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2 Moscow Journal of Cobinatorics and Nuber Theory. 0. Vol.. Iss p. The journal was founded in 00. ublished by the Moscow Institute of hysics and Technology with the support of Yandex and Microsoft. The ai of this journal is to publish original, high-uality research articles fro a broad range of interests within cobinatorics, nuber theory and allied areas. One volue of four issues is published annually. Website of our journal ØØÔ»»ÑÒغÔÝ ØºÙ Address of the Editorial Board Moscow institute of physics and technology (state university) Faculty of Innovations and High Technology, Laboratorny Korpus, k. 09, 9, Institutskii pereulok, Dolgoprudny, Moscow Region, Russia, 4700 E-ail ÑÒØÔÝ ØºÙ Адрес редакции Московский физико-технический институт (государственный университет) Факультет инноваций и высоких технологий Лабораторный корпус, к. 09, Институтский переулок, д. 9, г. Долгопрудный, Московская область, Российская Федерация, 4700 URSS ublishers ÆÑÓÚ Ý ÈÖÓ ÔØ ÅÓ ÓÛ ÊÙ ½½ Издательство «УРСС» ñ ¹ àû äòû ½½ Журнал зарегистрирован в Федеральной службе по надзору в сфере массовых коммуникаций, связи и охраны культурного наследия 3 сентября 00 г. Свидетельство ПИ ФС Формат 70 00/6. Печ. л. 5. Зак. ПЖ-43. Отпечатано в ООО «ЛЕНАНД». 73, Москва, пр-т Шестидесятилетия Октября, А, стр.. ISSN УРСС, 0 0 ID 58900,!7IF4F3-aaachc! S:./fig-eps/at-eng.eps S:./fig-eps/issn.eps All rights reserved.no part ofthisbookaybe usedor reproducedinanyannerwhatsoever without written perission of the publisher.

3 Moscow Journal of Cobinatorics and Nuber Theory 0, vol., iss.3, pp. 5 4, [pp ] Scale =.7 S:./fig-eps/Logo-MJCNT.eps A nuerically explicit version of the ólya Vinogradov ineuality Ditriy Frolenkov (Moscow) Abstract: A new nuerically explicit version of the ólya Vinogradov ineuality is proved in the paper. The proof is based on new ideas of V.A.Bykovskii and iproves on a recent result obtained by C. oerance. Keywords: Dirichlet character, character sus, ólya Vinogradov ineuality AMS Subject classification: L40 Received: ; revised: In the paper [3] C.oerance proved a new explicit version of the faous ólya Vinogradov ineuality for character sus. In this paper we present an iproveent of oerance s result. Our approach is based on a recent construction due to V.Bykovskii [], [3]. In [] and [3] this construction was used to obtain new upper bounds for the discrepancy of good lattice points sets. The present paper is organized as follows. In Section we give a brief survey of classic and recent results on the topic. In Section we forulate our ain result. In Section 3 we forulate two leas by oerance. In Section 4 we describe Bykovskii s construction. In Section 5 we coplete the proof of our ain result.

4 6 Ditriy Frolenkov (Moscow) [34. Introduction Let χ (od ) be a priitive Dirichlet character. ut S χ = ax 0M<N N n=m χ(n), T χ = ax N N a=0 χ(a). A character is defined to be even or odd if χ( ) = or χ( ) = respectively. In the case of even characters one has S χ = T χ. () In 98 ólya [] and Vinogradov [7] independently proved that for any nonprincipal Dirichlet character the ineuality S χ c Ô log () holds with an absolute constant c. In 007 Granville and Soundararajan [6] proved that for every priitive Dirichlet character χ (od ) of odd order g we have T χ Ô (log ) δ g/o(), where δ g = g sin g, ½. This result has recently been iproved by Goldakher[5]. He obtained the following estiate: T χ Ô (log ) δ go(). Under the General Rieann Hypothesis, Montgoery and Vaughan [0, Theore 3] proved that S χ Ô loglog. In fact, this is the best-possible result. aley [] proved that there exists an infinite class of uadratic characters χ n (od n ) such that S χn Ô n loglog n. An iportant proble is to find the ost precise for of the ineuality (). So far, two types of results have been obtained. Results of the first type don t include explicit bounds for the reainder ters, whereas results of the second type define all constants explicitly. As a rule, results of the first type are characterized by having ore accurate constants in the ain ter.

5 35] A nuerically explicit version of the ólya Vinogradov ineuality 7.. Asyptotic results Landau [9] proved that S χ Ô Ôlog o() if χ( ) = and S χ o() Ôlog if χ( ) =. Hildebrand [8] obtained that T χ 3 o() Ôlog if χ( ) = and T χ 3 o() Ôlog if χ( ) =. Later, Hildebrand [7] iproved his result in the case of even characters. He proved the estiate c T χ Ô Ôlog 3 o() for χ( ) =, where c =, if is cubefree; 4 3, otherwise. Granville and Soundararajan [6] obtained the following two ineualities: 69 c T χ 70 Ô Ôlog 3 o() if χ( ) = and T χ c o() Ôlog if χ( ) =. At the oent of writing, this is the best known asyptotic estiate.

6 8 Ditriy Frolenkov (Moscow) [36.. Nuerically explicit results In this section we discus nuerically explicit versions of the ólya Vinogradov ineuality. Qiu [4] proved that S χ 4 Ô log 0.38 Ô Ô 0.6 Ô. Sialarides [5], [6] obtained the estiates T χ 3 Ô log log 4 γ if χ( ) = and T χ Ô Ô log if χ( ) =. Dobrowolski and Willias [4] proved that for any non-principal Dirichlet character χ (od ) one has S χ Ô Ô log 3. log Bachan and Rachakonda [] iproved their result, proving that S χ Ô Ô log 6.5 3log3 for any non-principal Dirichlet character χ (od ). In [3] oerance proved that and S χ Ô 4 Ô 3Ô log loglog if χ( ) = (3) S χ Ô Ô Ô log loglog if χ( ) =. (4) To be ore precise, oerance showed that S χ Ô 4 Ô log loglog 4 Ô log4γ 3 n log 4 log if χ( ) =, (5) log

7 37] A nuerically explicit version of the ólya Vinogradov ineuality 9 and Ô log 4 S χ Ô Ô log loglog γ log 3 log 4 log log if χ( ) =, (6) where n = Ô log, = (4/) Ô log and is big enough (see [3]). Until now, these bounds have been the best known nuerically explicit versions of the ólya Vinogradov ineuality.. The ain result We prove the following theore, iproving (3) and (4) in the second ter. Theore. Let χ (od ) be a priitive character. Then. If χ( ) =, we have. If χ( ) =, we have S χ Ô log 4 Ô ( γ log C0 ) ψ (), S χ Ô log where γ is the Euler constant, C 0 = 4 5/ 5, and ψ ()= 4 C 0 8 Ô Ô γ log C 0 ψ (), exp 4Ô C 0, ψ ()= 3 C 0 Ô exp Ô. C 0 Siplecalculations showthatfor > exp(c 0 /4) thefirst andthesecond case of Theore iprove on (5) and (6) respectively. The proof of this theore is based on the ideas of Bykovskii [] and oerance [3].

8 30 Ditriy Frolenkov (Moscow) [38 3. oerance s leas In this section we forulate soe results fro oerance s paper [3]. Lea. For all real nubers x and positive integers n we have n j= sin jx j < log n γ log 3. n roof. See [3, Lea 3]. Lea. For all real nubers α, β and positive integers n we have n = cos α cos β < log n γ log 3 n. roof. See [3, 4]. 4. Bykovskii s ethod In this section we describe the ain construction fro the paper [], and then present our own odification of this construction. Let θ : [0, ½) [0,] be the following function: θ(t) = 0, if 0 < t ; t, if t ; t, if t ; 0, if t. For all t ¾ (0, ½) we have ½ j= ½ t θ j =.

9 39] A nuerically explicit version of the ólya Vinogradov ineuality 3 Let ω, ω : (0, ½) [0,] be defined as follows: ω(t) = ω (t) = ω(t) = ½ j=0 j= ½ t θ j = t θ j = 0, if 0 < t ; t, if t ;, if t,, if 0 < t ; t, if t ; 0, if t. For any positive and ¼ put S(u;, ¼ ) = If ¼, we can write ½ = ω ω sin u. ¼ S(u;, ¼ ) = S(u; ) S(u; ¼ ), where S(u; ) = ½ = ω sin u. (7) For any positive define G (u) = ½ n= ½ (n u) = ½ n= ½ exp n e(nu), (8) where e(u) = exp(iu). Bykovskii proved (see [, Lea ]) that S(u; ) 4G (u). We are going to iprove on this result.

10 3 Ditriy Frolenkov (Moscow) [40 Lea 3. For any u ¾ [0;] and > the ineuality is satisfied. S(u; ) C 0 G (u) roof. Take A ¾ [0;]. The optial value of the paraeter A will be coputed later. We are going to consider the following two cases separately.. Let u A. We can write sinu u (9) and sinu = u. (0) The proof of (0) can be found in [, Lea ]. By using partial suation and applying (0) we obtain >B/u sin u u u B B. () Let B ¾ [A;]. The optial value of the paraeter B will be calculated later. We can see that A u B u. Therefore, fro (7), (0), and () we have S(u; ) B/u sinu >B/u sin u u B. B/u If B /, then we have S(u; ) B B,

11 4] A nuerically explicit version of the ólya Vinogradov ineuality 33 and otherwise S(u; ) B logb /u<b/u By the definition of the function G (u) we have G (u) u B B = B logb u A. B. Thus, we can write S(u; ) C ( A )G (u), where C = B B, if A B ; ( B logb) B, if ax, A B.. Let u > A. The proof is going to be siilar to the proof of the second case of Lea in [], and therefore we are going to use the relevant notation fro []. Lea of [] states that for any ¼ > we have S(u;, ¼ ) = ½ n= ½ (η(n u;, ¼ ) η(n u;, ¼ )), where η(w;, ¼ ) = ¼ / x ω ω x ¼ e( wx) dx x. Applying the ideas of this lea, we obtain η(w;, ¼ ) 4 w 0 6 ¼.

12 34 Ditriy Frolenkov (Moscow) [4 Therefore, we can write which leads to For any integer n we have S(u;, ¼ ) ½ n= ½ S(u; ) 5 (n u) > This leads to the following estiate: S(u; ) 5 A ½ n= ½ 0 4 (n u) 6 ¼, ½ n= ½ (n u). A A (n u). (n u) < 5 G A (u). It is easy to prove that f (A) = C ( A ) is an increasing function if A ¾ [0;]. Therefore, we should use the value of A that satisfies the euation f (A) = 5. A If A Ô, then in B = Ô, AB/ B and therefore Solving the syste of euations f (A) = Ô ( A ), 0 A Ô. Ô ( A ) = 5 A 0 A Ô ()

13 43] A nuerically explicit version of the ólya Vinogradov ineuality 35 yields A = Ô 5 5/4. Substituting this value of A in the estiates that have been obtained earlier leads to Ô 5 S(u; ) f 5/4 G (u) = Ô 5 G (u), concluding the proof. Lea 4. For any ¾ N and > we have G a = exp. roof. Let us take δ (a) = x= ax e =, if a 0 (od ); 0, else, (3) and then apply (8) to write G a = = = = ½ n= ½ ½ n= ½ exp exp ½ n= ½ n= n exp exp n e n a = δ (n)= n n δ (n) = = exp exp, proving the lea.

14 36 Ditriy Frolenkov (Moscow) [44 5. roof of Theore Let 0 a < b, and let λ(x; a, b) be a function defined on [0;) as λ(x; a, b) =, if x = a;, if a < x < b;, if x = b; 0, otherwise. Its Fourier expansion can be written as λ(x; a, b) = b a S(x a) S(x b), where S(x) = ½ = sin x. For all 0 M < N we can write N a=m χ(a) = = a χ(a)λ ; M, N χ(m) χ(n) = χ(a) a M S a N S χ(m) χ(n). (4) The Gauss su τ(χ) is defined as τ(χ) = a χ(a)e. For a priitive Dirichlet character χ (od ) we have τ(χ) = Ô. (5)

15 45] A nuerically explicit version of the ólya Vinogradov ineuality 37 It has been proved by oerance (see euation (0) in [3]) that χ()τ(χ) = i χ(a)cos a, if χ is even; χ(a)sin a, if χ is odd. (6) 5.. The case of even characters It follows fro (4) that N χ(a) = χ(a) S a a N S χ(n). Let > 0. By the definition of the functions ω(t) and ω (t), we have ½ S(u) = = Hence, we can write N χ(a) = Fro (6) we also have sin u ω sin u ω χ(a) χ(a) = = sin u ω S(u; ). (7) sin a sin a N a N a χ(a) S ; S sin a sin a N χ()τ(χ) ω sin N ω, = ω ; χ(n). (8) =

16 38 Ditriy Frolenkov (Moscow) [46 and (5) iplies that χ(a) sin a sin a N Fro (9) and Lea 3 it follows that N χ(a) Ô By Lea and Lea 4 we have N ω sin N Ô C 0 G χ(a) Ô log γ log 3 C 0 C 0 exp sin N. (9) a. (0). () Let us take = C 0 Ô, obtaining N χ(a) Ô log Ô γ log C 0 C 0 4 Ô exp( 4Ô C 0 ). () Now applying () proves Theore in the case of even characters. N a=m 5.. The case of odd characters Fro (4) and (7) we have that χ(a)= χ(a) χ(a) sin a M sin a N ω a M a N S ; S ; χ(m)χ(n). (3)

17 47] A nuerically explicit version of the ólya Vinogradov ineuality 39 Applying (6) yields = i χ(a) sin a M sin a N ω = χ()τ(χ) cos M cos N and taking into account (5) and Lea allows us to write a=m χ(a) Ô Ô ω, sin a M sin a N ω cosm log γ log 3 Thus, fro (3), (4), and Lea 3 we have N Ô χ(a) log γ log 3 Applying Lea 4 yields N a=m χ(a) Ô cos N. (4) log γ log 3 C 0 C 0 exp C 0 G. a. Now let us take = C 0 Ô, obtaining N a=m χ(a) Ô log Ô γ log C 0 3 Ô C 0 exp Ô C 0.

18 40 Ditriy Frolenkov (Moscow) [48 This concludes the proof of Theore. Acknowledgeents. The research was supported by the grant RFBR a. Bibliography. G. Bachan, L. Rachakonda, On a proble of Dobrowolski and Willias and the ólya Vinogradov ineuality, Raanujan J. 5 (00), V.A.Bykovskii, The discrepancy of the Korobov lattice points, To appear in Izv. RAN. Ser. Mat. 3. V. A. Bykovskii, The discrepancy of the Korobov lattice points, rogra and Abstract Book p. 0. 7th Journées Arithétiues conference. 4. E.Dobrowolski, K.S.Willias, An upper bound for the su class of functions f, roc. Aer. Math. Soc. 4 (99), ah n=a f(n) for a certain 5. L. Goldakher, Multiplicative iicry and iproveents of the ólya-vinogradov ineuality, by Goldakher, Leo I., h. D., University of Michigan, 009, 09 pages; 33890, preprint is available in arxiv: v. 6. A. Granville, K. Soundararajan, Large character sus: pretentious characters and the ólya Vinogradov theore, Jour. AMS 0, (007), A. Hildebrand, Large values of character sus, J. Nuber Theory 9 (988), A. Hildebrand, On the constant in the ólya Vinogradov ineuality, Canad. Math. Bull. 3 (988), E. Landau, Abschätzungen von Charaktersuen, Einheiten und klassenzahlen, Nachrichten Königl. Ges. Wiss. Göttingen (98), H. L. Montgoery, R. C. Vaughan, Exponential sus with ultiplicative coefficients, Invent. Math. 43 (977), R.E.A.C.aley, A theore on characters, J.London Math. Soc. 7 (93), G. ólya, Über die Verteilung der uadratischen Reste und Nichtreste, Nachrichten Königl. Ges. Wiss. Göttingen (98), C. oerance, Rearks on the ólya Vinogradov ineuality, Integers (roceedings of the Integers Conference, October 009), A (0), Article 9, pp. 4. Z. M. Qiu, An ineuality of Vinogradov for character sus (Chinese), J. Shandong Univ., Nat. Sci. Ed. 6 (99), A. D. Sialarides, An eleentary proof of ólya Vinogradov s ineuality, eriod. Math. Hungar. 38 (999) A. D. Sialarides, An eleentary proof of ólya Vinogradov s ineuality,. eriod. Math. Hungar. 40 (000), 7 75.

19 49] A nuerically explicit version of the ólya Vinogradov ineuality 4 7. I. M. Vinogradov, On the distribution of power residues and non-residues, J. hys. Math. Soc. er. Univ. (98), 94 98; Selected works, Springer Berlin, 985, Ditriy Frolenkov Steklov Matheatical Institute Gubkina str., 8, 999 Moscow, Russia frolenkov adv@ail.ru