Distributions of Primes in Number Fields and the L-function Ratios Conjecture

Transcription

1 Distributions of Primes in Number Fields and the L-function Ratios Conjecture Casimir Kothari Maria Ross with Trajan Hammonds, Ben Logsdon Advisor: Steven J. Miller Young Mathematicians Conference August 10, 2018

2 Background We will be dealing with number fields Q(α)/Q.

3 Background We will be dealing with number fields Q(α)/Q. A number field has a ring of integers, which is the ring of all integral elements of K.

4 Background We will be dealing with number fields Q(α)/Q. A number field has a ring of integers, which is the ring of all integral elements of K. Example: Q(i) has ring of integers Z[i] = {a + bi : a, b Z}.

5 Background We will be dealing with number fields Q(α)/Q. A number field has a ring of integers, which is the ring of all integral elements of K. Example: Q(i) has ring of integers Z[i] = {a + bi : a, b Z}. An ideal p = a + bi of Z[i] is defined by a + bi = {r(a + bi) : r Z[i]}.

6 Background We will be dealing with number fields Q(α)/Q. A number field has a ring of integers, which is the ring of all integral elements of K. Example: Q(i) has ring of integers Z[i] = {a + bi : a, b Z}. An ideal p = a + bi of Z[i] is defined by a + bi = {r(a + bi) : r Z[i]}. We say an ideal p of a ring R is prime if it is not equal to R and ab p = a p or b p.

7 7 Intro L-Functions The Ratios Conjecture Computing the Variance Conclusions Refs Background To an ideal p = a + bi Z[i] we can associate an angle θ p by e iθp = a + bi a + bi.

8 Background To an ideal p = a + bi Z[i] we can associate an angle θ p by e iθp = a + bi a + bi. θ p determined up to rotation by π/2.

9 Background To an ideal p = a + bi Z[i] we can associate an angle θ p by e iθp = a + bi a + bi. θ p determined up to rotation by π/2. One can then study the smooth count of angles of prime ideals lying in a certain window: ψ K,X (θ) = ( ) N(a) Φ Λ(a)F K (θ a θ). X a Z[i] Φ a smooth compactly supported function Λ a generalization of Von-Mangoldt F K detects angles of size 2π/K.

10 History Hecke (1918): For K fixed, we have ψ K,X (θ) 4X K 0 Φ(x)dx.

11 History Hecke (1918): For K fixed, we have ψ K,X (θ) 4X K 0 Φ(x)dx. Rudnick, Waxman (2017): Calculated the mean value ψ K,X of ψ K,X (θ).

12 History Hecke (1918): For K fixed, we have ψ K,X (θ) 4X K 0 Φ(x)dx. Rudnick, Waxman (2017): Calculated the mean value ψ K,X of ψ K,X (θ). SMALL 2017 REU: Calculated Var(ψ K,X ) = 1 2π 2π 0 ψ K,X (θ) ψ K,X 2 dθ.

13 Our Setting Instead of considering the ring of integers of Q(i), we adjoin to Q the square root of a negative integer d.

14 Our Setting Instead of considering the ring of integers of Q(i), we adjoin to Q the square root of a negative integer d. d is chosen so that all ideals of the ring of integers are principal (generated by a single element) and Q( d) is said to have class number one.

15 Our Setting Instead of considering the ring of integers of Q(i), we adjoin to Q the square root of a negative integer d. d is chosen so that all ideals of the ring of integers are principal (generated by a single element) and Q( d) is said to have class number one. Let U be the number of units in Q( d)

16 Our Setting Instead of considering the ring of integers of Q(i), we adjoin to Q the square root of a negative integer d. d is chosen so that all ideals of the ring of integers are principal (generated by a single element) and Q( d) is said to have class number one. Let U be the number of units in Q( d) To each ideal p = a + bα d we may associate the angle θ p defined by e iθp = a + bα d a + bα d.

17 Our Setting Instead of considering the ring of integers of Q(i), we adjoin to Q the square root of a negative integer d. d is chosen so that all ideals of the ring of integers are principal (generated by a single element) and Q( d) is said to have class number one. Let U be the number of units in Q( d) To each ideal p = a + bα d we may associate the angle θ p defined by e iθp = a + bα d a + bα d. Thus, an analogous ψ K,X (θ) can be defined and the same questions can be asked in this scenario: we are interested in studying Var(ψ K,X ) in Z[α d ].

18 L-functions

19 L-functions An L-function is defined by a series of the form L(s, f ) = n=0 f (n) n s for s C, where f is some complex-valued multiplicative function

20 L-functions An L-function is defined by a series of the form L(s, f ) = n=0 f (n) n s for s C, where f is some complex-valued multiplicative function L(s, f ) has an Euler product

21 L-functions An L-function is defined by a series of the form L(s, f ) = n=0 f (n) n s for s C, where f is some complex-valued multiplicative function L(s, f ) has an Euler product Usually convergent on a half-plane

22 L-functions A Completed L-function Λ(s, f ) is the analytic continuation of an L-function L(s, f ) to a meromorphic function on the complex plane (i.e., complex differentiable everywhere except finitely many poles)

23 L-functions A Completed L-function Λ(s, f ) is the analytic continuation of an L-function L(s, f ) to a meromorphic function on the complex plane (i.e., complex differentiable everywhere except finitely many poles) Λ(s, f ) satisfies a functional equation of the form Λ(s, f ) = ε f Λ(1 s, f ) with ε f = ±1

24 Example 1: Riemann zeta function ζ(s) = n=0 1 n s converges for Re(s) > 1

25 Example 1: Riemann zeta function ζ(s) = n=0 1 n s converges for Re(s) > 1 Euler product: ζ(s) = (1 1p ) 1 s p prime

26 Example 1: Riemann zeta function ζ(s) = n=0 1 n s converges for Re(s) > 1 Euler product: ζ(s) = p prime (1 1p s ) 1 Completed zeta function: ξ(s) = π s/2 Γ(s/2)ζ(s)

27 Example 1: Riemann zeta function ζ(s) = n=0 1 n s converges for Re(s) > 1 Euler product: ζ(s) = p prime (1 1p s ) 1 Completed zeta function: ξ(s) = π s/2 Γ(s/2)ζ(s) Functional equation: ξ(s) = ξ(1 s)

28 Hecke characters and L-functions A Hecke character χ is a homomorphism χ : I C where I is a multiplicative group of fractional ideals of a field C is the multiplicative group of complex numbers χ satisfies certain properties

29 Hecke characters and L-functions If χ : I C is a Hecke character, then a Hecke L-function is defined by a series of the form L(s, χ) = a I where N(a) is the ideal norm of a χ(a) N(a) s

30 Hecke characters and L-functions If χ : I C is a Hecke character, then a Hecke L-function is defined by a series of the form L(s, χ) = a I where N(a) is the ideal norm of a Euler product is L(s, χ) = p prime χ(a) N(a) s ( 1 χ(p) ) 1 N(p) s

31 Set up for our Hecke character Q( d) imaginary quadratic field, class number 1

32 Set up for our Hecke character Q( d) imaginary quadratic field, class number 1 Z[α d ] = ring of integers, { 1+ d d 1 mod 4 α d = 2 d d 2, 3 mod 4

33 Set up for our Hecke character Q( d) imaginary quadratic field, class number 1 Z[α d ] = ring of integers, { 1+ d d 1 mod 4 α d = 2 d d 2, 3 mod 4 U = number of units in Z[α d ]

34 Set up for our Hecke character Q( d) imaginary quadratic field, class number 1 Z[α d ] = ring of integers, { 1+ d d 1 mod 4 α d = 2 d d 2, 3 mod 4 U = number of units in Z[α d ] Define Hecke character χ Uk on ideals a = a + bα d of Z[α d ] by χ Uk (a) = ( ) Uk a + bαd = e iukθa a + bα d

35 Hecke L-function Our main L-function of interest: L k (s) = a Z[α d ] χ Uk (a) N(a) s = a Z[α d ] e iukθa N(a) s

36 Hecke L-function Our main L-function of interest: L k (s) = a Z[α d ] χ Uk (a) N(a) s = a Z[α d ] e iukθa N(a) s Main goal is to compute Var(ψ K,X ), and we use Fourier and Mellin transforms to write Var(ψ K,X ) = 1 ( ) k 2 L k 4π 2 K 2 f (s) L k (s K L k L ) Φ(s) Φ(s )X s X s dsds k k 0 (2) (2)

37 The Ratios Conjecture

38 Background We are interested in computing the average R K (α, β, γ, δ) = 1 2K k <K k 0 L k ( α)l k( β) L k ( γ)l k( δ).

39 Background We are interested in computing the average R K (α, β, γ, δ) = 1 2K k <K k 0 L k ( α)l k( β) L k ( γ)l k( δ). Random Matrix Theory: Often used to model ratios of products of L-Functions, however don t capture lower order arithmetic terms.

40 Background We are interested in computing the average R K (α, β, γ, δ) = 1 2K k <K k 0 L k ( α)l k( β) L k ( γ)l k( δ). Random Matrix Theory: Often used to model ratios of products of L-Functions, however don t capture lower order arithmetic terms. The Ratios conjecture is a procedure for computing averages of ratios of L-functions that predicts these lower order terms, providing a better model.

41 Inputs to the Ratios Conjecture Approximate Functional Equation: L k (s) can be written as L k (s) = n<x a n n + ɛ k(s) a m + remainder s m1 s m<y where ɛ k (s) is a ratio of gamma factors that appear in the functional equation for L k (s).

42 Inputs to the Ratios Conjecture Approximate Functional Equation: L k (s) can be written as L k (s) = n<x a n n + ɛ k(s) a m + remainder s m1 s m<y where ɛ k (s) is a ratio of gamma factors that appear in the functional equation for L k (s). Generalized Möbius Functional Equation: One may also write 1 L k (s) = h=1 µ k (h) h s where µ k is an appropriate generalization of the Möbius function.

43 The Procedure Use the approximate functional equation for terms in the numerator (ignoring the remainder term) and the Möbius functional equation for terms in the denominator.

44 The Procedure Use the approximate functional equation for terms in the numerator (ignoring the remainder term) and the Möbius functional equation for terms in the denominator. Compute the expected value of the ɛ k (s) factors over k, and replace them with their averages in the expanded product.

45 The Procedure Use the approximate functional equation for terms in the numerator (ignoring the remainder term) and the Möbius functional equation for terms in the denominator. Compute the expected value of the ɛ k (s) factors over k, and replace them with their averages in the expanded product. Compute coefficient averages, and replace each component of the sum with its average.

46 The Procedure Use the approximate functional equation for terms in the numerator (ignoring the remainder term) and the Möbius functional equation for terms in the denominator. Compute the expected value of the ɛ k (s) factors over k, and replace them with their averages in the expanded product. Compute coefficient averages, and replace each component of the sum with its average. Extend the remaining sums out to infinity, and call the total G(α, β, γ, δ). The conjecture then gives a nice expression for the average R K (α, β, γ, δ) in terms of G.

47 Sketch of the Variance Calculation

48 Ratios Calculation We use the ratios conjecture to analyze the expression Var(ψ K,X) = X ( ) k 2 ( ) ( ) L k 1 L 4π 2 K f 2 R R K L k 2 + α k 1 L k 2 + β k 0 ( ) ( ) Φ α Φ 2 + β X α X β dadb where α = ɛ + ia β = ɛ + ib

49 Ratios Calculation We use the ratios conjecture to analyze the expression Var(ψ K,X) = X ( ) k 2 ( ) ( ) L k 1 L 4π 2 K f 2 R R K L k 2 + α k 1 L k 2 + β k 0 ( ) ( ) Φ α Φ 2 + β X α X β dadb where α = ɛ + ia β = ɛ + ib Note that L k L k ( α ) L k L k ( ) β = α L k ( α)l k( β) β L k ( γ)l k( δ) γ=α,δ=β

50 Ratios Calculation From ratios conjecture, we can write f k K ( ) k 2 L ( ) ( ) k 1 L K L k 2 + α k 1 L k 2 + β f k K ( ) k 2 K α β R K(α, β, γ, δ) γ=α,δ=β where R K (α, β, γ, δ) = 1 2K k <K k 0 L k ( α)l k( β) L k ( γ)l k( δ)

51 Ratios Calculation Following the steps of the ratios conjecture, we compute:

52 Ratios Calculation Following the steps of the ratios conjecture, we compute: G(α, β, γ, δ) = coefficient average from functional equation for L k (s)

53 Ratios Calculation Following the steps of the ratios conjecture, we compute: G(α, β, γ, δ) = coefficient average from functional equation for L k (s) ɛ k (s) K = gamma factor average from functional equation for L k (s) Note: K denotes average over k K

54 Ratios Calculation Then we have ( ɛ k (s) K = 1 2π 2 2s D U 2 K ) 2s 1 where D is the fundamental discriminant of the number field, i.e. D = 4d or D = d depending on d mod 4

55 Ratios Calculation Then we have ( ɛ k (s) K = 1 2π 2 2s D U 2 K ) 2s 1 where D is the fundamental discriminant of the number field, and G(α, β, γ, δ) = i.e. D = 4d or D = d depending on d mod 4 ζ(1 + 2α)ζ(1 + 2β)ζ(1 + α + β)ζ(1 + γ + δ) holomorphic function ζ(1 + α + γ)ζ(1 + α + δ)ζ(1 + β + γ)ζ(1 + β + δ)

56 Computing the Variance We use these averages to obtain expression for R K (α, β, γ, δ): R K (α, β, γ, δ) G(α, β, γ, δ) + ɛ k (1/2 + α) K G( α, β, γ, δ) + ɛ k (1/2 + β) K G(α, β, γ, δ) + ɛ k (1/2 + α)ɛ k (1/2 + β) K G( α, β, γ, δ)

57 Computing the Variance We use these averages to obtain expression for R K (α, β, γ, δ): R K (α, β, γ, δ) G(α, β, γ, δ) + ɛ k (1/2 + α) K G( α, β, γ, δ) + ɛ k (1/2 + β) K G(α, β, γ, δ) + ɛ k (1/2 + α)ɛ k (1/2 + β) K G( α, β, γ, δ) and plug back into the formula for variance: Var(ψ K,X) = X ( ) k 2 4π 2 K f RK(α, β, γ, δ) 2 R R K α β k 0 ( ) ( ) Φ α Φ 2 + β X α X β dadb γ=α,δ=β

58 The Variance Using contour integration, we arrive at our final expression for the variance: Theorem Assume the Ratios Conjecture. Then X K X Var(ψ K,X ) = K ( 2 log K log where X = K γ. X K (γ log K + 1) if γ < 1 (γ log K 3) ( )) if 1 < γ < 2, if γ > 2 π 2 4

59 Conclusions

60 Conclusions Variance and G(α, β, γ, δ) term seemed to be independent of specific number field, as coefficient averages remained the same, and contour integration led to cancellation. Dependence on the specific number field appeared in the Gamma factor averages. Further generalizations to higher class number seem doable, but there are obstacles to this such as producing a well-defined Hecke character on non-principal ideals that still captures the notion of angle.

61 References

62 B. Conrey and N. Snaith.: Applications of the L-functions ratios conjectures, Proc. Lond. Math. Soc. 94(3), (2007). E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I., Math. Z. 1 (1918), II, Math. Z. 6 (1920), Z. Rudnick, E. Waxman, Angles of Gaussian Primes. SMALL 2017 REU Variance of Gaussian Primes (2017). SMALL 2017 REU Ratios For Hecke L Functions - The Computation (2017).

63 Thank You!