A) Holomorphic continuation B) Polynomial continuation C) Meromorphic continuation D) Logarithmic continuation
A) Infinitely many poles B) One simple pole C) Two simple poles D) No poles
A) s = -1 B) s = 0 C) s = 1 D) s = 1/2
A) Functional equation B) Euler product formula C) Discriminant formula D) Class number formula
A) Gamma and beta factors B) Invariants of the unit group and class group C) Real and complex places D) Invariants of the prime ideals and residues
A) Riemann zeta function B) Gamma function C) Dirichlet L-series D) Hecke L-function
A) Bernhard Riemann B) John Tate C) Richard Dedekind D) Erich Hecke
A) Erich Hecke B) Leonhard Euler C) Carl Friedrich Gauss D) John Tate
A) 1 / s B) s - 1 C) -s D) 1 - s
A) Degree of K B) Class number of K C) Discriminant of K D) Unit group of K
A) Number of real places B) Discriminant of K C) Class number D) Number of complex places
A) Number of real places B) Degree over Q C) Number of conjugate pairs of complex places D) Total number of embeddings
A) r1 + 2 * r2 B) r1 + r2 C) 2 * r1 + r2 D) r1 * r2
A) pi-s/2 * Gamma(s/2) B) (2 * pi)-s/2 * Gamma(s/2) C) pi-s * Gamma(s) D) (2 * pi)-s * Gamma(s)
A) pi-s/2 * Gamma(s/2) B) (2 * pi)-s/2 * Gamma(s) C) pi-s * Gamma(s/2) D) (2 * pi)-s * Gamma(s)
A) Using the class number and unit group B) Using only the Riemann zeta function C) Using Hecke characters alone D) Using absolute discriminant, gamma factors, and zeta_K(s)
A) Lambda_K(s) = Lambda_K(1 - s) B) Lambda_K(s) = Lambda_K(s - 1) C) Lambda_K(s) = Lambda_K(-s) D) Lambda_K(s) = -Lambda_K(s)
A) Cancel poles of the gamma factors B) Shift the critical line C) Create new poles in the complex plane D) Determine the class number
A) Values where s = 1 B) Roots of the discriminant C) Common zeros of zeta_K and Lambda_K D) Poles of the gamma factors
A) -1 <= Re(s) <= 0 B) 0 <= Re(s) <= 1 C) 0 <= Re(s) <= 1/2 D) 1 <= Re(s) <= 2
A) Vertical line Re(s) = 1 B) Critical line Re(s) = 1/2 C) Imaginary axis D) Real line Re(s) = 0
A) Gamma function B) Riemann zeta function C) Hecke L-function D) Dirichlet series
A) Removes the poles, producing an entire function B) Shifts the poles to the real line C) Doubles the number of poles D) Turns poles into zeroes
A) To Re(s) = 1 B) To the imaginary axis C) To infinity D) To the real line
A) Xi_K(overline{s}) = Xi_K(s) B) Xi_K(s) = Xi_K(1 - s) C) Xi_K(overline{s}) = overline{Xi_K(s)} D) Xi_K(overline{s}) = -Xi_K(s)
A) As a differential equation B) As a matrix determinant C) As a Dirichlet series D) As a finite sum
A) Richard Dedekind B) Peter Gustav Lejeune Dirichlet C) Bernhard Riemann D) Carl Friedrich Gauss
A) A polynomial ring B) A Dedekind ring C) A Euclidean domain D) A field of fractions
A) Prime ideals B) Rational integers C) Quotient rings D) Principal ideals
A) Always one B) The non-negative integer generating the ideal C) Zero D) The square of the generator
A) Cauchy's bound B) Riemann's bound C) Minkowski's bound D) Dirichlet's bound
A) O(1) B) O(x) C) O(xs) D) O(log x)
A) It is logarithmic B) It is constant C) It is multiplicative D) It is additive
A) Divergent B) Absolutely convergent C) Conditionally convergent D) Asymptotically zero
A) r1 + r2 B) r1 + r2 - 1 C) r1 - r2 D) r2
A) r1 + r2 B) r2 C) r1 D) r1 + r2 - 1
A) Siegel B) Dedekind C) Riemann D) Taylor
A) Algebraically closed B) Arithmetically equivalent C) Conjugate fields D) Isomorphic
A) Perlis B) Bosma C) de Smit D) Artin
A) 1977 B) 2002 C) 1983 D) 1995
A) Spec L B) Spec O_K C) Spec Q D) Spec Z
A) The induced representation B) The regular representation C) The Galois representation D) The trivial representation
A) The regular representation B) The alternating representation C) The trivial representation D) The standard representation
A) Spec O_K B) Gal(L/K) C) Gal(M/L) D) Gal(M/K)
A) Selberg orthonormality B) Aramata–Brauer theorem C) Artin reciprocity D) Quadratic reciprocity
A) Irreducible representations B) Reducible representations C) Trivial representations D) Quadratic representations
A) A prime number B) An entire function C) A Galois group D) A Jacobi symbol
A) Artin reciprocity law B) Dedekind conjecture C) Aramata–Brauer theorem D) Quadratic reciprocity law
A) Artin reciprocity B) Selberg orthonormality conjecture C) Aramata–Brauer theorem D) Uchida and van der Waall theorem
A) The Riemann hypothesis B) The Dirichlet L-function formula C) The quadratic reciprocity law D) The Artin conjecture
A) The quadratic field conjecture B) The Riemann zeta conjecture C) The Selberg orthonormality conjecture D) The abelian extension conjecture
A) The critical line B) The functional equation C) Bernoulli numbers D) Algebraic K-theory
A) Imaginary numbers B) Zero and positive integers C) Negative odd integers D) Positive even integers
A) Nonzero rational numbers B) Imaginary numbers C) Transcendental numbers D) Negative integers
A) Bernoulli B) Riemann C) Stephen Lichtenbaum D) Siegel |