Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive integer n where d runs over all of the divisors of n such that 2d+1 is a prime. For example, 1-B₁ = 1/2+3/ 1+ B₂ 1 1/2 + 1/3 + 1 − B₁ = 1/2 + 1/3 + 1/7² 1 1 1+ B4 = 1- B5 = 2 3 2 3 11 1 + B6 = 2 + 3 + 5 +132-B₁ 7 This is known as the Staudt-Clausen theorem, which is found by Staudt (1840) and by Clausen (1840) independently. The latter was published in 'Astronomis- che Nachrichten', the oldest astronomical journal founded in 1821 by H. C. Schumacher (one of the friends and astronomical collaborators of Gauss), as a brief announcement of the result without proof. Later Schwering (1899) gave another proof using 1 1 (n-1)! + + + x + 1 2(x + 1)(x+2) n(x + 1)(x + 2)...(x + n) 1 1 B₁ B₂ = - - + 2x² x3 x³ + = + 17 15 + - lin + + - im B3 x7
Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive integer n where d runs over all of the divisors of n such that 2d+1 is a prime. For example, 1-B₁ = 1/2+3/ 1+ B₂ 1 1/2 + 1/3 + 1 − B₁ = 1/2 + 1/3 + 1/7² 1 1 1+ B4 = 1- B5 = 2 3 2 3 11 1 + B6 = 2 + 3 + 5 +132-B₁ 7 This is known as the Staudt-Clausen theorem, which is found by Staudt (1840) and by Clausen (1840) independently. The latter was published in 'Astronomis- che Nachrichten', the oldest astronomical journal founded in 1821 by H. C. Schumacher (one of the friends and astronomical collaborators of Gauss), as a brief announcement of the result without proof. Later Schwering (1899) gave another proof using 1 1 (n-1)! + + + x + 1 2(x + 1)(x+2) n(x + 1)(x + 2)...(x + n) 1 1 B₁ B₂ = - - + 2x² x3 x³ + = + 17 15 + - lin + + - im B3 x7
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Show that the Bernoulli number B₁ satisfies
B₁
= (-1) ² ( ²2² +²24 ²+1)
Σ
(mod 1)
2d
d
for any positive integer n where d runs over all of the divisors of n such that
2d + 1 is a prime.
For example,
1
1
1
1 1
1-B₁ =
1 + B₂ =
1- B3 = = 1/2+1/3+2/2+
2
3
2
3 5'
7'
1 1 1
1
1 1
1+ B4 =
+
+
1 - B5 =
+
2
3 5
2
3 11
1
1
1
1 + B6 =
1 1
+
+-+
+
2-B₁ =
2
3
5 7
13'
2 3
This is known as the Staudt-Clausen theorem, which is found by Staudt (1840)
and by Clausen (1840) independently. The latter was published in 'Astronomis-
che Nachrichten', the oldest astronomical journal founded in 1821 by H. C.
Schumacher (one of the friends and astronomical collaborators of Gauss), as
a brief announcement of the result without proof. Later Schwering (1899) gave
another proof using
1
1
(n-1)!
+
+
+
+...
x+1 2(x + 1)(x + 2)
n(x + 1)(x + 2)... (x + n)
1
1 B₁ B₂ B3
=
+
+
X
2x²
x3
x³
+
1 1
+-+
+
+
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