Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Average values of some non-multiplicative functions Greg Martin University of British Columbia joint work with Paul Pollack, Ethan Smith Canadian Number Theory Association XII Meeting University of Lethbridge June 22, 2012 slides can be found on my web page www.math.ubc.ca/ ∼ gerg/index.shtml?slides Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Outline Motivation: least quadratic nonresidues 1 Average least character nonresidues 2 Average least non-split prime in cubic number fields 3 Counting points on reductions of elliptic curves 4 Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Some constants that will appear The following values will be the average value of some function in this talk: ∞ 2 2 + 3 4 + 5 8 + 7 p k � 16 + · · · = 2 k ≈ 3 . 67464 1 k = 1 ( p + 1 ) − 1 ≈ 2 . 53505 � ℓ 2 � 2 ℓ prime p ≤ ℓ p prime 5 ℓ 3 + 6 ℓ 2 + 6 ℓ p 2 � � 6 ( p 2 + p + 1 ) ≈ 2 . 12110 3 6 ( ℓ 2 + ℓ + 1 ) p <ℓ ℓ prime p prime � �� � 2 1 1 � 1 − 1 + ≈ 0 . 50517 4 ( p − 1 ) 2 3 ( p − 2 )( p − 1 )( p + 1 ) p > 2 p prime Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Erd˝ os’s result Definition: least quadratic nonresidue � n � For q prime, n 2 ( q ) is the least number n such that = − 1 . q (Note that n 2 ( q ) is always a prime.) Theorem (Erd˝ os, 1961) � 1 ∞ � p k � � n 2 ( q ) = 2 k , lim π ( x ) x →∞ 2 < q ≤ x k = 1 where p k denotes the k th prime in increasing order. The average value of the least quadratic nonresidue modulo a k = 1 p k / 2 k ≈ 3 . 67464 . prime is the constant � ∞ Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) A surprising constant . . . A shiny result Time muffles the original éclat of a theorem. In 1967, in a Nottingham seminar, I did not get past the value of Erd˝ os’s limit . . . before Eduard Wirsing stopped me. “I don’t believe it!”, says he, looking at the expression for the constant, “I have never seen anything like it!” Peter Elliott Exercise ∞ ∞ p k 1 � � 2 k = 2 π ( n ) k = 1 n = 0 Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) . . . but a believable constant Definition: least quadratic nonresidue � n � For q prime, n 2 ( q ) is the least number n such that = − 1 . q Heuristic For a fixed prime p , asymptotically half the primes q satisfy � p � = − 1 . Using the number theorist’s conceit, q � p � p = 1 ) = 1 � � Prob ( = − 1 ) = Prob ( 2 . q q The statement n 2 ( q ) = p k is equivalent to � p k − 1 � p 1 � p 2 � p k � � � � = = · · · = = 1 and = − 1 . q q q q These k events should be independent, so we should have Prob ( n 2 ( q ) = p k ) = 2 − k . So the expected value of n 2 ( q ) should be � ∞ k = 1 2 − k p k . Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) 1 Evaluating � 2 < q ≤ x n 2 ( q ) , in one slide π ( x ) For n 2 ( q ) fixed, or small compared to x , this heuristic can 1 be made rigorous using quadratic reciprocity and the prime number theorem for arithmetic progressions: ∞ 1 p k � � n 2 ( q ) = 2 k + o ( 1 ) . π ( x ) 2 < q ≤ x k = 1 n 2 ( q ) small For medium-sized n 2 ( q ) , a similar approach using the 2 Brun–Titchmarsh theorem gives a suitable upper bound. For large n 2 ( q ) , Burgess’s bounds give 3 π ( x ) x 1 / 4 √ e + ε # 1 1 � � � n 2 ( q ) ≪ 2 < q ≤ x : n 2 ( q ) large , π ( x ) 2 < q ≤ x n 2 ( q ) large which can be shown to be o ( 1 ) by the large sieve. Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Considering all quadratic characters Definition: least character nonresidue for real characters For D a fundamental discriminant, n 2 ( D ) is the least number n � D � = − 1 . ( n 2 ( D ) is still always a prime.) such that n Theorem (Pollack, 2012) � − 1 � � ℓ 2 � � � p + 2 � � lim 1 n 2 ( D ) = 2 ( p + 1 ) , 2 ( ℓ + 1 ) x →∞ ℓ p <ℓ | D |≤ x | D |≤ x where � ℓ is over primes ℓ . The average value of the least character nonresidue for quadratic characters is ≈ 4 . 98085 . ℓ 2 p + 2 � �� D � � � �� D � � 2 ( p + 1 ) = ℓ Prob = − 1 � = − 1 p <ℓ Prob ℓ p 2 ( ℓ + 1 ) p <ℓ Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Considering all characters Definition: least character nonresidue For χ a Dirichlet character, n χ is the least number n such that χ ( n ) � = 1 and χ ( n ) � = 0 . ( n χ is still always a prime.) Theorem (M.–Pollack, 2012+) If we define ℓ 2 � ∆ = p ≤ ℓ ( p + 1 ) ≈ 2 . 53505 , � ℓ where the sum and product are taken over primes ℓ and p , then � − 1 � � � � � � � lim 1 = ∆ . n χ x →∞ q ≤ x χ (mod q ) q ≤ x χ (mod q ) Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Most characters quit right away Definition ℓ ( q ) is the least prime not dividing q . Note that n χ ≥ ℓ ( q ) . Proposition n χ − φ ( q ) ℓ ( q ) ≪ φ ( q )( log log q ) 3 � � � � 0 ≤ n χ − ℓ ( q ) = log q χ (mod q ) χ (mod q ) The proof involves sorting the χ according to whether n χ is equal to ℓ ( q ) , is medium-sized, or is large. The structure of the group ( Z / q Z ) × comes into play, as does the multiplicative order of ℓ ( q ) modulo q . Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) A sum of a non-multiplicative function � − 1 � � ℓ 2 � � � � � � = ∆ = lim 1 n χ � p ≤ ℓ ( p + 1 ) x →∞ q ≤ x q ≤ x χ (mod q ) χ (mod q ) ℓ The theorem now reduces to showing: � − 1 � � � � � φ ( q ) φ ( q ) ℓ ( q ) = ∆ lim x →∞ q ≤ x q ≤ x The function φ ( q ) ℓ ( q ) is certainly not multiplicative. However, if we sort q according to gcd ( q , Q ) where Q = � p ≤ z p , then both ℓ ( q ) and φ ( q ) / q are essentially determined as a function of gcd ( q , Q ) . We sum over all divisors of Q and (after four pages or so) obtain ∆ . Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Considering only primitive characters Theorem (M.–Pollack, 2012+) If we define p 2 − p − 1 ℓ 4 ∆ ∗ = � � ( p + 1 ) 2 ( p − 1 ) ≈ 2 . 15144 , ( ℓ + 1 ) 2 ( ℓ − 1 ) ℓ p <ℓ where the sum and product are taken over primes ℓ and p , then � − 1 � � � � � � � = ∆ ∗ . lim 1 n χ x →∞ q ≤ x q ≤ x χ (mod q ) χ (mod q ) χ primitive χ primitive Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) I’ve been talking in prose all this time? Theorem (Erd˝ os, 1961) � 1 ∞ � p k � � n 2 ( q ) = lim 2 k π ( x ) x →∞ 2 < q ≤ x k = 1 Among quadratic number fields with prime conductor: The average least inert prime is � ∞ p k 2 k . k = 1 Theorem (Pollack, 2012) � − 1 � � ℓ 2 � � � p + 2 � � lim 1 n 2 ( D ) = 2 ( ℓ + 1 ) 2 ( p + 1 ) x →∞ ℓ p <ℓ | D |≤ x | D |≤ x Among all quadratic number fields: ℓ 2 p + 2 The average least inert prime is � 2 ( p + 1 ) . p <ℓ 2 ( ℓ + 1 ) Average values of some non-multiplicative functions Greg Martin
Least quadratic nonresidues Least character nonresidues Primes in cubic fields Counting points on E ( F p ) Cubic number field result Definition: least non-split prime For K a number field, D K is the discriminant of K , and n K is the least rational prime that does not split completely in K . Theorem (M.–Pollack, 2012+) If we define 5 ℓ 3 + 6 ℓ 2 + 6 ℓ p 2 � � ∆ non-split = 6 ( p 2 + p + 1 ) ≈ 2 . 12110 , 6 ( ℓ 2 + ℓ + 1 ) ℓ p <ℓ where the sum and product are taken over primes ℓ and p , then � − 1 � � � � � lim 1 n K = ∆ non-split , x →∞ | D K |≤ x | D K |≤ x where the sums on the left-hand side are taken over (all isomorphism classes of) cubic fields K for which | D K | ≤ x . Average values of some non-multiplicative functions Greg Martin
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