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The Bernoulli Numbers
Kyler Siegel Stanford SUMO April 23rd, 2015
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Recall the formulas
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See https://gowers.wordpress.com/2014/11/04/sums-of-kth-powers/
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See https://gowers.wordpress.com/2014/11/04/sums-of-kth-powers/
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More generally:
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See The Book of Numbers by Conway and Guy
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Applying the identity, lots of terms cancel:
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which gives Faulhaber’s formula!
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Some History
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Johann Faulhaber
1k + 2k + ...+ nk
• Born 1580 in Ulm, Germany • Found a formula for , for ,
now known as Faulhaber’s formula • The mysterious coefficients in his formula would
later become known as the Bernoulli numbers
k 17
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• Born 1655 in Basel, Switzerland • Gave the first comprehensive treatment of the Bernoulli
numbers in Ars Conjectandi (1713)
Jacob Bernoulli
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Seki Takakazu
• Born ~1642 in Japan • Credited with independently discovering the
Bernoulli numbers
Tabulation of binomial coefficients and Bernoulli numbers
from Katsuyo Sampo (1712)
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The Analytical Engine
• Designed by Charles Babbage in 1837 • One of the first designs for a mechanical computer • Around 1842, the mathematician Ada Lovelace wrote
arguably the first computer program, which computes the Bernoulli numbers!
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Charles Babbage b. 1791 in London, England
Ada Lovelace b. 1815 in London, England
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Bernoulli numbers also come up naturally in the Taylor series of trigonometric functions:
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Denominators of Bernoulli numbers:The von Staudt-Clauen Theorem: the denominator of the 2nth Bernoulli number is the product of all primes p such that p-1
divides 2n.Ex:
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Infinite sum formulasEuler proved the following formulas:
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Connections with the Riemann zeta function:
• This sum is uniformly convergent on compact subsets of the complex half-space {Re(s) > 0} and hence defines an analytic function of s there
• In fact, admits an analytic continuation to
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In terms of the Riemann zeta function
Euler’s formulas can be written as
(for k a positive integer)
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The Riemann zeta function:
After extending the domain of the zeta function, one can also compute its values at negative
integers:
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We get rather bizarre formulas:
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Note that the Bernoulli numbers come up for both negative and positive s values
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Regular primes•An prime p is called regular if it does not divide the numerator of B^k for k=2,4,6,…,p-3
•It is conjectured that about 60.65% of all primes are regular •In 1850 Kummer proved Fermat’s Last Theorem for regular primes, i.e. a^p+ b^p = c^p
has no solutions, provided p is a regular prime. It would be ~150 years before this was
Ernst Kummer Born 1810 in Sorau, Prussia
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Bernoulli numbers in stable homotopy theory
Let denote the n-dimensional sphere.Sn
S1S2 S3
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Let denote the ith homotopy group of the n-dimensional sphere:
⇡i(Sn)
where for maps and we have if and only if f and be deformed into g through
continuous maps
f : Si ! Sn g : Si ! Sn
f ⇠ g
Ex: ⇡1(S2) has only one element
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There are more homotopy groups of spheres than
one might expect.
Ex: ⇡3(S2) = Z,
generated by the Hopf map h : S3 ! S2
http://en.wikipedia.org/wiki/Hopf_fibration
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Suspension of a topological space
http://en.wikipedia.org/wiki/Suspension_%28topology%29
For a topological space X, there is something called
the suspension of X, denoted ⌃X
Ex: ⌃Sn= Sn+1
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For a topological space X, there is something called
the suspension of X, denoted ⌃X
Given a map f : X ! Y ,we get a map ⌃f : ⌃X ! ⌃Y
f
⌃f
!
!
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The Freudenthal suspension theorem states that this is an isomorphism if n > k+1
This means that there is a map ⇡n+k(Sn) ! ⇡n+k+1(Sn+1)
Let denote for ⇧k ⇡n+k(Sn) n >> k
In general, the field of stable homotopy theory studies what happens to topology when you suspend everything in sight
a million times.
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Computing ⇡i(Sn) is an extremely di�cult open problem in topology.
We know the values for small values of i and n:
http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups
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Computing ⇧k is in principle easier, but still a very hard open problem.
Computing ⇡i(Sn) is an extremely di�cult open problem in topology.
We know the answer for small values of k:
http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_stable_homotopy_groups
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Bernoulli numbers enter the picture via the so-called J homomorphism
Let O(n) denote the orthogonal group consisting of all nxn matrices A such that ATA = AAT = In
O(n) is a Lie group. Roughly speaking this means that:• It is a group, i.e. we can multiply and invert
elements • It is a topological space, i.e there is a natural
notion of two elements being “close” to each other
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Let O(n) denote the orthogonal group consisting of all nxn matrices A such that ATA = AAT = In
O(n) is a Lie group. Roughly speaking this means that:
• It is a group, i.e. we can multiply and invert elements • It is a topological space, i.e there is a natural notion of two elements
being “close” to each other
Since O(n) is a topological space, it makes sense to talk about its homotopy groups
It turns out that is independent of k for n >> k⇡k(O(n))
Bott computed , and remarkably the answer is 8-periodic!
⇡k(O(1))
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Bott periodicity:
Raoul Bott b. 1923 in Budapest, Hungary
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J : ⇡i(O(n)) ! ⇡i+n(Sn)
Taking n to be rather large, we get a subgroup Im(J) ⇢ ⇧i
There is a homomorphism
Im(J) consists of classes of maps f : Sa ! Sb
such that f�1(p) for a random point p 2 Sb
is a (a� b)-dimensional sphere
|Im(J)| = 2· denominator
✓B2k
2k
◆
for Im(J) ⇢ ⇧4k�1
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Bernoulli also come up in formulas for the numbers of
“exotic spheres” in dimension n
|⇥4k�1| = R(k) · |⇧4k�1| ·B2k/2k
R(k) := 22k�3(22k�1 � 1) if k is even
where
R(k) := 2
2k�2(2
2k�1 � 1) if k is odd
Let ⇥n denote the group of n-dimensional exotic spheres
Milnor and Kervaire (1962) showed:
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Thanks for listening!