The sum of digits of primes in Z[i]

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1 The sum of digits of primes in Z[i] Thomas Stoll (TU Wien) Journées de Numération, Graz 2007 April 19, 2007 (joint work with M Drmota and J Rivat) Research supported by the Austrian Science Foundation, no9604

2 Gaussian primes Complex sum-of-digits Pointillism: Gaussian primes in the first quadrant

3 Gaussian primes Introduction Gaussian primes Complex sum-of-digits Gaussian primes p: (1) 1 + i and its associates, (2) the rational primes 4k + 3 and their associates, (3) the factors in Z[i] of the rational primes 4k + 1 Hecke prime number theorem: π i (N) := {p Z[i] non-associated : p 2 N} N log N Complex Von Mangoldt function Λ i : { log p, n = εp Λ i (n) = ν, ε unit, ν Z + ; 0, otherwise

4 The complex sum-of-digits function Gaussian primes Complex sum-of-digits [Kátai/Kovács, Kátai/Szabó] Let q = a ± i (choose a sign) with a Z + Then every n Z[i] has a unique finite representation λ 1 n = ε j q j, 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]

5 I Introduction Statements Tools and difference process Recall q = a ± i, a Z + and write e(x) = exp(2πix) Theorem For any α R with (a 2 + 2a + 2)α Z, a even, there is σ q (α) > 0 such that Λ i (n) e(αs q (n)) N 1 σq(α) n 2 N Theorem The sequence (αs q (p)), a even, running over Gaussian primes p is uniformly distributed modulo 1 if and only if α R \ Q

6 Statements Tools and difference process II Theorem Let a 2, a even, b Z 0, m Z +, m 2 and set d = (m, a 2 + 2a + 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 σq,m ) If (b, d) 1 then the set has cardinality O q,m (N 1 σq,m )

7 Statements Tools and difference process Inequalities à 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) max e(αs q (mn)) U for M x β 1, M Q < m 2 M x Q m 2 <t x m 2 M Q < m 2 x M Q m 2 < n 2 x Then x Q < n 2 x x Q m 2 < n 2 t a m b n e(αs q (mn)) U for x β 1 M x β 2 m 2 Λ i (n) e(αs q (n)) U(log x)2

8 Statements Tools and difference process Lemma (Van der Corput) Let z n C with n Z[i] and A, B 0 Then for all R 1 we have A< n <B z n 2 C 3 ( B A R ) + 2 B + A R r <2R ( 1 r ) 2R A< n <B A< n+r <B z n+r z n Now, start with S = Q µ 1 < m 2 Q µ b n e(αs q (mn)) Q ν 1 < n 2 Q ν

9 How to get the difference process: Statements Tools and difference process Denote f (n) = αs q (n) Then with Cauchy-Schwarz ineq, S 2 Q µ Q µ 1 < m 2 Q µ and with Van der Corput ineq, 2 b n e(f (mn)), Q ν 1 < n 2 Q ν S 2 Q 2(µ+ν) ρ + Q µ+ν max 1 r < q ρ Q ν 1 < n 2 Q ν e (f (m(n + r)) f (mn)) Q µ 1 < m 2 Q µ

10 The truncated sum-of-digits function Statements Tools and difference process We introduce the truncated sum-of-digits function of Z[i], defined by λ 1 λ 1 f λ (z) = f (ε j q j ) = α ε j, where λ Z and λ 0 j=0 Periodicity property: For any d Z[i], j=0 f λ (z + dq λ ) = f λ (z), z Z[i] Reason: Let d = x + iy Use the identities iq = aq + q 2, Q = (a 1) 2 q + (2a 1)q 2 + q 3

11 Statements Tools and difference process At only small cost: Carry propagation lemma Put λ = µ + 2ρ 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

12 Statements Tools and difference process At only small cost II: The addition automaton 0 1 a 2 1 a 2 a 2 0 [ ] P R 0 2a 1 (a 1) 2 a 2 Q 0 (a 1) 2 2a 1 a 2 0 2a a 2 2a a 2 0 a 2 2a + 2 2a 2 a 2 a 2 2a a 2 2a 2 2a 0 a 2 a 2 2a (a 1) 2 0 2a 1 0 Q a 2 2a 1 a 2 (a 1) 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 40i = (ε 0, ε 1, ε 2) = (8, 2, 7), and consider z i The corresponding walk is Q 8 3 Q 2 7 Q 7 2 Q 0 5 Q 0 5 P 0 1 [ ], thus z i = (3, 7, 2, 5, 5, 1)

13 Statements Tools and difference process At only small cost III: Proof Idea of proof of the carry propagation lemma : Assume mr = x + iy = 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)

14 Orthogonality relation Introduction Preliminaries Some estimates Let F λ = { λ 1 j=0 ε jq j : ε j N } be the fundamental region of the number system, which is a complete system of residues mod q λ with #F λ = Q λ Hence, z F λ e where tr (z) = 2R(z) Put ( tr (hzq λ ) ) = { Q λ, h 0 mod q λ ; 0, otherwise, λ F λ (h, α) = Q λ ( ϕ Q α tr (hq j ) ), j=1 where ϕ Q (t) = { sin(πqt) / sin(πt), t R \ Z; Q, t Z

15 Transformation Introduction Preliminaries Some estimates Let Then S 2(n) = 1 Q 2λ S 2(n) = u F λ Q µ 1 < m 2 Q µ e (f λ (m(n + r)) f λ (mn)) e(f λ (u) f λ (v)) v F λ e h F λ k F λ Q µ 1 < m 2 Q µ = h F λ k F λ F λ (h, α)f λ ( k, α) ( h(m(n + r) u) tr q λ + tr Q µ 1 < m 2 Q µ e ( tr ) k(mn v) q λ (h + k)mn + hmr q λ )

16 Preliminaries Some estimates of F λ Lemma For all α R, ξ C and a 3 we have λ 1 α tr (ξq j ) 2 λ 2 (a 2 2(a 2 + 1) 2 + 2a + 2)α 2 j=0 Corollary There exists a constant C a > 0 only depending on a such that F λ (h, α) exp( C a λ (a 2 + 2a + 2)α 2 ) uniformly for all h Z[i], α R and integers λ 0

17 Preliminaries Some estimates of G λ Define G λ (b, d, α) := h F λ h b mod d F λ (h, α), G λ (α) = G λ (0, 1, α) = Lemma For a 2, b Z[i], α R, 0 δ λ there is η Q < 1 2 and G λ (b, q δ, α) = h F λ h b mod q δ F λ (h, α) Q η Q(λ δ) F δ (b, α) h F λ F λ (h, α) In particular, G λ (α) = G λ (0, 1, α) Q η Qλ

18 Preliminaries Some estimates II of G λ Lemma For a 2, b Z[i], α R, 0 δ λ, k Z[i] and k q λ δ, q k we have G λ (b, kq δ, α) 2 k 2η 5 Q η 5(λ δ) F δ (b, α) Lemma For a even we have F λ (h, α) 2 = F δ (b, α) 2 h F λ h b mod q δ and thus, in particular, h F λ F λ (h, α) 2 = 1

The sum of digits of polynomial values in arithmetic progressions

The sum of digits of polynomial values in arithmetic progressions Thomas Stoll Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France E-mail: stoll@iml.univ-mrs.fr