Citation Tsukuba journal of mathematics (198.

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1 TitleOn the sum of digits of primes in i Author(s) Kaneiwa, Ryuji; Shiokawa, Iekata Citation Tsukuba journal of mathematics (198 Issue Date URL Rights This document is downloaded at: Barrel - Otaru University of Commerce Academ

2 TSUKUBA J. MATH. Vol. (1980) ON THE SUM OF DIGITS OF PRIMES IN IMAGINARY QU ADRATIC FIELDS By Ryuji KANEIWA and Iekata SmOKA W A 1. Introduction. 2 be a fixed integer. Any positive integer n can be uniq uely written in the form (1) n each (lj is one of 0, 1,..., 1'-1 and (2) [-t~i-';~ ] 1, [uj is the integral part of the real number u. We put 1. Kata! [lj proved, assuming the validity of density hypothesis for the Riemann zeta function, that in the sum p runs through the prime numbers. The second-named author [6J proved, without any hypothesis, the result of Katai with an improved remainder term (3) O(X(_log log x )112). log x His method is to appeal to a simple combinatorial inequality (see Lemma in the deepest result on which he depends is the prime number theorem in a weak form (4) Heppner independently proved a more general result by making use a Chebyshev's inequality to the sum independent random variables [5J p. 387, Theorem 2): Let B be a set of positive integers such that Received 29, Revised 12, 1980.

3 128 Ryuji KANEIWA and Iekata SmOKA \VA Then log x B(x) =o(log x), B(x)= 2J 1. n S."r neb 2J s(n)::::=- 1'-1 T(;5.J: 2 neb I og l' x B(x)(1 This together with (,1) implies (3). In the present paper we shall show that the estimate (3) is also valid, in some sense, for primes in each imaginary quadratic field Q(-V=1n), 17l is any positive square free integer. 2. Representation of integers in Q( -V=<1<7t) in the scale of r. Let 0 be the ring of all integers in Q( -v=-ln). Any a 0 can be expressed in a unique way as m=j a=a+bev (a, b::::=z), -V =11i if -<7n 2,3 (mod 4), 1 if (mod 4), 2 and Z denotes as usual the set of all rational integers. expessions So by means of the given by (1), we can define coordinatewisely the representation of a scale of 1'; i. e. 0 in the (5) (x= (6) 12= lz(a)=max la\), k(\bl)}, 12(0)=1, and sgn i C i if C =1'= 0, otherwise. We define We write 1:= {ai-lxvi a, bez;

4 On the sum of digits of primes in imaginary quadratic fields 129,./72= {-a+bw I a+bwt::: {-a-bwla+bweo so that for the expressions (5) We denote by fbi the set of all 'digits' aj needed all a c.jl j. Then {c+dwjc, 1 J '", and card y" (1 So we may say that the r-adic expression (5) aeoo is a kind of representation in the scale of For any fixed (3 EO fbi we denote by (3) the number of i3 appearing in the expression (5) of an integer a EO By definition (7) 5(0:)=,BF(a, /3) and (8) C-j-c/(v)=F(-a+bw, -c-l-dw) The norm (I bw, -c-dw)=--=f(a-bw, c--dw) o is a rational integer if (mod a a/)+ m+1 if --m=.::: 1 4 so that for cr 0 (9) J.V(a) 1<. log r.-:: ( 1, (mod 4J Cl is a constant depending only on 111, since by definition hea) 1 (we mean max (log 0, and 2 max (log ia., log Ibl)-log log (1 +m.) 'J:: 11 3 (mod log (2-: _) if ---m:=1 (mod

5 130 Ryuji KANEIWA and Iekata SHIOKA WA 3. A prime number theorem (A. Mitsui [3J, [4J). An integer a 0 is said to be prime if (a) is an prime ideal in Q( v"-l~i7,). Let (}b (}2 be two real numbers such that <(}2<2rr. Then (10) ~ tt: prime iv(, J:5:.,f,' 1~arg"~502 ((}2-(}1)W 2rrh dt log t +O(x exp (-cz(log X)~j5(log log x) h is the class number of Q( v-and W=j:2 :: m=3, otherwise We note that a weaker estimate O(x/(log our theorem. is sufficient for the proof of 4. A combinatorial lemma (1. Shiokawa [6J). Let PI,"', l3 g be given g symbols and let A} be the set of all sequences of these symbols of length} l. Denote by FjCa,,3) the number of any fixed symbol P appearing in a sequence ac::ai. Then for any 2 with 0<2<1/2 there exist a positive integer jo independent of c such that the number of sequences a./jj satisfying I F/Ct,,8) is less that jg} exp ( for all C 3 is an absolute constant. 5. Theorem. any (}ll (}2 satisfying (j),1 =0 7[. Then for I <(}2~rpj+-l for some j we have (11) O(x(_~o~ log :~)1!2) I log x Aj~\ l+w if j=l, +(0 if ) -1-w if } I-OJ if j=4, and the O-constant depends at most on rand }rl

6 On the sum of digits of primes in imaginary quadratic fields Proof of Theorem. By (7) and (8) we may assume j=l. We define for ae and pe (12) D(a,,8)= I Pea, Put for brevity Then by (7) and (12) {aeo I a: prime, N(a) x, 0 1 (13) s(a)= ~ 19 ~ F(a, /3) /3E!:Bl (JE:c'(X) By (9) and (10) we have (14) Put D(a)=D(a, po), po is any fixed integer in!if1' vve have from (9), (10), and (12) (15) D(a);;::; DCa) (,r) =O( ~ (log =O( D(a) (ret: (.T) O(xlog -1 O(log X 1). Besides, using (9), 1, I( Applying no\v the lemma in 4 with and N we get 1 <' exp (- for all ) JOI which leads to (16) 1-0(1)+ ~ jr 2 } exp io<j:&l(x)

7 132 Ryuji KANEIWA and Iekata SmoKA W A 0(1) /2 O(x(log the O-constant is uniform In 2. If we take a constant 0<2<1/2 in such a way that we obtain from (15) and (16) (log large enough and choose s= C 1 log log x o(x(. log log X )lh). log x r) with This together with (13) and (14) yealds the theorem. References C 1 ] K{ltai, 1., On the sum of digits of prime numbers. Ann. Univ. Sci. Budapest Roland Edtvds nom. Sect. Math., 10 (1967), [2 J Heppner, E., Uber die summe der Ziffern naturlicher Zahlen. Ann. Univ. Sci. Budapest Roland Edtvos nom. Sect. Math., 19 (1976), 41-,13. [3 J!Vlitsui, T., Generalized prime number theorem. Japanese J. Math., 26 (1956), '1 J Mitsui, T., Research Institute for Mathematical Sciences Kyoto Univ., "Kokyuroku" 294 (1977), in Japanese. [5 J Renyi, A., Probability Theorey, North-Holland, Amsterdam-London, J Shiokawa, 1., On the sum of digits of prime numbers. Proc. Japan Acad., 50, no. 8, 55155,1. Ryuji Kaneiwa Institute of Liberal Arts Otaru University of Commerce Otaru, Hokkaido 047 Japan Iekata Shiokavla Department of Mathematics Keio University Yokohama 223 Japan