Remarks on the Pólya Vinogradov inequality

Transcription

1 Remarks on the Pólya Vinogradov ineuality Carl Pomerance Dedicated to Mel Nathanson on his 65th birthday Abstract: We establish a numerically explicit version of the Pólya Vinogradov ineuality for the sum of values of a Dirichlet character on an interval While the techniue of proof is essentially that of Landau from 98, the result we obtain has better constants than in other numerically explicit versions that have been found more recently Keywords: Dirichlet character, Gauss sum, Pólya Vinogradov ineuality Introduction Let χ be a non-principal Dirichlet character to the modulus and let Sχ = max M,N a=m The Pólya Vinogradov ineuality independently discovered by Pólya and Vinogradov in 98 asserts that there is a universal constant c such that Sχ c /2 log This ineuality is remarkable for several reasons: it is relatively easy to prove, it is surprisingly strong, and it has many applications To indicate its strength, it is known that Sχ /2 for χ primitive, a result of Montgomery and Vaughan [0]; this shows that apart from the log factor in, the ineuality is best possible Montgomery and Vaughan [9] have also shown on the Riemann Hypothesis for Dirichlet L-functions that Sχ /2 log log This contrasts with a result of Paley from 932 that Sχ /2 log log for infinitely many uadratic characters χ There has been exciting recent work on improving the order of magnitude in In [5], Granville and Soundararajan show that one can save a small power of log in when χ has odd order This result has been improved by Goldmakher [4], who has shown that for each fixed odd number g >, for χ of order g, Sχ /2 log g+o, g = g sin g, Given the usefulness of the Pólya Vinogradov ineuality it is interesting to prove a version of it that is completely numerically explicit Qiu [2] has 200 Mathematics Subject Classification: L40, L26 The author was supported in part by NSF grant DMS

2 2 Carl Pomerance shown that for χ primitive, Sχ 4/ 2 /2 log /2 plus some lesser, but explicit terms In a series of two papers, the later being [3], Simalarides obtained the leading coefficient / in the case where the interval is [0,N] and χ = In [2], Bachman and Rachakonda show that Sχ < 3log 3 /2 log + 65 /2, the emphasis being on the method that was previously introduced by Dobrowolski and Williams Working through the details of their proof several years ago unpublished, I was able to make the small improvement to M+N a=m+ 2 3log 3 /2 log N / /2 for 22,500 and N /2 When N/ /2 is not too large, this ineuality gives an explicit bound with very good constants I recently learned that an almost identical result appeared in Vinogradov [4] It is interesting that some early papers on the Pólya Vinogradov ineuality, though not advertised as such, had numerically explicit estimates, or estimates that could be easily made numerically explicit For example, in [8], Landau recapitulates Pólya s argument, getting for χ primitive, Sχ /2 log + 2 /2 log log + 2 /2, 2 see page 83 To achieve this, on page 8 one should not replace the denominator n + x with nx Landau goes on to improve the leading coefficient / in 2 to /2 /2 in the case of even characters and /2 in the case of odd characters A character is even or odd if it is an even or odd function, respectively, and this is determined by whether χ = or, respectively With a little effort, explicit secondary terms can be worked out here And with this effort, one would have an explicit ineuality that majorizes the others that are described above In [6], Hildebrand mentions that an unpublished observation of Bateman allows an improvement of the constant /2 /2 in the case of even characters to 2/ 2 Namely, Bateman replaces an estimate for n sin mθ /m in Landau s paper with a better one found in the book of Pólya Szego [] In this note we carefully work through the Landau Bateman argument with an eye towards a numerically explicit final result We prove the following theorem Theorem For χ a primitive character to the modulus >, we have 2 Sχ 2/2 log + 4 2/2 log log /2, χ even, 2 /2 log + /2 log log + /2, χ odd

3 Let Remarks on the Pólya Vinogradov ineuality 3 S 0 χ = max N, so that we always have Sχ 2S 0 χ via the triangle ineuality In the case of even characters, this may be reversed, that is, Sχ = 2S 0 χ Thus, the above ineualities for Sχ above may be divided by 2 for χ even when considering an initial interval Let a=0 2 Preliminaries C n x = j= cos jx j It is easy to see that for all x, C n x C n 0 = n j= /j, which tends to infinity like log n as n tends to infinity However, we are concerned with a lower bound for C n x As a function of x, C n x is periodic with period 2 and satisfies C n 2 x = C n x Thus, it suffices to consider its minimum on the interval [0,] In 93, Young [6] showed that C n x for all n and x This is best possible, since C = However, if n >, one can do slightly better, as shown by Brown and Koumandos [3]; they have C n x 5/6 in this case This too is best possible, since C 3 = 5/6 We begin with the following simple result which is useful when n is large Lemma 2 For all real numbers x and all positive integers n, we have C n x > log 2 2 n Proof As noted above, we may assume that 0 x In [6] it is shown that the minimum of C n x on this interval occurs at n + 2 x n = 2 n + also see [], Problem 27 in Part VI First assume that n is odd Then x n = and j C n x n = j j= This alternating series converges to log 2 The last term is negative, so the finite sum is slightly below the limiting value by an amount less than the first of the terms from n + to infinity, which is /n + Thus, C n x > log 2 /n + Now assume that n is even Then, n is odd, so that C n x = C n x + which proves the lemma cos nx n > log 2 n + cos nx n log 2 2 n,

4 4 Carl Pomerance We remark that while it is not hard to improve the expression 2/n in the lemma, the number log 2 is best possible since C n log 2 as n We use Lemma 2 to get an upper bound for the series S n x = j= sin jx, j following the outline in [], Problems 34 and 38 in Part VI Lemma 3 For all real numbers x and positive integers n, we have S n x < 2 log n + 2 γ + log 2 + 3, n where γ is the Euler Mascheroni constant Proof We begin with the Fourier expansion for sin θ as recorded in [], Problem 34 in Part VI: sin θ = 2 4 cos 2mθ 4m 2 Thus, S n x = 2 j= j= j 4 Using Lemma 2, we thus have S n x < 2 j + 4 log 2 + 2/n 4m 2 4m 2 = 2 j= j= cos 2mjx j j + 2 log n It remains to note that the harmonic series satisfies j < log n + γ + n 3 j= For a real number x and positive integer n, let sin jx R n x = j j=n+ In [8, page 8], Landau follows Pólya s argument, getting an estimate for the sum of R n x at eually spaced points in 0,] Here we do slightly better Lemma 4 If > is an integer, then 2a R n < 2 n + log + log

5 Remarks on the Pólya Vinogradov ineuality 5 Proof Following the argument in [8], we have for 0 < x and every positive integer n, R n x n + sinx/2 Since R n 2 x = R n x, the same estimate holds when < x < 2 Thus, 2a R n n + sina/ 4 On 0, the function cscx is concave up Thus, for a an integer with a, we have so that sina/ < csca/ < /2 /2 a+/2/ a /2/ csc t dt, csc t dt = 2 + cos/2 log sin/2 Using cos t < and sint > t t 3 /6 for t > 0, we have sina/ < 2 4 log /2 2 /6 And using log/ t < log + 2t < 2t for 0 < t < /2, we have sina/ < 2 log + log Thus, the result follows from The lemma may be slightly improved using Alzer and Koumandos [, Lemma 2]: the expression log4/+ 2 /2 2 = o may be replaced with γ log/2 = An earlier result of Watson [5] had γ log/2 + o Let be an integer with 3 and let M,N be numbers with 0 M < N < Let Φ,M,N = Φ be the function on [0,2 satisfying /2, if x = 2M/ or x = 2N/ Φx =, if 2M/ < x < 2N/ 0, otherwise Then, as in Landau [8], page 82, the real Fourier series for Φx is a 0 + a m cos mx + b m sin mx,

6 6 Carl Pomerance where a 0 = N M/ and for m, a m = sin 2Nm m b m = m sin 2Mm cos 2Nm cos 2Mm, Further, this series converges to Φx uniformly on [0,2 Recall that if χ is a Dirichlet character to the modulus, then the Gauss sum τχ is defined as τχ = ea/, where ex = e 2ix If χ is a primitive character with modulus, then τχ = /2, χmτχ = 6 eam/, 7 for every integer m, see [7, 32 and 34] Suppose that χ is primitive with modulus > and so 3 Using that cos is even and sin is odd, we have for every integer m, sin 2am = /2<a</2 sin 2am = 0 8 when χ is even, and cos 2am = /2<a</2 cos 2am = 0 9 when χ is odd Thus, from 7, 8, 9, and using ex = cos 2x+isin 2x, cos 2am, if χ is even, χmτχ = 0 i sin 2am, if χ is odd 3 The case of even characters Suppose that χ is even and primitive mod with > and so 5 We treat the case when the interval [M,N] is an initial interval; that is, M = 0 As remarked at the end of the Introduction, after multiplying by 2, our estimate will stand for Sχ

7 Remarks on the Pólya Vinogradov ineuality 7 We introduce the function Φx discussed above With a m,b m the coefficients in 6, we have for any positive integer n, = 2a 2 χn + Φ = 2 χn + a 0 + a=0 + m=n+ = 2 χn + A n + B n, say a m cos 2am + b m sin 2am a m cos 2am + b m sin 2am Using 8 and 0, A n = Thus, by 7, 6, and Lemma 3, a m χmτχ A n = /2 a m χm /2 2 2/2 log n + γ + log n 2Nm m sin Entering the coefficients 6 in the expression for B n we get B n = = Thus, by Lemma 4, m=n+ 2N am sin + 2am sin m m 2a + R n R n 2N a B n 2N a R n + 2a R n = 2 = 2 2a R n < log + log n + 2a R n

8 8 Carl Pomerance Putting these estimates together, we have 2 + A n + B n a=0 < /2 log n + γ + log n log + log n + We now choose n = 2 /2 log, so that < 2 2/2 a= log + log log /2 log 4 + γ n + log4/ + log < 2/2 log + 2 2/2 log log /2, log 2 for 2 An inspection of the cases of even primitive characters with conductor < 2 shows that the estimate holds here as well In fact, each such S 0 χ is smaller than the least significant term 3 4 /2 according to calculations kindly supplied by Jonathan Bober and Leo Goldmakher This proves Theorem in the case of even characters 4 Odd characters Now we consider the case of an odd primitive character χ We consider a general interval [M,N] in [0, As with the even case, we have a=m = 2 χm + 2 χn + + 2N a R n = 2 χm + 2 χn + C n + D n, say Using 9, 0, and 6 we have C n = iτχ χm m a m cos 2am R n 2M a cos 2Nm cos 2Mm, + b m sin 2am

9 so that by 7, we have Remarks on the Pólya Vinogradov ineuality 9 C n /2 cos mα cos mβ, m where α = 2N/ and β = 2M/ Now m cos mα cos mβ = 2 mα + β mα β m sin sin 2 2 /2 /2 mα + β mα β 2 sin2 sin2 m 2 m 2 = 2 cos mα + β 2m log n + γ + log n, /2 using Lemma 2 and 3 for the last step Thus, C n /2 log n + γ + log n cos mα β 2m The expression for D n is estimated as with B n, getting D n log + log + 2 n /2 We now take n = 4/ /2 log Putting the estimates together, we have for an odd primitive character χ, 2 /2 log + 4 /2 log log + log + γ + log n a=m + /2 + log4/ 2 + log log Collecting the significant terms, we have 2 /2 log + /2 log log + /2, a=m whenever 45 A check over smaller cases shows that this ineuality holds there as well According to the calculations supplied by Jonathan Bober and Leo Goldmakher, Sχ < /2 for primitive odd χ with 2; the value of the right side of at = 3 exceeds 6 yet Sχ < 6 for 28; and the value of the right side exceeds 0 at = 29 yet Sχ < 0 for 44 This completes the proof of Theorem in the case of odd characters

10 0 Carl Pomerance Acknowledgments I gratefully thank Jonathan Bober and Leo Goldmakher for their calculations of Sχ for small moduli I am also grateful to Adolf Hildebrand and Igor Shparlinski for some helpful conversations And I thank Ke Gong for informing me of [4] References [] H Alzer and S Koumandos, On a trigonometric sum of Vinogradov, J Number Theory , [2] G Bachman and L Rachakonda, On a problem of Dobrowolski and Williams and the Pólya Vinogradov ineuality, Ramanujan J 5 200, 65 7 [3] G Brown and S Koumandos, On a monotonic trigonometric sum, Monatsh Math , 09 9 [4] L Goldmakher, Multiplicative mimicry and improvements of the Pólya Vinogradov ineuality, arxiv:095547v [mathnt] 30 Nov 2009 [5] A Granville and K Soundararajan, Large character sums: pretentious characters and the Pólya Vinogradov theorem, J Amer Math Soc , [6] A Hildebrand, On the constant in the Pólya Vinogradov ineuality, Canad Math Bull 3 988, [7] H Iwaniec and E Kowalski, Analytic number theory, American Math Soc, Providence, 2004 [8] E Landau, Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen, Nachrichten Königl Ges Wiss Göttingen 98, [9] H L Montgomery and R C Vaughan, Exponential sums with multiplicative coefficients, Invent Math , [0] H L Montgomery and R C Vaughan, Mean values of character sums, Can J Math 3 979, [] G Pólya and G Szegő, Problems and theorems in analysis Volume II: Theory of functions Zeros Polynomials Determinants Number Theory Geometry Springer- Verlag, Berlin, Heidelberg 998 [2] Z M Qiu, An ineuality of Vinogradov for character sums Chinese, Shandong Daxue Xuebao Ziran Kexue Ban 26 99, [3] A D Simalarides, An elementary proof of Pólya Vinogradov s ieuality, II, Period Math Hungar , 7 75 [4] I M Vinogradov, A new improvement of the method of estimation of double sums Russian, Doklady Akad Nauk SSSR NS , [5] G N Watson, The sum of a series of cosecants, Philos Mag 3 96, 8 [6] W H Young, On a certain series of Fourier, Proc London Math Soc 2 93, Carl Pomerance Department of Mathematics, Dartmouth College Hanover, NH , USA carlpomerance@dartmouthedu