The sum of digits function in finite fields

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1 The sum of digits function in finite fields Cécile Dartyge, András Sárközy To cite this version: Cécile Dartyge, András Sárközy. The sum of digits function in finite fields. Proceedings of the American Mathematical Society, American Mathematical Society, 013, 141 (1), < /S >. <hal > HAL Id: hal htts://hal.archives-ouvertes.fr/hal Submitted on 1 Mar 016 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 (7/5/01, 16h15) The sum of digits function in finite fields Cécile Dartyge (Nancy) and András Sárközy (Budaest) Abstract. We define and study certain sum of digits function in the context of finite fields. We give the number of olynomial values of F q with a fixed sum of digits. We also state a result for the sum of digits of olynomial values with generator arguments. 1. Introduction Let g N be fixed with g >. If n N, then reresenting n in the number system to base g: r 1 (1 1) n = c j g j, 0 6 c j 6 g 1, c r > 1, we write j=0 r 1 (1 ) S(n) = c j. Many aers have been written on the connection between this sum of digits function S(n) and the arithmetic roerties of n (for examle [1], [3], [4], [5], [6], [7], [8], [13]). In articular, Mauduit and Sárközy [1] studied the arithmetic structure of the integers whose sum of digits is fixed, while Mauduit and Rivat [10], [11] obtained some asymtotic formulae for the number of squares and also for the number of rimes whose sum of digits is even resect. odd. In this aer our goal is to study the analogs of some of these roblems in finite fields. Indeed, let be a rime number, q = r with r >, and consider the field F q. Let B = {a 1,..., a r } be a basis of the linear vector sace formed by F q over F, i. e., let a 1, a,..., a r be linearly indeendant over F. Then every x F q has a unique reresentation (1 3) x = with c j F. Write (1 4) s B (x) = j=0 r c j a j r c j. An imortant secial case is when the basis B consists of the first r owers of a generator of F q : B = {a 1, a,..., a r } = {1, z, z,..., z r 1 }. 010 Mathematics Subject Classification: Primary 11A63; Secondary 11L99. Key words and hrases: sum of digits function, finite fields, character sums, generators, rimitive elements. Research artially suorted by the Hungarian National Foundation for Scientific Research, Grants K7731 and K10091 and the Agence Nationale de la Recherche, grant ANR-10-BLAN 0103 MUNUM.

3 Cécile Dartyge and András Sárközy Then (1 3) becomes (1 5) x = r c j z j 1. (1 4) and (1 5) are of the same form as (1 1) and (1 ), thus we may consider (1 3) as the finite field analog of the reresentation (1 1), and we may call c 1,..., c r in (1 3) as digits, and s B (x) can be called as sum of digit function. If we consider the generators (or rimitive elements) as finite fields analogs of rimitive roots of F, then we end u with the finite fields analogs of some roblems mentioned above : How many squares are there in F q with a fixed sum of digits, and more generally, how many values f(x) of a olynomial f have a fixed sum of digits? How many generators of F q whose sum of digits is a fixed value? In this aer our goal is to study these roblems. Let c F. We define Q c as the set of the squares of F q such that their sum of digits is equal to c: Q c = We rove the following result : Theorem 1.1. For all c F, we have: n o x F q : s B (x) = c and y F q such that y = x. Q c r 1 6 q. Let f F q [] be of degree n with (n, q) = 1. We are now interested in the cardinality of the sets: D(f, c) = {x F q : s B (f(x)) = c}. While Theorem 1.1 can be roved elementarily, we will need Weil s Theorem [14] to estimate D(f, c) : Theorem 1.. Let f F q [] be of degree n with (n, q) = 1. Then for all c F, we have D(f, c) r 1 6 (n 1) q. The main term of this estimate is larger than the one of Theorem 1.1 because here the values f(x) are taken with multilicities. We denote by G the set of the generators (or rimitive elements) of F q and for c F we consider the sets G(f, c) = {g G : s B (f(g)) = c}. Theorem 1.3. Let f F q [] be of degree n with (n, q) = 1. Then for all c F we have ϕ(q 1) G(f, c) 6 (n 1)τ(q 1) q where τ(n) denotes the divisor function.

4 . The sum of digits of the squares The sum of digits function in finite fields 3 In this section we rove Theorem 1.1. First we suose that c F. We use the quadratic character to detect the elements of Q c : Q c = 1 h r 1 + γ c j a j i, (c 1,c,...,c r ) F r c 1 +c + +c r =c where γ is the quadratic character. We relace c r by c (c c r 1 ): Q c = r (c 1,...,c r 1 ) F r 1 γ ca r + r 1 c j (a j a r ). To make the c j indeendent, it is convenient to switch to the additive characters via the Gaussian sums. We recall that if χ is a multilicative character of F q and ψ is an additive character of F q then the Gaussian sum of χ and ψ is defined by ( 1) G(χ, ψ) = x F q χ(x)ψ(x), (see [9]). Then we can switch to additive characters with the following formula for all x F q : χ(x) = 1 G(χ, ψ)ψ(x). q ψ Since c 6= 0, ca r + P r 1 c j(a j a r ) F q for all c 1,..., c r 1 F. Then we obtain for Q c : ( ) Q c = r G(γ, ψ) q ψ (c 1,...,c r 1 ) F r 1 r 1 Y ψ(ca r ) ψ(c j (a j a r )). If ψ(a j ) 6= ψ(a r ), then ψ(a j a r ) is a -th root of the unity ; there exists λ j {1,..., 1} such that ψ(a j a r ) = e(λ j /) with the standard notation e(t) = ex(iπt). In this case, cj λ j ψ(c j (a j a r )) = e = 0. c j F c j F Thus, in the right hand side of ( ), the summation over (c 1,..., c r 1 ) is not 0 if and only if ψ(a j ) = ψ(a r ) for all 1 6 j 6 r 1. This means that ψ is a ower of ψ 1 where ψ 1 is the additive character defined by ψ 1 (a j ) = e(1/) for all 0 6 j 6 r 1. Then we have: Q c = r j=0 G(γ, ψ j 1)ψ j 1 (c). Next we use the classical fact ([9] Theorem 5.1) that G(χ, ψ) 6 q if (χ, ψ) 6= (χ 0, ψ 0 ) the coule of the trivial multilicative resectively additive character. We obtain Q c r 1 q 6. If c = 0 then we have to remove the term with c 1 = c =... = c r 1 = 0 in ( ). This gives an extra error term q/ and we obtain Q 0 r 1 6 q.

5 4 Cécile Dartyge and András Sárközy 3. The sum of digits of the olynomial values Let f F q [] of degree n such that (n, q) = 1. We are now interested in the cardinality of the sets D(f, c). The character ψ 1 defined in the revious section is connected with the sum of digits function s B by the formula sb (x) ψ 1 (x) = e. Thus we have: D(f, c) = 1 1 h=0 hc ψ1 h (f(x))e x F q. The main term is rovided by h = 0: D(f, c) = q E(h), h=1 with E(h) = hc ψ1 h (f(x))e. x F q We will use the following theorem of Weil ([14], see also [9] Theorem ) to obtain an uer bound for the terms E(h): Theorem 3.1(Weil). Let g F q [] be of degree n > 1 with (n, q) = 1 and ψ a nontrivial additive character of F q. Then ψ(g(x)) 6 (n 1) q. x F q By this Theorem, we deduce that E(h) 6 (n 1) q for all 1 6 h 6 1. Thus we obtain: D(f, q c) 6 (n 1) q. 4. The sum of digits of olynomial values with rimitive element arguments Like the revious section, we consider a olynomial f F q [] of degree n with (n, q) = 1, but we study now the sets G(f, c). By the same argument as in the revious section we have (4 1) G(f, c) = 1 1 hc ψ1 h (f(g))e h=0 g G By using Weil s Theorem 3.1 we will deduce the following bound for additive character sums with rimitive element arguments..

6 The sum of digits function in finite fields 5 Lemma 4.1. Let g F q [] be of degree n with (n, q) = 1. Let Ψ be a non trivial additive character of F q. Then ψ(f(g)) 6 (n 1)τ(q 1) q + g G ϕ(q 1) q 1. Proof. The roof follows the argument of Lemma.3 of [] which gives a similar uer bound for sum with multilicative characters over F. Let g 0 be a rimitive root of F q. Then we have: ψ(f(g)) = ψ(f(g0 k )). g G 16k<q (k,q 1)=1 Then like in [], we use the Möbius function to handle the corimality condition and next we remark that g kd 0 is eriodic in k with eriod (q 1)/d: (4 ) ψ(f(g)) = g G d q 1 (q 1)/d µ(d) k=1 ψ(f(g kd 0 )) = d q 1 µ(d) d ψ(f(x d )). x F q When d q 1, the degree of f( d ) is corime with q and we can aly the Theorem 3.1. This ends the roof of the Lemma 4.1, the ϕ(q 1)/(q 1) term is the contribution of x = 0 excluded in (4 ). It remains to insert Lemma 4.1 in (4 1) and this gives G(f, c) This ends the roof of Theorem 1.3. ϕ(q 1) 6 (n 1)τ(q 1) q + 1. References [1] W. D. Banks, A. Conflitti and I. E. Sharlinski, Character sums over integers with restricted g-ary digits, Illinois J. Math. (3) 46 (00), [] C. Dartyge and A. Sárközy, On additive decomositions of the set of rimitive roots modulo, Monatsh. Math. (to aear). [3] C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiles d entiers, Ann. Inst. Fourier Grenoble/ 55 (005), [4] C. Dartyge and G. Tenenbaum, Congruences de sommes de chiffres de valeurs olynomiales, Bull. London Math. Soc. 38 (006), no. 1, [5] M. Drmota, C. Mauduit and J. Rivat, The sum of digits function of olynomial sequences, J. London Math. Soc. 84 (011), [6] E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres, Acta Arith. 77 (1996), [7] E. Fouvry and C. Mauduit, Sommes des chiffres et nombres resque remiers, Math. Ann. 305 (1996), [8] A. O. Gelfond, Sur les nombres qui ont des roriétés additives et multilicatives données, Acta Arith. 13 (1967/1968), [9] R. Lidl and H. Niederreiter, Finite Fields, Encycloedia of Mathematics and its Alications, vol. 0, Addison-Wesley Publishing Comany, (1983). [10] C. Mauduit and J. Rivat, La somme des chiffres des carrés. Acta Math., vol. 03, 009, [11] C. Mauduit and J. Rivat, Sur un roblème de Gelfond : la somme des chiffres des nombres remiers. Annals of Math., vol. 171, n 3, 010, [1] C. Mauduit and A. Sárközy, On the arithmetic structure of the integers whose sum of digit is fixed, Acta Arith. 81 (1997), [13] T. Stoll, The sum of digits of olynomial values in arithmetic rogressions, Functiones et Aroximatio, to aear. [14] A. Weil, Sur les courbes algébriques et les variétés qui s en déduisent, Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann, Paris, (1948).

7 6 Cécile Dartyge and András Sárközy Cécile Dartyge Institut Élie Cartan Université de Lorraine,BP Vandœuvre Cedex, France András Sárközy Deartment of Algebra and Number Theory Eötvös Loránd University 1117 Budaest, Pázmány Péter sétány 1/C Hungary