Let be $\chi$ (non-trivial) Dirichlet charakter of conductor $f$. Then I know that $$B_{n,\chi}=f^{n-1}\sum_{a=1}^f\chi(a)B_n\left(\frac{a}{f}\right).$$ Assume $\chi(-1)=1$ and plug $n=2,$ then $$B_{2,\chi}=f\sum_{a=1}^f\chi(a)\left(\frac{a^2}{f^2}-\frac{a}{f}+\frac{1}{6}\right).$$ Using $\sum_{a=1}^f\chi(a)=0$ I get $$B_{2,\chi}=\frac{1}{f}\sum_{a=1}^f\chi(a)a^2-\sum_{a=1}^f\chi(a)a.$$ I would like to show that the second sum equals to $0,$ but I don't know how to do it. I haven't used the assumption $\chi(-1)=1$ yet, but I don't see why it is helpful.
2026-02-22 21:46:34.1771796794
Second generalized Bernoulli number $B_{2,\chi}$
103 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- Show that $(x,y,z)$ is a primitive Pythagorean triple then either $x$ or $y$ is divisible by $3$.
- About polynomial value being perfect power.
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Reciprocal-totient function, in term of the totient function?
- What is the smallest integer $N>2$, such that $x^5+y^5 = N$ has a rational solution?
- Integer from base 10 to base 2
- How do I show that any natural number of this expression is a natural linear combination?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
Related Questions in CHARACTERS
- Show that for character $\chi$ of an Abelian group $G$ we have $[\chi; \chi] \ge \chi(1)$.
- Properties of symmetric and alternating characters
- Counting characters of bounded conductors
- The condition between $\chi(1)$ and $[G:H]$ which gives us a normal subgroup.
- How to realize the character group as a Lie/algebraic/topological group?
- Show $\widehat{\mathbb{Z}}$ is isomorphic to $S^1$
- Confusion about the conductor for Dirichlet characters
- Relation between characters of symmetric group and general linear group
- Information on semilinear groups.
- Let $ f $ be an irreducible polynomial in $ \mathbb{F }_q [x] $, why $ f ^\frac{s}{deg (f)} $ has degree term $ s-1 $?
Related Questions in BERNOULLI-NUMBERS
- Infinite sum containing the Bernoulli numbers $B_{2n}$
- Stirling's series for Log Gamma
- Sign convention for Bernoulli numbers
- Contour integral calculation
- Bernoulli numbers [A classical introduction to modern number theory]
- convergence of an iterated series which is had Bernoulli numbers
- Evaluate $\lim\limits_{n\to\infty}(f(n+1)-f(n))$ where $f(n)=|B_{2n}|^{1/2n}$
- Justify an approximation of $\sum_{n=1}^\infty|G_n|\log\left(\frac{n+1}{n}\right)$, where $G_n$ is the $n$th Gregory coefficient
- Show that $n+1$ is prime if (Denominator(Bernoulli Number($n$)))/($n+1$) is an integer
- Bounds for Periodic Bernoulli Polynomials via Fourier Series
Related Questions in BERNOULLI-POLYNOMIALS
- Riemann zeta meromorphic cont. using Abel summation formula
- Formula for a sequence defined on $K_1(x,y) := y+0$ if $x \geq y$ and $y-1$ otherwise
- First Order Differential Equations Applied Question
- Constant determinant of matrix of Bernoulli polynomials
- Bounds for Periodic Bernoulli Polynomials via Fourier Series
- Prove following statements concerning Bernoulli polynomials
- Second generalized Bernoulli number $B_{2,\chi}$
- Bernoulli equation when not homogenous
- Help with Fourier Series $\sum_{j=1}^{\infty} \frac{1}{j^{2k}}\sin{2\pi j x}$
- Is the series $\sum_{n=0}^{\infty}\frac{B_n(z)}{2^n}$ convergent?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
$$\sum_{a=1}^f\chi(a)a=\sum_{a=1}^{f-1}\chi(a)a=\sum_{a=1}^{f-1}(f-a)\chi(f-a)=\sum_{a=1}^{f-1}(f-a)\chi(a)$$ $$\Rightarrow2\sum_{a=1}^{f-1}\chi(a)a=f\sum_{a=1}^{f-1}\chi(a)=0$$