AN UPPER BOUND FOR THE SUM EStf+i f(n)

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1 proceedings of the american mathematical society Volume 114, Number 1, January 1992 AN UPPER BOUND FOR THE SUM EStf+i f(n) FOR A CERTAIN CLASS OF FUNCTIONS / EDWARD DOBROWOLSKJ AND KENNETH S. WILLIAMS (Communicated by William Adams) Abstract. For a certain class of functions /: Z > C an upper bound is obtained for the sum 5ZnÍa+i /(") This bound is used to give a proof of a classical inequality due to Pólya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols J2x=i {(Rgx + S)/P)» where g is a primitive root of the odd prime p, 1 < H < p - 1 and RS is not divisible by p. 1. Introduction Let /: Z C be a function satisfying the following three conditions: (1.1) there exists a positive real number A such that \f(n)\ < A for all n Z; (1.2) there exists a positive integer k such that f(n + k) = f(n) for all n ez; (1.3) there exists a positive real number B such that * n=l for all positive integers h. /(n + r),2 <Bhk In 2 we show that such a function / satisfies the following inequality: Theorem 1. Let a and H be integers with 1 < H < k. (1.1)-(1.3), we have Then, if f satisfies (1.4) /(») <^f^vklogk + 3A^/k. Received by the editors May 30, 1990 and, in revised form, July 20, Mathematics Subject Classification (1985 Revision). Primary 11L40, 26D15. Key words and phrases. Inequality, character sum, Pólya-Vinogradov inequality. Research of the first author was supported by the Centre for Research in Algebra and Number Theory, Carleton University, Ottawa, Ontario, Canada. Research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada Grant A American Mathematical Society /92 $1.00+ $.25 per page

2 30 EDWARD DOBROWOLSKI AND K. S. WILLIAMS In 3 we apply Theorem 1 with f(n) = x(n) > where ^ is a nonprincipal Dirichlet character modulo k, to obtain the following form of the famous inequality first proved independently by Pólya [8] and Vinogradov [10] 70 years ago. Theorem 2. Let a and H be integers with H > 1. If x is a nonprincipal character modulo k then (1.5) X(n) Vklogk+3V] What is particularly interesting about our proof of Theorem 2 is that it does not require the value of the modulus of the Gauss sum for a primitive character X modulo k ; namely, (1.6) k-i x(n)exp(2nin/k) n=0 = Vk. All that is required is for x to satisfy an inequality of the type (1.3). The result (1.6) is used in the proofs of the Pólya-Vinogradov inequality given in [5, 6, 8, 9, 10] (see also [1, Theorem 13.15, p. 299; 2, Theorem 5.1, p. 320]. No attempt has been made to find the best possible constants multiplying the terms Vklogk and \fk in (1.4). It may therefore be possible to improve the constant 1/(2log2) in (1.5). The form of the Pólya-Vinogradov inequality having the smallest constant multiplying the term Vklogk is due to Hildebrand [5]. Under certain additional assumptions, Burgess [4] has significantly improved the estimate J2ta+i X(n) = 0( Jklogk). In 4 we use Theorem 1 to estimate the sum of Legendre symbols H x=l Rgx + S where H is an integer with l<h<p-l,pisan odd prime, g is a primitive root (mod p), and R, S are integers with RS ^ 0 (mod p). We prove Theorem 3. H x=i Rgx + S < y/p- llogq?- 1) + 3v^-l. We conclude the introduction by sketching briefly the idea of the proof of Theorem 1. Many values of h are chosen so that relatively many of the inner sums in the inequality (1.3) are partial sums of the sum (1.4) and can be combined to cover most of the range of summation of (1.4) a considerable number of times. An application of the Cauchy-Schwarz inequality then gives inequality (1.4). 2. Proof of Theorem 1 If 1 < k < 9 then inequality (1.4) is trivial as /(«) v/5 < \f(n)\<ah<ak<3avk<-j- Vklogk + 3AVk. n=a+\

3 AN UPPER BOUND FOR Y," t,"+ A") Thus we may suppose that k > 10. If H < y/k then inequality (1.4) is again trivial as /(") Thus we may also suppose that H > \fk. Let (2.1) S = {a+l,a + 2,...,a + H}, and set (2.2) s = < AH < A^/k < 3AVk < ^^-^fklogk + 3A\fk. log A: 1.21og2j ' _H ~2S' (2.3) hj = 2*-J-l[c], j = 0,l,...,s-l. We note that 5 > [log 10/()] = 1, and that (2.4) 2s < VX < 2i+1, 1 < c < 2\fk. From (2.2)-(2.4), we see that the hj are positive integers satisfying (2.5) H>2ho, H-h0-hi-hj-i>2hj (j = 1,..., s- 1). Next we construct (H - h0 + l)s sets S j (i = 1,..., H ho + 1 ; j = 1,..., s) such that for k = I,..., s, (2.6) S,i U U S tk consists of exactly ho~\-r- /z^-i consecutive integers of S. For / = 1,..., H - ho + 1 we set (2.7) 5",,i = {a + i, a + i+ 1,..., a + i + h0-1}. Clearly each S, i is a sequence of ho consecutive integers of S. Now suppose that S 11, St, 2,., S, jt, where k is an integer satisfying 1 < k < s-l, have been constructed such that for j I,..., k the set Sf, i U U Si t consists of exactly ho H-h /z/-i consecutive integers of S. We show how to construct Sitk+\ so that Sj, i U U S^jt+i consists of exactly ho -\-+ hk consecutive integers of S. Let Clearly we have L = {ss: s < minis,,] U---u5/jfc)}, R = {s S: s> max(si,i U US()jt)}. L + Ä = 5 -,SiiiU...US/,fc = H-(ho + hi + + Vi) > 2hk, so that max( L, \R\) >hk. If \L\ > \R\, so that \L\ >hk,we Sifjt+i = {mm(si,iu USitk) -hk,..., min(5/,i u usiit) - 1}, whereas, if \L\ < \R\, so that \R\ > hk, we set Sitk+i ={max(5f>iu---u5,fk)+l,...,max.(si,lu---usitk) + hk}. In both cases S^k+i consists of hk consecutive integers of 51 such that 5,-, i U ' USitk+i comprises ho-\-\-hk consecutive integers of S. This completes the required construction. set 3I

4 32 EDWARD DOBROWOLSKl AND K. S. WILLIAMS Next, for i = 1,..., H - ho + 1, we set and observe that so that s \s-s,\= s-ljs^j j=l Si = Si,iü-.-öSiiS, = H-(ho + h\ hs-i) = c2s -(25" l)[c] = c2s - (2s - l)[c] = (c - [c])2s + [c] <2s + c< \fk + 2\fk, (by (2.4)) (2.8) \S-Si\<7»fk, i=l,...,h-h0 + l Now, for / = 1,..., H - ho + 1, we have /(«) /(«) nes < «65, + /(») n S-Si so, appealing to (1.1) and (2.8), we obtain (2.9) /(«: < /(«: I «= 2+1 n S, + 3A\/k (i=\,...,h-ho + \). Summing (2.9) over /, we obtain (H-ho+r f(n) «=a+l = ;=1 /(») «=a+l ï =i /(«) n S, + 3A\/k H-h0+l i=i /(«) «es, + 3Ay/k(H-h0+ 1), so that (2.10) /(») < 1 H-h0+l i=l /(«: «6S, + 3/1 vx.

5 AN UPPER BOUND FOR Yf*". f(n) 33 Next, making use of the Cauchy-Schwarz so that (2.11) E(=1 /(«) «es, 1 //-Ao+1 II «ÜT // - Ao + 1 s * E =i j=i /(««es,,y inequality, we have 1/2 if-ao+1 i \ s 2 *l E i E E i=l j=l 1=1 7=1 //-Ao+1 = v/(//-/t0 + i)s /=! j=i /(«) «es, < y/s ^ /(») «es,-,, /(«) «es,,, s 1=1 7=1 /(») «es,,y 2\ ]/2 2\!/2 1/2 Furthermore, we have (2.12) s E E 1=1 7 = 1 /(«) «es,,; =1 7=1 6=a 5 -hj-, w*) /(«) «es,,, S,,; = {A+1,...,&+«,_,} /(* + where Nj(b) = number of / (1 < i < H - ho + 1) such that S j = {b + I,..., b + hj-i}. As Nj(b) < 2j~l, j = 1,..., s, we have i -hj-, w 5 -hj. se*-1 /(* + ' ) E 7=1 =a 5 a+ ^E^-E s k -E^'-'E 7=1 ft=l //-A0+l i -hj- hj-»7-2 /(* + / / =i A,_, /( + r) <Er'%'^ (by (1.3)), 7 = 1 (ash-hj-i (by (1.2)) 2 <H <k)

6 34 EDWARD DOBROWOLSKI AND K. S. WILLIAMS that is (2.13) i -hj-, E "m /(è + r) r=\ <sbk2s~l[c]. Hence, by (2.10)-(2.13), (2.5), (2.3), and (2.2), we have <. ^ y/sbk2s-l[c] + 3AVk y/h -h0+l yfb^k2^-^'2y/[c]+3ay/k y/h - ho + 1 " V^Ä7 < ^==v/ßv/rc2(i-1)/2v/[<5] + 3/lv/fc " v^yär 2(^)->+3^v/c = sv~b~vk + 3A\fk < \ÍB^k^^ + 3A\fk, as asserted. 3. Proof of Theorem 2 Let a and H be integers with H > 1, and let / be a nonprincipal character modulo k. Now ],, #(«) = 0 if n runs through any complete residue system modulo k, so that E^f+i *(«) = EJES?1*'*1 *(n) Hence we may assume that H < k. The function f(n) = x(n) satisfies (1.1) with A = 1, (1.2), and (1.3) with B 1 (see [3]). Theorem 2 now follows immediately from Theorem Proof of Theorem 3 Let p be an odd prime, g a primitive root (mod p), R and S integers with RS =i 0 (mod p), and H an integer satisfying 1 < H < p - 1. Next we define (4.1) f(n) Rgn+S where ( p) denotes the Legendre symbol, so that /: Z -» {-1, 0, 1}. Clearly / satisfies (1.1) with A = 1 and (1.2) with k = p - 1. Next we show that / satisfies (1.3) with B 1. First we recall that for a ^ 0 (mod p) we have p-i (4-2) m=0 am2 + bm + c )-{ -(a/p), (p-l)(a/p), if b2 4ac (mod p), if b2 = 4ac (mod p).

7 Then we have p-i n=l /(«+ r) AN UPPER BOUND FOR V<"+", fin) * 'n=a+\ J x ' 35 p-l A A = «=1 i=l A p-l = EE r,s=l «=1 ü^b+,,-r-s\ frgn+'+s" P J \ P J (R2gr+s)g2n + {RS{gr + gs))gn + $2 y* yl f(r2gr+s)m2 + (RS(gr + gs))m + S2 r,s=l m=l A (P-l = E E r,s=l \m=0 h E(p-i)+ í(r2gr+s)m2 + (RS(gr + gs))m + S2 \ P h (-(-i)r+i)-/*2 r,j=l - 1 = A(p-i)- (-i)r+5- (-i)r,r,i=l r,i=l r=s = h(p-l)-(2[h/2]-h)2 + h-h2 <h(p-l), so that (1.3) holds with B = 1. Theorem 3 now follows from Theorem 1. References 1. Tom M. Apóstol, Introduction to analytic number theory, Springer-Verlag, New York, R. Ayoub, An introduction to the analytic theory of numbers, Math. Surveys Monogr. vol. 10, Amer. Math. Soc, Providence, R.I., D. A. Burgess, On a conjecture of Norton, Acta Arith. 27 (1975), _, On character sums and L-series, Proc. London Math. Soc. (3) 12 (1962), Adolf Hildebrand, On the constant in the Pólya-Vinogradov inequality, Canad. Math. Bull. 31 (1988), E. Landau, Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen, Nachrichten Königl. Ges. Wiss. Gottingen (1918), K. K. Norton, On character sums and power residues, Trans. Amer. Math. Soc. 167 (1972), G. Pólya, Über die Verteilung der quadratischen Reste und Nichtreste, Gott. Nachr. (1918), I. Schur, Einige Bemerkungen zur vorstehenden Arbeit des Herrn G. Pólya, Nachrichten Königl. Ges. Wiss. Göttingen (1918), I. M. Vinogradov, Über die Verteilung der quadratischen Reste und Nichtreste, J. Soc. Phys. Math. Univ. Permi 2 (1919), 1-4. Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Kl S 5B6 Current address, E. Dobrowolski: Department of Mathematics, College of New Caledonia, Prince George, British Columbia, Canada V2N 1P8