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