A lower bound for biases amongst products of two primes

Transcription

1 9: 702 Hough Res Number Theory DOI 0007/s R E S E A R C H A lower bound for biases amongst products of two primes Patrick Hough Open Access * Correspondence: patrickhough2@uclacuk Department of Mathematics, University College London, Gower Street, London WCE 6BT, England Abstract We establish a conjectured result of Dummit, Granville and Kisilevsky for the maximum bias in P2 races, which compare the number of P2 s x whose prime factors, evaluated at a given quadratic character, take specific values Keywords: Primes in arithmetic progressions, Prime races Background The study of prime races has long been a popular one and has attracted the attention of many great minds in number theory Its conception as a subject can be dated back to 853 by a correspondence between Tchbychev and M Fuss in which the former notes: There is a notable difference in the splitting of the prime numbers between the two forms 4n + 3, 4n + : the first form contains a lot more than the second The obvious generalisations of this to other moduli have been studied at great length and the most recent work on this can be found in [] In these works, one is concerned with the probability that, given x, there are more primes p x that lie in one particular residue class than lie in any other with respect to the modulus under consideration A comprehensive exposition of the main results on these primes races can be found in [2] Whilst these results present some of the most predictable such prime races, it has not yet been possible to find a race in which one team leads indefinitely for sufficiently large x It turns out that far more concrete results can be given when we consider semi-primes, integers that are a product of exactly two primes In 205, Dummit et al [3] showed that significantly more than a quarter of all odd integers of the form pq up to x, withp and q both prime, satisfy p q 3 mod 4 More generally they proved the following Theorem Theorem [3] Let χ be a quadratic character of conductor d For η = or we have #{pq x : χ d p) = χ d q) = η} 4 #{pq x : pq, d) = } = + η L χd + o)), wherel χd := p χ d p) p ) We focus on just how large this bias can get if we restrict the conductor of our quadratic character by d x The following prediction was made in relation to this problem The Authors) 207 This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made

2 : 702 Hough Res Number Theory 9 Page 2 of Conjecture [3] There exists d xsuchthat #{pq x : χ d p) = χ d q) = η} 4 #{pq x : pq, d) = } + log + O) It is important to note that in proving such a claim, one must prove a uniform version of Theorem since the proof there assumes that x is allowed to be very large compared to d Whilst this is not a straightforward task, once achieved, all that remains is to show that L χd can be found suitably large for d x It turns out that this final step is easily accomplished using a slight adaptation of a result from Granville and Soundararajan s work on extremal values of L, χ)[4] In this paper, we prove Conjecture and indeed give a slightly stronger result in the form of the following Theorem Theorem 2 Fix ɛ>0 For large x there exist at least Dx) ɛ integers d Dx) for which #{pq x : χ d p) = χ d q) = η} 4 #{pq x : pq, d) = } + Here, Dx) = exp[clog x) /2 ] log + O) In 928, Littlewood gave [5] an upper bound on the value of L, χ) as follows Theorem 3 Littlewood 99) Assume the Generalised Riemann Hypothesis GRH) For any non-principal primitive character χ mod q), one has L, χ) 2e γ + o)) log log q This upper bound, when combined with a lower bound derived from [4], allows us to claim that, under GRH, we have found the maximum bias amongst these races Theorem 4 Assume GRH Then #{pq x : χ d p) = χ d q) = η} max d x 4 #{pq x : pq, d) = } log Using a second derived result from [4], the initial legwork of proving a uniform version of Theorem allows us to show just how small this bias can get Theorem 5 Fix ɛ>0 For large x there exist at least Dx) ɛ integers d Dx) for which #{pq x : χ d p) = χ d q) = η} 4 #{pq x : pq, d) = } Here, Dx) = exp[clog x) /2 ] log + O) Finally, we conducted a computational search for conductors d that give rise to particularly biased prime races of this kind In order to find these large biases, is it crucial to study the quantity L χd Indeed, if χ d p) = for a large proportion of the small primes p or alternatively χ d p) =+, then we expect to have race with a correspondingly large bias Such characters are closely linked to prime generating polynomials and it was the work of Jacobson Jr and Williams that gave us the best results in this area [6] We

3 : 702 Hough Res Number Theory 9 Page 3 of are thus able to give, what we believe to be, the lowest known value of L χd conductor d of our character is where the , and L d/ ) 208 In the words of Fiorilli and Martin [], this conductor gives rise to the most unfair known such prime race Moreover, the corresponding value of L, χ d ) 044, which is so important in determining L χd, is the lowest calculated for a real Dirichlet character χ Indeed, extremely small values of L, χ) have many links with prime generating polynomials 2 Proof of Theorem 2 We begin by defining the notion of a Siegel zero for a Dirichlet L-function Ls, χ) associated with the real Dirichlet character χ of modulus qazeroofls, χ) is called a Siegel zero if, for some suitable positive constant c, Ls, χ) has a real zero β such that c log q β As mentioned, a proof of Theorem 2 requires that we give a uniform version of Theorem Moreover, we need that ) holds for d Dx) := exp[clog x) /2 ] We focus on the case in which η = sincethecaseη = istackled using much the same reasoning Establishing this result, assuming Ls, χ d ) has no Siegel zero, will make up the majority of the proof To complete the proof we show that, for large x, there exist many d Dx) corresponding to real Dirichlet characters, such that L χd log + O), 2) where Ls, χ d ) has no Siegel zero The following application of Theorem from [4] is sufficient in order to make this last step Throughout this paper we will denote the j-fold iterated logarithm by log j where appropriate Theorem 2 Application of Theorem from [4]) Let D be large and τ = log 2 D) 2log 3 D)Then D τ) D ɛ, for fixed ɛ>0, where D τ) is the proportion of fundamental discriminants with d Dforwhich L, χ d ) e γ τ Proof The proof of this result is simply computational Fix ɛ>0we take τ = log 2 D) 2log 3 D)inTheoremof[4] By Theorem and then Proposition of [4] we have where C is given explicitly in Proposition of [4]): D log 2 D) 2log 3 D)) = exp Now we note that log 2 D) 2log 3 D)) + O [ + O elog 2 D) 2log 3 D) C log 2 D 2log 3 D)) )) log 3 D) log 2 D) )) > + O log 2 D) )) ], for D sufficiently large log 2 D)

4 : 702 Hough Res Number Theory 9 Page 4 of Therefore, ) D τ) > exp e C logd) log 2 D)) 3, > D ɛ, for D sufficiently large Theorem 2 ensures that there are, for sufficiently large D, atleastd ɛ fundamental discriminants d D for which L, χ d ) e γ log 2 D) 2log 3 D)) This follows from the well established fact that there are 6 π 2 x + Ox 2 +ɛ ) fundamental discriminants d with d x Note that we may write where L χ = p p χp) p = m μm) m log Lm, χ m ) = log L, χ) + Eχ), 22) log ) + ) = p p Eχ) p log + ) ) = 0898 p p Therefore for sufficiently large D, there are at least D ɛ fundamental discriminants d D for which L χd log 3 D) + O) In light of a result by Page that ensures there is never more than one conductor d x for which Ls, χ d ) has a Siegel zero, Theorem 2 combines with 22)togive2) as required Corollary 22 Fix ɛ>0 For sufficiently large D, there are at least D ɛ fundamental discriminants d Dforwhich L χd log 3 D) + O), such that Ls, χ d ) does not have a Siegel zero 2 Proofof uniformversion of )with Ls, χ) having no Siegel zero We write #{ab x : χa) = χb) = } 4 #{ab x :ab, d) = } = a, = + a, = + + χa)) + χb))/ log x 2x + O)) where we have used the following result due to Landau = 2x + O)) log x a, a, ) / χa) + χb) + χab) a,, a,, χa) + χb) + χab)), 2

5 : 702 Hough Res Number Theory 9 Page 5 of Here we count each of the products ab and ba in the sum separately, which accounts for the factor of 2 used here in contrast with the the usual formulation of this result Let us consider the first sum here a, χa) = χa) b x/a + a x/b χa) χa) 24) To get a useful bound for the terms in this expression it is clearly necessary to understand the quantity χn) 25) n x n prime We now incorporate the notion of a Siegel zero by way of a theorem from [7], pg 95 Theorem 23 [8] If c is a suitable positive constant, there is at most one real primitive character χ to a modulus d xforwhichls, χ) has a real zero β satisfying β> c log x 26) Such a zero is called a Siegel zero Davenport provides us with the next insight In Chapter 4 it is shown that, restricting the conductor of our character χ d by d exp[clog x) 2 ] and assuming that the corresponding L-function has no zero lying in the region defined by 26) wehave,aftersome computation, n x n prime χn) a x/b x log x) N, 27) for any given N > 0, where the implicit constant depends only on NUsing27), we may boundthesecondtermof 24) as follows: x χa) log x) N+ b x, where N > log x) N Using the prime number theorem, the third term of 24) can be similarly bounded, χa) x/2 log x) N x/2 log x x log x) N+ The first term may be written as χa) b x/a = χa)lix/a) + O x log x) N Here we have used the following result, proved by La Valle Poussin in 899 πx) = Lix) + O e alog x)/2), ), where we take N > 0 28)

6 : 702 Hough Res Number Theory 9 Page 6 of for some positive constant a These techniques, by symmetry, yield the same result for the second sum of 2 Summarising, we have shown that χa) + χb)) = 2 ) x χa)lix/a) + O log x) N, where we take N > 0 a, a, We now deal with the remaining term in 2 Namely χab) = χa) χb) + χb) χa) χa) b x/a b x/a a x/b χb) Applying 27) to each the three terms allows us to bound them by x/log x) N with N > Indeed, the first term becomes x χa) χb) = log x) N+ a x, where N > 0 log x) N Simply interchanging a and b yields the same bound for the second term and the third term is dealt with simply by taking the product of the two sums after applying 27) Thus, we have for N > 0, #{ab x : χa) = χb) = } 4 #{ab x :ab, d) = } = + Now let us examine χa)lix/a) = The last term can be written x χa) alogx/a)) 2 + O log x x + O)) + O))log x) N xχa) a logx/a) + O x x log x) 2 x log x) 2, a using χa), χa)lix/a) ) χa) alogx/a)) 2 29) where we have used the following result = + O) a a x We may also change the summand of the first term, only incurring a small error term so that it becomes xχa) a log x,

7 : 702 Hough Res Number Theory 9 Page 7 of where we have used the result log p = log x + O) p p x p prime The difference between this sum and the original can be bounded by x log a alog x) 2 x log x, which represents the error incurred in changing the summand of the first term in 29) The result of this is that χa)lix/a) = x χa) log x a x ) χa) x log x a + O 20) log x Applying 27) to the second term here, 29) becomes χa)lix/a) = x Lχ + O) ) log x a> x The above calculations imply that, for large x and d exp[clog x) /2 ] where Ls, χ d )has no exceptional zero, #{ab x : χa) = χb) = } L χ + O) 4 #{ab x :ab, d) = } = + + O/ ), 2) as required Now, denoting b = O/ ), we have + O/ ) = b = + b + b2 + For large x, b < /2and + b + b 2 + < + 2b = + Ob) Thus, = + O/ ), + O/ ) and 2) becomes #{ab x : χa) = χb) = } 4 #{ab x :ab, d) = } = + L χ + O) + OL χ / ), = + L χ + O), 22) as required, where we have used 22) and that L, χ) < c log x for some positive constant c Proof of Theorem 2) Defining Dx) = exp[clog x) /2 ], Corollary 22 gives us that there are at least Dx) ɛ integers d Dx) such that the corresponding Dirichlet character χ d is real and L χd log 3 Dx)) + O), where Ls, χ d ) had no Siegel zero In light of 22), we have at least Dx) ɛ integers d Dx) for which χ d is a real Dirichlet character and #{ab x : χa) = χb) = } 4 #{ab x :ab, d) = } is at least as large as + log + O)

8 : 702 Hough Res Number Theory 9 Page 8 of 3 Proof of theorem 4 We begin by showing that 22) can be obtained, for large x, on the range d x if we assume the Generalised Riemann Hypothesis GRH) The crucial advantage in making this assumption is that it allows us to write θx, χ) x /2 logdx)) 2, This bound can be found in [7] pg 25) and is far stronger than that obtained in the absence of GRH We thus have the corresponding bound, obtained using partial summation, χn) x /2 log x), for d x 3) n x n prime The procedure for arriving at 22) is then much the same as in the proof of Theorem 2 The only notable difference is in bounding the first term of 24) where we use the following result of Helge Von Koch 90) that assumes the Riemann Hypothesis πx) = Lix) + O x log x) ThenextstepistonotethatTheorem2 gives us that, for large x, there exists d x such that e γ log 2 x 2log 3 x) L, χ d ), Also, Theorem 3ensures that L, χ d ) GRH 2e γ + o)), for any d x In light of 22), we thus have max L χ d log + O) 32) d x Now, combining 32)with22), we have that #{ab x : χ d a) = χ d b) = } log max d x + 4 #{ab x :ab, d) = }, and Theorem 4 follows 4 Proof of Theorem 5 Following exactly the same steps as seen in the proof of Theorem 2, we have that for large x and d exp[clog x) /2 ], where Ls, χ d ) has no Siegel zero, #{ab x; χa) = χb) = } 4 #{ab x; ab, d) = } = + L χ + O) 4) Now we use a second application of Theorem from [4] Theorem 4 Application of Theorem from [4]) Fix ɛ>0 For sufficiently large D, there are at least D ɛ fundamental discriminants d Dforwhich L χd log 3 D) + O), such that Ls, χ d ) does not have a Siegel zero

9 : 702 Hough Res Number Theory 9 Page 9 of Proof of Theorem 4) The proof of this result follows is much the same way as in the proof of Theorem 2 Again, taking τ = log 2 D) 2log 3 D)inTheoremof[4], we have that for fixed ɛ and sufficiently large D, D τ) D ɛ, where D τ) is the proportion of fundamental discriminants with d D for which L, χ d ) π 2 6e γ τ Now recall the result of Page which ensures that there is never more than one conductor d D for which Ls, χ d ) has a Siegel zero Finally, using 22)wehavethatforsufficiently large D, there are at least D ɛ fundamental discriminants d D such that L χd log 3 D) + O) Proof of Theorem 5) Defining Dx) = exp[clog x) /2 ], Theorem 4 gives us that there are at least Dx) ɛ integers d Dx) such that the corresponding Dirichlet character χ d is real and L χd log 3 Dx)) + O), where Ls, χ d ) had no Siegel zero In light of 22), we have at least Dx) ɛ integers d Dx) for which χ d is a real Dirichlet character and #{ab x : χa) = χb) = } 4 #{ab x :ab, d) = } is at least as small as log + O) 5 Computational aspects It was Euler in 772 who first came across what became Euler s Polynomial x 2 + x + 4 What is so remarkable is that for the integers 0 x 39, the value of x 2 +x +4 is prime If we now consider the general quadratic f x) = ax 2 + bx + c, then, for a given integer x, the value of this quadratic is divisible by p if and only if ax 2 + bx + c 0 mod p, which is equivalent to saying that 2ax + b) 2 b 2 4ac mod p ) f ) Said differently p = 0 or This suggests that if a given quadratic produces a high ) density of primes for integers x m, then the character corresponding to χn) = f ) n might produce a high density of values for primes n m We now turn to an article by Jacobson and Williams [6] Their work is based upon a conjecture of Hardy and Littlewood [9], namely Conjecture F This implies that, for a polynomial of the form f x) = x 2 +x+a, A Z with discriminant, the asymptotic density of prime values of f is related to

10 : 702 Hough Res Number Theory 9 Page 0 of a quantity C ) They also suggest that the larger the value of C ), the higher the asymptotic density of primes for any polynomial of discriminant We thus restrict to polynomials of the form f x) = x 2 + x + A IfwealsodenotebyP A n), the number of primes produced by f A x) for 0 x n then Conjecture F can be written as follows Conjecture 5 Conjecture F simplified), Hardy and Littlewood [9]) where and P A n) C )L A n), L A n) = 2 n 0 dx log f A x), p ) C ) = p p 3 Jacobson and Williams predict that the polynomial x 2 + x + A has the highest asymptotic density of prime values for any polynomial of this type currently know, where A is given by ) We would thus expect the character with conductor d given by χn) = d/n) where d = 4A to yield a low value of L χd Firstly,wenotethatd mod 4 is square-free and thus d/n) defines a real primitive character A simple piece of SageMath yields L d/) 208 Furthermore, the data confirms this bias Defining r d x) := #{pq x : χ dp) = χ d q) = η} 4 #{pq x : pq, d) = }, where d is the conductor of the character χ d, we have that, on taking d = 4A where A is given by 5)andη =, r d 0 3 ) 3847, r d 0 4 ) 2974, r d 0 5 ) 2394, r d 0 6 ) 2067 This is the largest bias calculated for such primes races amongst products of two primes In light of 22), it is no surprise that this conductor also gives rise to what we believe to be the lowest known value of L, χ d ), where χ d is a general Dirichlet character of modulus d L, χ d ) 044s Table gives some values of r d x) for a few moduli d which were of interest when searching for extreme biases The modulus 5 is included for comparison and was the one used to introduce the search for greater biases in [3] The modulus 63 is the discriminant of Euler s polynomial and 5,47,453 yielded the highest bias found when performing a naive search for several hours over all integers When searching for small values of L, χ), Lehmer et al showed in [0] that the modulus 49,07,823,33 gave rise to the then) lowest known value of L, χ) 0695) The final row represents the prime

11 : 702 Hough Res Number Theory 9 Page of Table The values of r d x) over small x for moduli of interest that arose during our investigation Modulus, d r d 0 3 ) r d 0 4 ) r d 0 5 ) r d 0 6 ) 5 860, ,47, ,07,823, A race corresponding to the character that we have extracted from Jacobson and Williams work Acknowledgements This paper comes as a result of my MSc dissertation at University College London I give greatest thanks to my supervisor Prof Andrew Granville I also thank Ronnie George and Tarquin Grossman for their general exuberance Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Received: 8 October 206 Accepted: June 207 References Fiorilli, D, Martin, G: Inequities in the Shanks-Rnyi prime number race: an asympototic formula for the densities J die reine angawandte Math 676, Granville, A, Martin, G: Prime Number Races arxiv:math/ ) 3 Dummit, D, Granville, A, Kisilevsky, H: Big biases amongst products of two primes Mathematika 62, ) 4 5 Granville, A, Soundararajan, K: The distribution of values of L, χ d ) Geom Funct Anal 3, Littlewood, JE: On the class number of the corpus P k) Proc London Math Soc 27, ) 6 Jacobson Jr, MJ, Williams, HC: New quadratic polynomials with high density of prime values Math Comput 72, ) 7 Davenport, H: Multiplicative Number Theory, 3rd edn Springer, New York 2002) 8 Page, A: On the number of primes in an arithmetic progression Proc London Math Soc 39, ) 9 Hardy, GH, Littlewood, JE: Partitio numerorum III: on the expression of a number as a sum of primes Acta Math 44, Lehmer, DH, Lehmer, E: Integer sequences having prescribed quadratic characters Math Comput 240), )