Numerically explicit version of the Pólya Vinogradov inequality

Transcription

1 Nuerically explicit version of the ólya Vinogradov ineuality DAFrolenkov arxiv:07038v [athnt] Jul 0 Abstract In this paper we proved a new nuerically explicit version of the ólya Vinogradov ineuality Our proof is based on the new ideas of VA Bykovskii and iproves a recent ineuality obtained by C oerance In paper [] C oerance proved a new explicit version of the faous ólya Vinogradov ineuality for character sus In the present paper we obtained an iproveent of oerance s result Our approach is based on a recent construction due to V Bykovskii [], [3] In [], [3] this construction was used to obtain new upper bounds for 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 Introduction Let χ od be a priitive Dirichlet character ut S χ = ax χn 0 M<N, T χ = ax N n=m a=0 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[] independently proved that for any nonprincipal Dirichlet character ineuality The research was supported by the grant RFBR a S χ c log

2 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 T χ log δg +o where δ g = g sin g, This result has been recently iproved by Goldakher see [7] He obtained the following result T χ log δg+o Under the General Rieann Hypothesis Montgoery and Vaughan [5, Theore 3] proved that S χ loglog In fact it is the best-possible result aley [4] proved that there are an infinite class of uadratic characters χ n od n for which S χn n loglog n An iportant proble is to find the ost precise for of ineuality There are two types of results The results of the first type do not take care about the explicit bounds for the reainder ters The results of the second type give all constants explicitly Usually the results of the first type have better constants in the ain ter Asyptotic results and Landau [3] proved that Hildebrand [4] obtained and S χ S χ T χ T χ log +o if χ = log +o if χ = log 3 +o if χ = log 3 +o if χ = Later Hildebrand [5] iproved his result for even characters He showed that the estiate c log T χ 3 +o if χ = where c = {, if is cubefree; 4, else, 3

3 holds Granville and Soundararajan [6] obtained two ineualities 69 c log T χ o if χ = and T χ c +o log if χ = Up to now this result is the best-known one Nuerically explicit results In this section we discus nuerically explicit versions of the ólya Vinogradov ineuality Qiu [6] proved that S χ 4 log Sialarides [7], [8] obtained estiates T χ 3 log+ log 4 γ if χ = and T χ log+ + if χ = Dobrowolski and Willias [9] proved that for any nonprincipal Dirichlet character χ od one has S χ log+3 log Bachan and Rachakonda [0] iproved their result and proved S χ log+65 3log3 for any nonprincipal Dirichlet character χ od In [] oerance proved that and S χ 4 3log+ loglog+ if χ = 3 S χ log+ loglog+ if χ = 4 Up to now these bounds are the best-known nuerically explicit versions of the ólya Vinogradov ineuality 3

4 Main result We prove the following theore which iproves both 3 and 4 in the second ter Theore Let χ od be a priitive character then If χ = then If χ = then S χ log+ 4 +γ+logc0 +ψ, S χ log+ where γ is the Euler constant, C 0 = 4 5/ +5 and ψ = + 4 C ψ = + 3 C 0 + +γ+log C 0 +ψ, exp C 0 exp C 0 Easy calculations show that for > 8000 the first case of Theore is better than 3 and for > 8000 the second case of Theore is better than 4 The proof of this theore is based on the ideas of Bykovskii [] and on the ideas of oerance [] 3 oerance s leas In this section we forulate all necessary results fro the oerance s paper [] Lea For all real nubers x and positive integers n, we have n j= roof See [, Lea 3] sinjx j < logn+ γ+log+ 3 n Lea For all real nubers α, β and positive integers n, we have roof See [, 4] n = cosα cosβ < logn+γ+log+ 3 n 4

5 4 Bykovskii s ethod In this section we describe the ain construction fro the paper [] We give a slight odification Let θ : [0, [0,] be the following function 0, if 0 < t ; t, if t ; θt = t, if t ; 0, if t For all t 0, one has j= t θ = j Let ω,ω : 0, [0,]be the following functions t 0, if 0 < t ; ωt = θ = t, if t ; j j=0, if t, ω t = ωt = j= = t θ = j, if 0 < t ; t, if t ; 0, if t For any positive nubers and put Su;, = ω ω sin u If then where For any positive nuber let G u = Su;, = Su; Su;, Su; = = + n+u = ω sin u 5 where eu = expiu Bykovskii proved see [, Lea ] that We iprove this result Su; 4G u Lea 3 For any u [0;] and > the ineuality exp n enu, 6 is valid Su; C 0 G u 5

6 roof Take A [0;] The optial value of paraeter A will be calculated later We should consider two cases Let u A We see that and sinu u 7 sinu u 8 = The proof of 8 can be found in [, Lea ] By partial suation and 8 we have sin u u u B B 9 > B u Let B [A;] The optial value of paraeter B will be calculated laterwe will define B later We see that A u B u So by 5, 8, 9 we have Su; B u sinu + > B u sin u B u u + B If B then Su; B+ B, else Su; + u < B u u + B +logb B + B = = B+logB+ B By the definition of function G u we have So where G u + u +A Su; C +A G u, { B+ C =, if A B ; B B+logB+, if ax,a B B 6

7 Let u > A Our proof has uch in coon with the proof of the second case in [, Lea ] So we use the notation of [, Lea ] It was shown in [, Lea ] that for any > we have Su;, = ηu u ;, ηu+ u ;,, where ηw;, = / By the ideas of [, Lea ] we have So Hence ηw;, 4 w Su;, Su; 5 For any integer nuber n we have Therefore Su; 5 It is easy to prove that x x ω ω e wx dx x n+u + 6 n+u n+u > A +A + n+u + A + n+u < 5 + A G u f A = C +A is an increasing function when A [0;] So we should take A such that f A = 5 + A If A then in Hence A B B+ = B f A = +A, 0 A 7

8 Solving the syste { +A = 5 + A, 0 A, 0 we have So Lea is proved 5 A = 5/4 5 Su; f G 5/4 u = + 5 G u Lea 4 For any N and > the forula is valid a G = + exp a= roof Set δ a = e x= { ax, if a 0 od ; = 0, else, then by 6 we have = a G = a= + n= Lea is proved exp n a= δ n exp n 5 roof of Theore = e n a = + exp n= n Let 0 a < b and λx;a,b be the function on [0; satisfying, if x = a;, if a < x < b; λx;a,b =, if x = b; 0, else Writing its Fourier expansion we have λx;a,b = b a+ Sx a Sx b exp n δ n = = + exp exp 8

9 where For any 0 M < N one has = = a=m a= S a= Sx = = sin x λ a; M, N + χm+χn = a M a N S + χm+χn The Gauss su τχ is defined as τχ = e a= a For a priitive Dirichlet character χ od one has It was proved in [, 0] that χτχ = a= 5 The case of even characters It follows fro that = a= a= τχ = 3 cos a, if χ is even; i 4 sin a, if χ is odd a= S a S a N Let > 0 by the definition of functions ωt,ω t we have Su = sin u ω + = sin u ω = + χn sin u ω = +Su; 5 Hence = a= + a= a= S a ; sin a sina N a N S ; ω + + χn 6 9

10 By 4 one has a= By 3 one has sin a sina N a= ω = sin a sina N ω χτχ sin N ω sin N 7 Thus, by 7 and by Lea 3, sin N + a= a= C 0 G a + 8 By Lea 7 and Lea 4 we have log+γ+log+ 3 + C 0 + C 0 a= Now we take = C 0 So log+ +γ+logc C 0 a= exp + 9 exp C We take into account to coplete the proof of Theore in the case of even characters 5 The case of odd characters By and 5 we have Using 4 = sin a M sin a N ω a=m a= + a M a N S ; S ; a= = i a= sin a M sin a N ω = χτχ cos M cosn ω + + χm+χn 0

11 Using 3 and Lea we have a= sin a M sin a N ω log +γ+log+ 3 Thus, by, and by Lea 3 we have log +γ+log+ 3 + a=m By Lea 4 we have a=m Now we take = C 0 So a=m Theore is proved cosm cosn a= log +γ+log+ 3 + C 0 + C 0 log+ +γ+log C 0 C 0 G a + exp C 0 exp C 0 +

12 References [] ólya G Über die Verteilung der uadratischen Reste und Nichreste Nachrichten KöniglGesWissGöttingen 98,pp -9 [] Vinogradov I M On the distribution of power residues and non-residues, J hys Math Soc er Univ 98,pp 94-98; Selected works, Springer Berlin,985,pp [3] Landau E Abschätzungen von Charaktersuen, Einheiten und klassenzahlen Nachrichten KöniglGesWissGöttingen 98,pp [4] Hildebrand A On the constant in the ólya Vinogradov ineuality, Canad Math Bull 3 988, [5] Hildebrand A Large values of character sus, J Nuber Theory 9 988, 7-96 [6] Qiu Z M An ineuality of Vinogradov for character Chinese, J Shandong Univ, Nat Sci Ed 6 99, 58 [7] Sialarides A D An eleentary proof of ólya Vinogradov s ineuality eriod MathHungar [8] Sialarides A D An eleentary proof of ólya Vinogradov s ineuality, eriod MathHungar [9] Dobrowolski E and Willias KS An upper bound for the su a+h n=a+ fn for a certain class of functions f, roc Aer Math Soc 499, 9-35 [0] Bachan G and Rachakonda L On a proble of Dobrowolski and Willias and the ólya Vinogradov ineuality, Raanujan J 5 00, 65-7 [] oerance C Rearks on the ólya Vinogradov ineuality Integers roceedings of the Integers Conference, October 009, A 0, Article 9, pp [] Bykovskii V A The disperancy of the Korobov lattice points To appear in Izv RAN Ser Mat [3] Bykovskii V A The discrepancy of the Korobov lattice points rogra and Abstrct Book p0-7th Journées Arithétiues confernce [4] aley REAC A theore on characters, JLondon Math Soc 7 93, 8-3 [5] Monjtgoery H L and Vaughan R C Exponential sus with ultiplicative coefficients, Invent Math , 69-8 [6] Granville A and Soundararajan K Large character sus: pretentious characters and the ólya Vinogradov theore, Jour AMS Vol 0, Nuber 007, [7] Goldakher L Multiplicative iicry and iproveents of the ólya-vinogradov ineuality by Goldakher, Leo I, hd, UNIVERSITY OF MICHIGAN, 009, 09 pages; 33890, preprint is available in arxiv:095547v

13 DA Frolenkov Departent of Nuber theory Moscow State University e-ail: frolenkov 3