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Dedekind zeta function - Quiz
Contributed by: Collier
  • 1. What kind of continuation do Dedekind zeta functions have to the complex plane?
A) Polynomial continuation
B) Logarithmic continuation
C) Holomorphic continuation
D) Meromorphic continuation
  • 2. How many poles does the Dedekind zeta function have in its analytic continuation?
A) Two simple poles
B) No poles
C) One simple pole
D) Infinitely many poles
  • 3. At what value of s does the Dedekind zeta function have its pole?
A) s = -1
B) s = 1
C) s = 0
D) s = 1/2
  • 4. Which formula gives the residue of the Dedekind zeta function at its pole?
A) Class number formula
B) Euler product formula
C) Functional equation
D) Discriminant formula
  • 5. What arithmetic data does the class number formula combine?
A) Real and complex places
B) Invariants of the prime ideals and residues
C) Invariants of the unit group and class group
D) Gamma and beta factors
  • 6. The construction of analytic continuation usually uses the fact that the Dedekind zeta function is what type of function for a trivial Hecke character?
A) Hecke L-function
B) Gamma function
C) Dirichlet L-series
D) Riemann zeta function
  • 7. Who was the first to prove the analytic continuation and functional equation for Hecke L-functions in 1917?
A) Erich Hecke
B) Bernhard Riemann
C) John Tate
D) Richard Dedekind
  • 8. Whose doctoral thesis simplified the complicated methodology used by Hecke in 1950?
A) Leonhard Euler
B) Carl Friedrich Gauss
C) Erich Hecke
D) John Tate
  • 9. The functional equation for the Dedekind zeta function relates its values at s and which other value?
A) 1 / s
B) 1 - s
C) -s
D) s - 1
  • 10. What does delta_K denote in the functional equation?
A) Unit group of K
B) Class number of K
C) Degree of K
D) Discriminant of K
  • 11. What does r1 denote in the context of the number field K?
A) Number of complex places
B) Class number
C) Discriminant of K
D) Number of real places
  • 12. What does r2 denote in the context of the number field K?
A) Number of conjugate pairs of complex places
B) Number of real places
C) Degree over Q
D) Total number of embeddings
  • 13. How is the degree of K over Q expressed using r1 and r2?
A) r1 + 2 * r2
B) r1 + r2
C) 2 * r1 + r2
D) r1 * r2
  • 14. What is the definition of the real gamma factor Gamma_R(s)?
A) (2 * pi)-s/2 * Gamma(s/2)
B) pi-s * Gamma(s)
C) (2 * pi)-s * Gamma(s)
D) pi-s/2 * Gamma(s/2)
  • 15. What is the definition of the complex gamma factor Gamma_C(s)?
A) pi-s/2 * Gamma(s/2)
B) (2 * pi)-s * Gamma(s)
C) pi-s * Gamma(s/2)
D) (2 * pi)-s/2 * Gamma(s)
  • 16. How is Lambda_K(s) constructed?
A) Using Hecke characters alone
B) Using only the Riemann zeta function
C) Using absolute discriminant, gamma factors, and zeta_K(s)
D) Using the class number and unit group
  • 17. What equation does Lambda_K(s) satisfy?
A) Lambda_K(s) = Lambda_K(-s)
B) Lambda_K(s) = Lambda_K(1 - s)
C) Lambda_K(s) = -Lambda_K(s)
D) Lambda_K(s) = Lambda_K(s - 1)
  • 18. What do the trivial zeroes of the Dedekind zeta function do?
A) Determine the class number
B) Shift the critical line
C) Cancel poles of the gamma factors
D) Create new poles in the complex plane
  • 19. What are nontrivial zeroes of the Dedekind zeta function?
A) Values where s = 1
B) Poles of the gamma factors
C) Common zeros of zeta_K and Lambda_K
D) Roots of the discriminant
  • 20. In what vertical strip must nontrivial zeros lie, according to the functional equation and Euler product?
A) 1 <= Re(s) <= 2
B) 0 <= Re(s) <= 1
C) 0 <= Re(s) <= 1/2
D) -1 <= Re(s) <= 0
  • 21. With respect to which line are the nontrivial zeros symmetric?
A) Imaginary axis
B) Critical line Re(s) = 1/2
C) Real line Re(s) = 0
D) Vertical line Re(s) = 1
  • 22. The Xi function defined for the Dedekind zeta function is analogous to the Xi function defined from which function?
A) Riemann zeta function
B) Hecke L-function
C) Dirichlet series
D) Gamma function
  • 23. What does the function Xi_K(s) do to the poles of Lambda_K?
A) Removes the poles, producing an entire function
B) Doubles the number of poles
C) Turns poles into zeroes
D) Shifts the poles to the real line
  • 24. Where does the Xi_K function move the critical line?
A) To the real line
B) To Re(s) = 1
C) To the imaginary axis
D) To infinity
  • 25. What functional equation does Xi_K satisfy in terms of complex conjugates?
A) Xi_K(overline{s}) = overline{Xi_K(s)}
B) Xi_K(overline{s}) = Xi_K(s)
C) Xi_K(overline{s}) = -Xi_K(s)
D) Xi_K(s) = Xi_K(1 - s)
  • 26. How can the Dedekind zeta function be defined?
A) As a matrix determinant
B) As a differential equation
C) As a Dirichlet series
D) As a finite sum
  • 27. Who is the Dedekind zeta function named after?
A) Peter Gustav Lejeune Dirichlet
B) Bernhard Riemann
C) Carl Friedrich Gauss
D) Richard Dedekind
  • 28. What type of ring is the ring of integers of a number field?
A) A polynomial ring
B) A field of fractions
C) A Euclidean domain
D) A Dedekind ring
  • 29. For general number rings, what are the main objects of interest instead of prime elements?
A) Quotient rings
B) Rational integers
C) Principal ideals
D) Prime ideals
  • 30. What is the absolute norm equal to for rational integers?
A) Zero
B) Always one
C) The non-negative integer generating the ideal
D) The square of the generator
  • 31. What mathematical bound is used along with summing over all ideal classes to show the growth bound?
A) Dirichlet's bound
B) Minkowski's bound
C) Riemann's bound
D) Cauchy's bound
  • 32. What is the growth bound for the sum of I(n) for n less than or equal to x?
A) O(x)
B) O(log x)
C) O(xs)
D) O(1)
  • 33. What is true about the norm function with respect to the multiplication of ideals?
A) It is constant
B) It is logarithmic
C) It is additive
D) It is multiplicative
  • 34. What kind of product is the Euler product in the region Re(s) > 1?
A) Asymptotically zero
B) Divergent
C) Absolutely convergent
D) Conditionally convergent
  • 35. What is the multiplicity of the trivial zeros of ζ_K(s) at non-zero even negative integers?
A) r1 + r2
B) r2
C) r1 - r2
D) r1 + r2 - 1
  • 36. What is the multiplicity of the trivial zeros of ζ_K(s) at odd negative integers?
A) r1 + r2 - 1
B) r2
C) r1
D) r1 + r2
  • 37. Who showed that ζ_K(s) is a non-zero rational number at odd negative numbers when K is totally real?
A) Taylor
B) Riemann
C) Dedekind
D) Siegel
  • 38. What are two fields called if they have the same Dedekind zeta function?
A) Arithmetically equivalent
B) Isomorphic
C) Conjugate fields
D) Algebraically closed
  • 39. Who showed in 1977 that two number fields are arithmetically equivalent based on inertia degrees?
A) Perlis
B) de Smit
C) Artin
D) Bosma
  • 40. What year did Bosma and de Smit use Gassmann triples to find arithmetically equivalent non-isomorphic fields?
A) 2002
B) 1995
C) 1977
D) 1983
  • 41. The Dedekind zeta function is a special case of the Hasse-Weil zeta function for which scheme?
A) Spec L
B) Spec Z
C) Spec Q
D) Spec O_K
  • 42. What specific representation of the Galois group yields the Dedekind zeta function as an Artin L-function for a finite extension L/K?
A) The trivial representation
B) The Galois representation
C) The induced representation
D) The regular representation
  • 43. For a finite Galois extension L/K, the Dedekind zeta function of L is the Artin L-function for which representation of Gal(L/K)?
A) The regular representation
B) The trivial representation
C) The standard representation
D) The alternating representation
  • 44. In the non-Galois extension formula, what group is denoted by H?
A) Gal(M/L)
B) Spec O_K
C) Gal(M/K)
D) Gal(L/K)
  • 45. What allows the factorization of the Dedekind zeta function to be described in terms of Hecke L-functions when L/K is an abelian extension?
A) Selberg orthonormality
B) Artin reciprocity
C) Aramata–Brauer theorem
D) Quadratic reciprocity
  • 46. Primitive Hecke characters correspond to what kind of representations of the abelian Galois group?
A) Reducible representations
B) Trivial representations
C) Irreducible representations
D) Quadratic representations
  • 47. In the zeta function formula for a quadratic field, what is the character chi used as?
A) A Galois group
B) A Jacobi symbol
C) A prime number
D) An entire function
  • 48. The fact that the zeta function of a quadratic field is a product of the Riemann zeta function and a Dirichlet L-function is an analytic formulation of what law?
A) Quadratic reciprocity law
B) Artin reciprocity law
C) Aramata–Brauer theorem
D) Dedekind conjecture
  • 49. For general Galois extensions, the Dedekind conjecture follows from which theorem?
A) Artin reciprocity
B) Selberg orthonormality conjecture
C) Uchida and van der Waall theorem
D) Aramata–Brauer theorem
  • 50. Which of the following implies the general case of the Dedekind conjecture?
A) The Dirichlet L-function formula
B) The quadratic reciprocity law
C) The Artin conjecture
D) The Riemann hypothesis
  • 51. Which conjecture besides the Artin conjecture can the general case of the Dedekind conjecture follow from directly?
A) The abelian extension conjecture
B) The Selberg orthonormality conjecture
C) The quadratic field conjecture
D) The Riemann zeta conjecture
  • 52. What allows one to distinguish trivial and non-trivial zeros of Dedekind zeta functions?
A) The critical line
B) Algebraic K-theory
C) The functional equation
D) Bernoulli numbers
  • 53. There are attempts to generalize the relation between Bernoulli numbers and values of the Riemann zeta function at what types of numbers?
A) Negative odd integers
B) Positive even integers
C) Zero and positive integers
D) Imaginary numbers
  • 54. According to Siegel, for a totally real field, the values of zeta_K(s) at negative integers are what kind of numbers?
A) Imaginary numbers
B) Nonzero rational numbers
C) Negative integers
D) Transcendental numbers
  • 55. Who conjectured specific values for the rational numbers of zeta_K(s) at negative integers in terms of algebraic K-theory?
A) Stephen Lichtenbaum
B) Siegel
C) Bernoulli
D) Riemann
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