LINNIK S THEOREM MATH 613E UBC

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1 LINNIK S THEOREM MATH 63E UBC FINAL REPORT BY COLIN WEIR AND TOPIC PRESENTED BY NICK HARLAND Abstract This report will describe in detail the proof of Linnik s theorem regarding the least prime in an arithmetic progression Some background information is included, and references are given to fill in any omitted details This report also surveys both the conjectured and computed results for Linnik s constant as well as a brief discussion of the more recent improvements Introduction A classical theorem of Dirichlet says that the primes are evenly distributed amongst the equivalence class modulo q This naturally leads to many questions about the distribution For eample, one may ask how large is the least prime congruent to a modulo q, denoted pa, q We can first approach this question using the Prime Number Theorem for arithmetic progressions Define θ; q, a := p p a mod q log p Then the theorem tells us that for < log q A there eists a c such that θ; q, a = + O ep c log Notice that close to 0, θ; q, a = 0 and that as increases θ; q, a remains zero until is greater than the first prime congruent to a modulo q Thus if we can find a value of such that θ; q, a > 0 then it must be the case that pa, q We set θ; q, a > 0 and using a specific positive O-constant D we solve for We have that D ep c log < log < c logd

2 COLIN WEIR As < q, we may simplify the above to read log < c logdq > ep c log Dq > Dq c log Dq q C log q, for a computable constant C So we obtain that pa, q q C log q Notice however that this bound is super-polynomial in q However, if we assume the Riemann Hypothesis, then we can improve the error term in the Prime Number Theorem to θ; q, a = + O / log A similar calculation with this epression yields pa, q log q So under the Riemann Hypothesis we get a result polynomial in q of order q +ɛ But an even stronger result of q +ɛ is conjectured Therefore, before Linnik, the best unconditional lower bounds on pa, q were etremely distant from the conditional ones That was how the problem stood until Linnik s works [Lin44a] and [Lin44b] in 944 where he proves Linnik s Theorem: There eists an effectively computable constant L > 0 such that min {p :, p a mod q} =: pa, q q L Linnik himself never computed an eplicit value for L but since his work many people have An overview of these results is included in Table L Name Ref 0000 Pan [Pan57] 777 Chen [J65] 80 Jutila [Jut77] 0 Graham [Gra8] 6 Wang [Wan86] 55 Heath-Brown [HB9] 5 Xylouris [Xyl09] Table Estimates For Linnik s Constant In this paper we will prove the statement of Linnik s theorem as written above with no attempt to eplicitly compute the constant The net section contains some required background information and motivation for the approach of the proof Section 3 contains a few key propositions, the proofs of which are covered in [IK04] With these results we prove Linnik s Theorem in Section 4 We close with a brief discussion of some of the more recent improvements mainly due to Heath-Brown

3 LINNIK S THEOREM 3 Preliminaries The proof of Linnik s theorem relies on a few key ideas First we need to derive an arithmetic function that has support on primes congruent to a modulo q We will find this function while constructing a summatory function related to certain L-functions We will eventually derive bounds from an investigation of the zeros of these L-functions Finally, these bounds are manipulated to give bounds on the least prime in an arithmetic progression We begin by investigating the Dirichlet series of a character modulo q Specifically, we consider n Ls, = n s n= which we rewrite as an Euler product Ls, = pp s The Euler product and Dirichlet series can be shown to converge absolutely for Rs > Using the Euler product, we derive a formula for the logarithmic derivative of Ls, Taking the logarithm, we see that log Ls, = log pp s Using the Taylor Epansion of log, = log Ls, = = log pp s log pp s k k pk p ks and since the series is absolutely convergent, differentiating gives L s, Ls, = = n p k s p k logp k Λnnn s where Λ is defined to be the function that makes the above equality true; specifically { logp n = p k for some k Z Λn = 0 otherwise

4 4 COLIN WEIR The Mellin Inversion formula provides an epression for the sum of Λn in terms of L-functions; +it L Λnn = lim s, s T πi it L s ds n Net, we use Perron s Formula to approimate the integral above up to a bound T and then move the line of integration to arrive at the following epression: ψ, := Λnn = δ ρ ρ + O logq T n Imρ T where ρ runs over the zeros of Ls,, and δ = 0 unless is the principal character in which case δ = see [IK04, Prop 55] To demonstrate the idea behind the proof of Linnik s Theorem, let s take a closer look at the function ψ, when is the principal character, which we will denote 0 In this case Ls, 0 is just the Riemann Zeta function ζs Similar to θ, q; a, for values of near 0, ψ, 0 = 0 Furthermore, as increases, ψ, remains zero until we reach the first prime power, where it takes the value log So if we were looking for a bound on how big the smallest prime power is potentially with the power, one way to do it is to prove that ψ, 0 > 0 for all greater than some n With only a little more work we can describe how small the smallest prime is That is, if one knows something about the zeros of ζs they could describe the behavior of ψ, 0 and consequently the growth of primes This idea from the proof of the Prime Number Theorem is essentially the idea behind the proof of Linnik s theorem as well However, we don t want to eamine the function ψ,, which increments at all prime powers, but a different function that only increments on prime powers congruent to a modulo q With this in mind, we define the following function: ψ; q, a := aψ, Using the orthogonality of characters we see that ψ; q, a = aψ, = = = = a n Λnn Λn n Λn n n n a mod q Λn n a na

5 LINNIK S THEOREM 5 Notice that ψ; q, a is positive for all greater than the first prime congruent to a modulo q Furthermore, by substituting equation into the definition of ψ; q, a we obtain the eplicit formula ψ; q, a = a Imρ T ρ ρ + O logq T Hence we have an epression for ψ; q, a in terms of the zeros of L-functions So if we can describe the behavior of the zeros of Ls, for all characters modulo q, then we can describe the number of prime powers congruent to a modulo q With this information it is a relatively simple matter to describe all primes congruent to a modulo q 3 Required Results The proof of Linnik s Theorem is in essence an effort to describe the growth of ψ; q, a via the eplicit formula As such, we need to make use of results about the distribution and density of the zeros of Ls, for all characters We therefore turn our attention to their product We denote L q s := Ls, For / α and T we define Nα, T, to be the number of zeros ρ = β +iγ of Ls, counted with multiplicity in the rectangle α < σ, t T And we write N q α, T := Nα, T, for the number of zeros of L q s counted with multiplicity in the rectangle Following the approach of [IK04] we will make use of the following three results Theorem 3 - The Zero-Free Region: Let σ := Rs There is a positive constant c, effectively computable, such that L q s has at most one zero in the region σ c / logqt, If such a zero eists, then it is simple and the zero of Ls, where is real, and non-principal Theorem 3 - The Log-Free Density Theorem: There are effectively computable constants c, c > 0 such that for any / < α <, T, N q α, T cqt c α Theorem 33 - The Deuring-Heilbron Phenomenon: There is a constant c 3 such that if there does eist an eceptional zero ρ = β with Lρ, = 0 and

6 6 COLIN WEIR c / logqt β, then for all characters modulo q, Ls, has no other zeros in the region log β logqt σ c 3, t T logqt Notice that Theorem 33 is about how the zeros of all characters modulo q are repelled by the presence of an eceptional zero of one character The proofs of these theorems will not be discussed in this paper However we note that the zero-free region is due to Landau and the proof can be found in [IK04, Thm 56] Theorems 3 and 33 are due to Linnik in [Lin44a] and [Lin44b] respectively, however the proofs of these results are also the subjects of Sections 8 and 83 in [IK04] 4 Proof of Linnik s Theorem We present the proof as demonstrated by Nick Harland He followed the approach used in [IK04], which borrows heavily from [Gra8] Recall that we aim to prove the following Linnik s Theorem: There eists an effectively computable constant L > 0 such that min {p :, p a mod q} =: pa, q q L We begin by establishing some notation Let c, c, c, c 3, c 4, ρ, β be as in Theorems 3,3, and 33 Assume without loss of generality that 0 < c, c 3 < and c, c > To keep the calculations clean, we set R = /c We will prove Linnik s theorem through a series of propositions regarding the behavior of ψ; q, a, beginning with the following Proposition 4 Let q 4c Then ψ; q, a = a β where θ is some function with θ < 4, and β log q + θc η/ + O q η := { c3 log β log q log q if β eists c log q otherwise Proof We begin with the eplicit formula for primes in arithmetic progression; Equation ψ; q, a = ρ a ρ + O logq T Imρ R

7 LINNIK S THEOREM 7 Let 0 < T R and consider the magnitude of a portion of the summation above: ρ a ρ β ρ β T/ T by bringing the absolute values inside and bounding ρ Notice that this sum can be represented by a Riemann-Stieltjes integral Namely, β T/ = T α d N q α, T = T N q, T + T N q, T + T α logn q α, T dα Now we substitute in the bound on N q α, T from Theorem 3, giving that the above is at most T cqt c c qt c + T log α dα [ qt T cqt c c + T log c 0 qt c ] log Noticing that, c T qt c c T log = c T qt c qt c log qt c qt c / log log qt c < c T qt c / log log < c T qt c / < 0 we obtain that a Finally, using qt c+/3, a T T ρ ρ ρ ρ c T c T log log c log qt log qt c 6cc + /3 T 3

8 8 COLIN WEIR Therefore, looking back at the eplicit formula, ψ; q, a = + a ρ log ρ +O R + error of summing only to T instead of R We can bound the error term using equation 3 In particular, equation 3 give us the bound on the error for each time we reduce our sum from R to R Repeating, we have a bound on the error from reducing the sum from /R to /4R, and then to /8R, etc, until we reach T Thus, taking an infinite sum, the total error is bounded by Consequently, i= R = O i T i i= ψ; q, a = + = φ + a a ρ ρ + O log R + ρ ρ + O T log R + T i= i and if T = q then ψ; q, a = + a ρ ρ + O log q q Now that we have the same error term as the statement of the proposition, we turn our attention to the sum up to T over the zeros of the Dirichlet characters If an eceptional zero eists, we remove it from the sum Then, using the values of η given in the statement of the proposition, a ρ uneceptional ρ ρ = a ρ uneceptional η α dn q α, T β = Nq /, T + log η α N q α, T dα

9 LINNIK S THEOREM 9 Appealing to Theorem 3 and making a similar calculation to the one above, we arrive at the bound Nq /, T + log η α N q α, T dα cq c + clog η clog log q c 4c η/, which gives the desired bound on ψ; q, a q c η q α c α dα We have established an estimate for ψ; q, a to work from The last step in the proof is to use the estimate from Proposition 4 to create lower bounds for ψ; q, a To do that we proceed in two different ways depending on whether or not an eceptional zero eists The simpler situation is when an eceptional zero does not eist In that case we have the following result Proposition 4 If β does not eist, then if q 4c ψ; q, a = + θc ep c log 4 log q then log q q Proof Set η = log q in Proposition 4 In the more difficult case where an eceptional zero does eist we have the following Proposition 43 Suppose β eists, and suppose that δ := β c { } Set log q ν := ma 4c, 4 c, 4 log8c c 3 log c If q ν, then ψ; q, a δ log q + O q 3c Proof We consider two pieces of the epression for ψ; q, a in Proposition 4 First, using the assumptions on and δ we have a β β δ β β δ β q νδ = δ q νδ Using that e / +, which can be proved via Taylor series, we get that the above is greater than νδ log q + νδ log q δ νδ log q + νc / δ = δ log q /ν + c / δ

10 0 COLIN WEIR Finally, substituting in for ν we get that the above is greater than δ log q c 4 /4 + c / δ = 4δ log q 3c δ Secondly, log q log c log q ν/ q ν/ q c 3 logδ log q 4 log q q log8c logδ Recalling that c >, we can can drop the absolute values around log c Hence, the above is equal to q log8c logδ log q log c log q = e log8c logδ log q log c log 8c log c = δ log qδ log q Substituting for δ yields log 8c log c δ log qδ log q δ log q c δ log q c 8c = δ log q 4cc log 8c log c c Using these inequalities in Proposition 4 we find ψ; q, a 4δ log q δ δ log q log q + O 3c c q = δ log q log q δ O 3c q Factoring out δ log q/3c gives the result As δ log q q we can combine the results of Propositions 43 and 4 to conclude for q ν ψ; q, a q Recall that ψ; q, a increments at primes and prime powers We would like to infer a bound on θ; q, a := p p a mod q logp,

11 LINNIK S THEOREM which increments only at primes Comparing ψ; q, a and θ; q, a we find that ψ; q, a θ; q, a = logp < logp As such, we define θ := p< i= p i p a mod q log p and obtain ψ; q, a θ; q, a < i= i= log θ n = i= p i θ n as θ = 0 for < Hence, to relate ψ; q, a and θ; q, a we need to investigate the growth of θ /n Using Riemann-Stieltjes integration we see that θ /n = /n log y dπy = π / log / /n Using the trivial bound πy > we can simplify the integral yielding θ /n < π /n log /n /n πy y dy y dy = π /n log /n And so by the Prime Number Theorem we see that θ /n /n Therefore, ψ; q, a θ; q, a /n / Finally, we see that θ; q, a = ψ; q, a + O / n= q / + C/ q / for q L for some eplicitly computable number L Recalling that θ; q, a is zero until is greater than a prime congruent to a modulo q, we conclude that pa, q q L, where L is a effectively computable constant Linnik s Theorem is proved 5 Concluding Remarks One could work through the proof we have provided and compute a numerical value for L This was first done by Pan [Pan57] who achieved a value of L = 0000 and later more carefully L = 5448 The more recent result of Heath-Brown achieves L = 55 using essentially similar but considerably better methods To get a sense of his improvements, we note that he is able to use c = 0348, which would only give us a value of ν > 5 We conclude this report with a brief discussion on the types of improvements that Heath-Brown made

12 COLIN WEIR The proof we provided was an investigation of ψ; q, a; essentially a search for a value of to which ψ; q, a > 0 Besides making use of modern improvements of Theorems 3, 3 and 33, Heath-Brown investigates a slightly different function He defines log n Σ := Λnnn s f, log q n= where f is a continuous function with support on [0, 0 He similarly finds an eplicit formula for Σ involving L-functions and also looks for bounds based on their zeros Heath-Brown also argues using a zero free region, but with at most four zeros in it He credits the idea for this improvement to Graham, who in [Gra8] proves the following Theorem 5 If q is sufficiently large, then L q s = Ls, has at most two distinct zeros ρ = σ + it in the region σ and at most four zeros in the region σ 0069 log q + t, 0769 log q + t However, using his new eplicit formula with the function f, Heath-Brown improves Graham s bounds and proves the following three theorems Theorem 5 If q is sufficiently large, then L q s has at most one zero in the region σ 0348, t log q Such a zero, if it eists, is real and simple, and corresponds to a non-principal real character Theorem 53 If q is sufficiently large, then L q s has at most two zeros, counted with multiplicity, in the region σ 0696, t log q Moreover, for large enough q, there eists a character non-vanishing for σ 070, t, log q for all characters with, mod q such that Ls, is

13 LINNIK S THEOREM 3 Theorem 54 If q is sufficiently large, there eist characters, Ls, is non-vanishing for σ 0857, t log q for all characters with,,, mod q such that Notice that these three theorems encapsulate statements about both zero-free regions and Deuring-Heilbronn phenomena In fact, Heath-Brown argues using variational calculus that the function f was chosen optimally to maimize the bounds given in these theorems One more improvement comes from taking a closer look at the zeros of individual L-functions Specifically, Heath-Brown counts the characters whose L-function has a zero in the region λ/ log q He denotes this Nλ and shows Nλ + ɛ 67 e 73λ 6λ 30 e 5 6λ for λ /3 log log log q He credits this as following directly from a bound on character sums given by Burgess [Bur86] Heath-Brown s value of L = 55 also relies heavily on numerical computations; and admittedly Heath-Brown made no effort to compute these values to the finest precision Indeed, he closes his paper with a list of nine small technical improvements that could be made to improve his result This was essentially what gave Xylooris a value of L = 5 in [Xyl09] References [Bur86] D A Burgess, The character sum estimate with r = 3, J London Math Soc , no, 9 6 [Gra8] S Graham, On Linnik s constant, Acta Arith 39 98, no, [HB9] D R Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an [IK04] arithmetic progression, Proc London Math Soc , no, H Iwaniec and E Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol 53, American Mathematical Society, Providence, RI, 004 [J65] Chen J, On the least prime in an arithmetical progression, Sci Sinica 4 965, [Jut77] Matti Jutila, On Linnik s constant, Math Scand 4 977, no, 45 6 [Lin44a] U V Linnik, On the least prime in an arithmetic progression I The basic theorem, Rec Math [Mat Sbornik] NS , [Lin44b], On the least prime in an arithmetic progression II The Deuring-Heilbronn phenomenon, Rec Math [Mat Sbornik] NS , [Pan57] C D Pan, On the least prime in an arithmetical progression, Sci Record NS 957, 3 33 [Wan86] W Wang, On the least prime in an arithmetic progression, Acta Math Sinica 9 986, no 6, [Xyl09] T Xylouris, On linnik s constant, arxiv: , 009 address: cjweir@ucalgaryca