The sum of digits of primes in Z [ i ] Thomas Stoll (TU Wien) Journ - PowerPoint PPT Presentation

Introduction Main results Fourier analysis The sum of digits of primes in Z [ i ] Thomas Stoll (TU Wien) Journ´ ees de Num´ eration, Graz 2007 April 19, 2007 (joint work with M. Drmota and J. Rivat) Research supported by the Austrian Science Foundation, no.9604. Th. Stoll Sum of digits of Gaussian primes

Introduction Gaussian primes Main results Complex sum-of-digits Fourier analysis Pointillism: Gaussian primes in the first quadrant Th. Stoll Sum of digits of Gaussian primes

Introduction Gaussian primes Main results Complex sum-of-digits Fourier analysis Gaussian primes Gaussian primes p : (1) 1 + i and its associates, (2) the rational primes 4 k + 3 and their associates, (3) the factors in Z [ i ] of the rational primes 4 k + 1. Hecke prime number theorem: N π i ( N ) := { p ∈ Z [ i ] non-associated : | p | 2 ≤ N } ∼ log N . Complex Von Mangoldt function Λ i : � log | p | , ε unit , ν ∈ Z + ; n = ε p ν , Λ i ( n ) = 0 , otherwise. Th. Stoll Sum of digits of Gaussian primes

Introduction Gaussian primes Main results Complex sum-of-digits Fourier analysis The complex sum-of-digits function [K´ atai/Kov´ acs, K´ atai/Szab´ o] Let q = − a ± i (choose a sign) with a ∈ Z + . Then every n ∈ Z [ i ] has a unique finite representation λ − 1 � ε j q j , n = j =0 where ε j ∈ { 0 , 1 , . . . , a 2 } are the digits and ε λ − 1 � = 0. Let s q ( n ) = � λ − 1 j =0 ε j be the sum-of-digits function in Z [ i ]. Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis Main results I Recall q = − a ± i , a ∈ Z + and write e ( x ) = exp(2 π i x ). Theorem For any α ∈ R with ( a 2 + 2 a + 2) α �∈ Z , a even, there is σ q ( α ) > 0 such that � Λ i ( n ) e ( α s q ( n )) ≪ N 1 − σ q ( α ) . | n | 2 ≤ N Theorem The sequence ( α s q ( p )) , a even, running over Gaussian primes p is uniformly distributed modulo 1 if and only if α ∈ R \ Q . Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis Main results II Theorem Let a ≥ 2 , a even, b ∈ Z ≥ 0 , m ∈ Z + , m ≥ 2 and set d = ( m , a 2 + 2 a + 2) . If ( b , d ) = 1 then there exists σ q , m > 0 such that p ∈ Z [ i ] : | p | 2 ≤ N , � # s q ( p ) ≡ b mod m } d m ϕ ( d ) π i ( N ) + O q , m ( N 1 − σ q , m ) . = If ( b , d ) � = 1 then the set has cardinality O q , m ( N 1 − σ q , m ) . Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis Inequalities ` a la Vaughan and van der Corput Lemma (Vaughan) Let β 1 ∈ (0 , 1 3 ) , β 2 ∈ ( 1 2 , 1) . Further suppose that for all a n , b n with | a n | , | b n | ≤ 1 , n ∈ Z [ i ] and all M ≤ x we uniformly have (put Q = a 2 + 1 ) � � � � � � � � M ≤ x β 1 , max e ( α s q ( mn )) ≤ U for � � � � x x Q | m | 2 < t ≤ | m | 2 � � M Q | m | 2 < | n | 2 ≤ t x Q < | m | 2 ≤ M � � � � � � � � x β 1 ≤ M ≤ x β 2 . � � ≤ U a m b n e ( α s q ( mn )) for � � � � � x x � M Q | m | 2 < | n | 2 ≤ Q < | m | 2 ≤ M | m | 2 � � Then � � � � � � � ≪ U (log x ) 2 . Λ i ( n ) e ( α s q ( n )) � � � � x Q < | n | 2 ≤ x � � Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis Lemma (Van der Corput) Let z n ∈ C with n ∈ Z [ i ] and A , B ≥ 0 . Then for all R ≥ 1 we have 2 � � � � � B − A � � 1 − | r | � · B + A � � � � � z n ≤ C 3 + 2 z n + r z n . � � R R 2 R � � A < | n | < B | r | < 2 R A < | n | < B � � A < | n + r | < B Now, start with � � � � � � � � S = b n e ( α s q ( mn )) . � � � � Q µ − 1 < | m | 2 ≤ Q µ Q ν − 1 < | n | 2 ≤ Q ν � � Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis How to get the difference process: Denote f ( n ) = α s q ( n ). Then with Cauchy-Schwarz ineq., 2 � � � � | S | 2 ≤ Q µ � � � � b n e ( f ( mn )) , � � � � Q µ − 1 < | m | 2 ≤ Q µ Q ν − 1 < | n | 2 ≤ Q ν � � and with Van der Corput ineq., | S | 2 ≪ Q 2( µ + ν ) − ρ � � � � + Q µ + ν � � � � max e ( f ( m ( n + r )) − f ( mn )) . � � 1 ≤| r | < | q | ρ � � Q ν − 1 < | n | 2 ≤ Q ν Q µ − 1 < | m | 2 ≤ Q µ � � Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis The truncated sum-of-digits function We introduce the truncated sum-of-digits function of Z [ i ], defined by λ − 1 λ − 1 � f ( ε j q j ) = α � f λ ( z ) = ε j , j =0 j =0 where λ ∈ Z and λ ≥ 0. Periodicity property: For any d ∈ Z [ i ], f λ ( z + dq λ ) = f λ ( z ) , z ∈ Z [ i ] . Reason: Let d = x + i y . Use the identities i q = aq + q 2 , Q = ( a − 1) 2 q + (2 a − 1) q 2 + q 3 . Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis At only “small” cost: Carry propagation lemma Put λ = µ + 2 ρ . Lemma Let a ≥ 2 . For all integers µ > 0 , ν > 0 , 0 ≤ ρ < ν/ 2 and r ∈ Z [ i ] with | r | < | q | ρ denote by E ( r , µ, ν, ρ ) the number of pairs ( m , n ) ∈ Z [ i ] × Z [ i ] with Q µ − 1 < | m | 2 ≤ Q µ , Q ν − 1 < | n | 2 ≤ Q ν and f ( m ( n + r )) − f ( mn ) � = f λ ( m ( n + r )) − f λ ( mn ) . Then we have E ( r , µ, ν, ρ ) ≪ Q µ + ν − ρ . Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis At only “small” cost II: The addition automaton 0 1 . . . . . . a 2 − 1 a 2 a 2 0 [ • ] P R ( a − 1) 2 0 2 a − 1 0 . . . . . . . . Q . . . . ( a − 1) 2 a 2 2 a − 1 a 2 0 2 a 0 a 2 − 2 a + 2 a 2 − 2 a + 2 0 2 a 0 . . . . . . . . . . . . . . . . . . . . . . . . a 2 − 2 a a 2 2 a − 2 a 2 a 2 2 a − 2 a 2 a 2 − 2 a ( a − 1) 2 0 2 a − 1 0 . . . . . . . . . . . . − Q a 2 a 2 ( a − 1) 2 2 a − 1 1 0 . . . . . . a 2 a 2 − 1 [ • ] − R − P 0 a 2 Performs addition by 1 ( P ), by − a − i ( R ) and by a − 1 + i ( Q ). Example: Let a = 3 and z = 58 − 40 i = ( ε 0 , ε 1 , ε 2 ) = (8 , 2 , 7), and consider z + 2 + i . The corresponding walk is 8 | 3 2 | 7 7 | 2 0 | 5 0 | 5 0 | 1 → [ • ] , Q → − Q → Q → − Q → Q → P − − − − − − − − − − − − thus z + 2 + i = (3 , 7 , 2 , 5 , 5 , 1). Th. Stoll Sum of digits of Gaussian primes

Introduction Statements Main results Tools and difference process Fourier analysis At only “small” cost III: Proof Idea of proof of the “carry propagation lemma”: Assume mr = x + i y = − y ( − a − i ) + ( x + ay ) with y < 0, x > − ay . Write f ( mn + mr ) − f ( mn ) = f ( mn + mr ) − f ( mn + ( a + i ) + mr ) + f ( mn + ( a + i ) + mr ) − f ( mn + 2( a + i ) + mr ) + . . . + f ( mn − ( y + 1)( a + i ) + mr ) − f ( mn − y ( a + i ) + mr ) + f ( mn − y ( a + i ) + mr ) − f ( mn − y ( a + i ) + mr − 1) + f ( mn − y ( a + i ) + mr − 1) − f ( mn − y ( a + i ) + mr − 2) + . . . + f ( mn − y ( a + i ) + mr − x − ay + 1) − f ( mn ) . Th. Stoll Sum of digits of Gaussian primes

Introduction Preliminaries Main results Some estimates Fourier analysis Orthogonality relation j =0 ε j q j : ε j ∈ N} be the fundamental region of the Let F λ = { � λ − 1 number system, which is a complete system of residues mod q λ with # F λ = Q λ . Hence, � Q λ , h ≡ 0 mod q λ ; � tr ( hzq − λ ) � � = e 0 , otherwise, z ∈F λ where tr ( z ) = 2 ℜ ( z ). Put λ | F λ ( h , α ) | = Q − λ � α − tr ( hq − j ) � � ϕ Q , j =1 where � | sin( π Qt ) | / | sin( π t ) | , t ∈ R \ Z ; ϕ Q ( t ) = Q , t ∈ Z . Th. Stoll Sum of digits of Gaussian primes

Introduction Preliminaries Main results Some estimates Fourier analysis Transformation Let � S ′ 2 ( n ) = e ( f λ ( m ( n + r )) − f λ ( mn )) . Q µ − 1 < | m | 2 ≤ Q µ Then 1 � � S ′ e ( f λ ( u ) · f λ ( v )) · 2 ( n ) = Q 2 λ u ∈F λ v ∈F λ � tr h ( m ( n + r ) − u ) + tr k ( mn − v ) � � � � · e q λ q λ h ∈F λ k ∈F λ Q µ − 1 < | m | 2 ≤ Q µ � tr ( h + k ) mn + hmr � � � � = F λ ( h , α ) F λ ( − k , α ) . e q λ h ∈F λ k ∈F λ Q µ − 1 < | m | 2 ≤ Q µ Th. Stoll Sum of digits of Gaussian primes

Introduction Preliminaries Main results Some estimates Fourier analysis Fourier analysis of F λ Lemma For all α ∈ R , ξ ∈ C and a ≥ 3 we have λ − 1 � 2 ≥ λ − 2 � 2 . � ( a 2 + 2 a + 2) α � � α − tr ( ξ q j ) � � � � 2( a 2 + 1) 2 j =0 Corollary There exists a constant C a > 0 only depending on a such that � ( a 2 + 2 a + 2) α � 2 ) � � | F λ ( h , α ) | ≤ exp( − C a λ uniformly for all h ∈ Z [ i ] , α ∈ R and integers λ ≥ 0 . Th. Stoll Sum of digits of Gaussian primes