numbers:
Algebraic numbers

Rational numbers
Integers
Natural numbers

Numbers which are zeros ofPolynomials with rationalcoefficients.

Rational numbers:
Numbers which can bewritten as a fraction

Integers & Natural
numbers: Numbers whicharise from counting, adding andsubtracting.

Transcendental numbers (超越数)• Real numbers which are not algebraic are called
transcendental numbers.
Real numbers
Algebraic numbers• There are a lot of transcendentalnumbers.

• In 1882 F. Lindemann proved that 𝜋is transcendental.

• This means you can not find a polynomial P(x) with
𝜋 as its root, i.e. P(𝜋) = 0.

Transcendental numbers (超越数)
Exercise:
Suppose c is rational andA and B are transcendental.

Real numbers

BAi) Is cA transcendental?
ii) Is A² transcendental?
iii) Is A+B transcendental?

Algebraic numbers

Transcendental numbers (超越数)• There are a lot of numbers where we do not know if they are
algebraic or transcendental.

? ?

?

• To introduce some of them I first need to explain infinite series.

Finite sums & Infinite Series (無限級数)• A series is a sum of numbers.
• The symbol
is used for short notation
Example:1) The sum of the first 100 natural numbers:

2) The sum of the first six odd numbers

Finite sums & Infinite Series (無限級数)

Exercise: Write the following sums with the symbol

1)2)Finite sums & Infinite Series (無限級数)

Exercise: Write the following sums with the symbol

1)2)Finite sums & Infinite Series (無限級数)
What happens if we calculate

for different b?

We see that this sum approaches 2 and we write

This sum is called an infinite Series.

Finite sums & Infinite Series (無限級数)
In general one can proof (not hard!) the following:This sum is called the geometric series.
幾何級数

Finite sums & Infinite Series (無限級数)
But what happens if we calculate

What isfor different b?

??

Infinite Series (無限級数)
This problem was first solved by L. Euler in 1735

He gave a formula for all sums of this type witheven exponents.
Example:Leonhard Euler

Riemann zeta values (リーマンゼータ値)
More general Euler considered for arbitrary k = 2,3,4,5,…
the numbers

which are called Riemann zeta values.

What about?
Bernhard Riemann

Riemann zeta values (リーマンゼータ値)
偶数

Due to Euler we know that for even k
the values
are transcendental.

Real numbers
Algebraic numbers

Rational numbers
Integers
Natural numbers

Riemann zeta values (リーマンゼータ値)
For odd k nobody knows where to put
奇数Real numbers
Algebraic numbers

Rational numbers
Integers
Natural numbers?
?

in this picture….

Riemann zeta values (リーマンゼータ値)
For odd k nobody knows where to put

in this picture….

奇数

Real numbers
Algebraic numbers

?

?

?

Rational numbers

Is not rational
Integers
Natural numbers
Roger Apéry (1979)

Multiple zeta values (多重ゼータ値)• In my research I consider a generalization of theRiemann zeta values.
• For this we need to explain a new notation forinfinite series:

Sum over the numbers n=1,2,…

= Sum over numbers n with n>0Multiple zeta values (多重ゼータ値)• The double zeta values are defined by
二重ゼータ値

0<1<20<1<30<2<30<1<4

Sum over all numbers
n1 and n2 with 0<n1 < n2

• The multiple zeta values are then defined by

0<2<4Multiple zeta values (多重ゼータ値)• These numbers satisfy a lot of linear relations
Examples:• One of the goals is to understand all these relations

Mathematics
Why should you study Mathematics?
••

It is fun!
Japan is a good place to study mathematics!

I like Mathematics because….
•••…there are no limits.
…there is just one “right” and “wrong”.
…it is international.

Mathematics – No Limits
Imagine you study biology and you are interested in cats
生物学and elephants.

How to decide the topic for your thesis?
主題

学位論文

Mathematics – No Limits

You can not write a thesis on bothsince there are no catephants!

Mathematics – No Limits• In Mathematics there are no such limits.
• I was also interested in two different topics(Multiple zeta values and Eisenstein series).

Multiplezeta values

EisensteinseriesMathematics – No Limits
My Master thesis & PhD Thesis was devoted to Multiple
Eisenstein series which combined both topics.
Multiple Eisenstein series!

Multiplezeta values

EisensteinseriesMathematics – No Limits• There are endless open problems in mathematics• You can come up with your own questions and objects• There is a lot of freedom since you do not need tothink about an application before you start working ona problem.

Mathematics – International• During my research I was able to travel a lot.
• I met a lot of people from different countries.

• Usually everybody in mathematical research canspeak (simple) English! LEARN ENGLISH!

Where to start….?
• Study mathematics!
• Also try to read English textbooks or research papers!
Other nice books to start:(www.arxiv.org)

Fermat's Last Theorem - Simon Singh

Story about a nice mathematical problem, which was solved 1994.
This book is also available in Japanese.
And there is an English documentary (video) about this:
http://www.dailymotion.com/video/x3wrbsb
Proofs from THE BOOK - Martin Aigner, Günter M. Ziegler

A collection of beautiful mathematical problems and proofs.

Thank you for your attention!