Group automorphisms with prescribed growth of periodic points, and small primes in arithmetic progressions in intervals
aa r X i v : . [ m a t h . D S ] S e p GROUP AUTOMORPHISMS WITH PRESCRIBEDGROWTH OF PERIODIC POINTS, AND SMALLPRIMES IN ARITHMETIC PROGRESSIONS ININTERVALS
ALAN HAYNES AND CHRISTOPHER WHITE
Abstract.
We investigate the question of which growth rates arepossible for the number of periodic points of a compact group au-tomorphism. Our arguments involve a modification of Linnik’sTheorem, concerning small prime numbers in arithmetic progres-sions which lie in intervals. Introduction
In this paper we investigate the problem of determining which growthrates are possible for the number of periodic points of a compact groupautomorphism. We begin with a brief description of the history andmotivation for pursuing this line of inquiry, which turns out to beclosely connected to Lehmer’s Conjecture concerning Mahler measuresof algebraic numbers (an excellent survey of which is [13]).Suppose T is a (continuous) automorphism of a compact group G ,let h ( T ) denote the topological entropy of T , and for each n ∈ N let F n ( T ) be the number of points of period n , F n ( T ) := { g ∈ G : T n g = g } . We assume that our compact groups are Hausdorff, therefore the au-tomorphisms T are homeomorphisms. As mentioned in [10], Lehmer’sConjecture is equivalent to the statement thatinf { h ( T ) : h ( T ) > } > , where the infimum is taken over all compact group automorphisms T .If G is a metric space and T is expansive then we also have (see [11]) h ( T ) = lim n →∞ log F n ( T ) n . (1.1) Mathematics Subject Classification.
Motivated by this observation, Ward proved in [14] that for any
C > T for which the limit on theright hand side of (1.1) exists and is equal to C . However this does notimply Lehmer’s Conjecture because, although the groups G in Ward’sconstruction are metrizable, the corresponding automorphisms T arenot expansive and have h ( T ) = 0.Although it is a further departure from Lehmer’s Conjecture, theproblem of constructing compact group automorphisms with prescribedgrowth rates of periodic points is a natural one which is interesting inits own right. In this paper we will demonstrate how a slightly modifiedversion of Ward’s basic method, which we call the F -method, can beused to prove the following theorem. Theorem 1.1.
Suppose that r : N → R is a multiplicative functionsatisfying r ( p a ) − r ( p a − ) > . · a log p, for all primes p and for all a ∈ N . Then there exists a compact groupautomorphism T with F n ( T )exp( r ( n )) ≍ . Here and throughout the paper the notation f ( n ) ≍ g ( n ) means thatthere exist constants c , c > c | g ( n ) | < | f ( n ) | < c | g ( n ) | for all n ∈ N . With a small change to account for finitely many prime powers, The-orem 1.1 can be applied with r ( n ) = Cn to recover Ward’s result con-cerning the logarithmic growth rate of the number of periodic points.However if we are only interested in the logarithmic growth rate then,by arguing directly in the proof of the theorem, we can make the fol-lowing improvement. Theorem 1.2.
Suppose that r : N → R is a multiplicative functionsatisfying r ( p a ) − r ( p a − ) > . · a log p, for all primes p and for all a ∈ N . Then there exists a compact groupautomorphism T with lim n →∞ log F n ( T ) r ( n ) = 1 . These two theorems follow from the more general Theorem 2.1 be-low, where the assumption that r be multiplicative is relaxed. How-ever, within the context of group automorphisms constructed using the ERIODIC POINTS AND PRIMES IN PROGRESSIONS 3 F -method, some non-trivial arithmetical conditions on r must be en-forced. We illustrate this fact in our proof of the following theorem,which demonstrates that even very quickly growing functions r can failto be the logarithmic growth rates for the number of periodic points ofany group automorphism constructed using the F -method. Theorem 1.3.
Given any function t : N → R , we can find a function r : N → [0 , ∞ ) with t ( n ) = o ( r ( n )) , such that no automorphism T constructed with the F -method satisfies lim n →∞ log F n ( T ) r ( n ) = 1 . To resolve any potential confusion, we mention that this theoremcontradicts a remark at the end of [14]. The source of this contradictionis a small error in [14, Equation (9)]. The error is easily repaired anddoes not change the statement of the main result in that paper, but itdoes have some bearing on the first remark at the end of the paper.Theorem 1.3 shows that there are logarithmic growth rates r ( n )which tend to infinity arbitrarily quickly, which cannot be obtainedusing the F -method. It is also true that ‘polynomial’ growth rates, ofthe form r ( n ) = k log n , for k ∈ N , cannot be obtained (this will beproven in Section 4). However it turns out that some growth rateswhich are only marginally faster than polynomial are attainable. Toillustrate this, we will prove the following theorem. Theorem 1.4.
For x > and k ∈ N , let k x denote the k th iteratedexponential of x , k x := x x ·· x |{z} k , and define the function ι : N → R by ι ( n ) := min { k ∈ N : n < k e } . There exists a compact group automorphism T with lim n →∞ log F n ( T ) ι ( n ) log n = 1 . The F -method (defined precisely in the next section) requires us toselect a sequence of prime numbers { p n } ∞ n =1 , with each p n = 1 mod n .In order to control the growth of the quantities F n ( T ) we choose theprimes p n to lie in intervals whose position and length depend upon n (and upon our desired value for r ( n )). Ward was able to complete hisproof (which uses a slightly different setup then ours) by appealing toLinnik’s Theorem [12] on primes in arithmetic progressions. Linnik’s ALAN HAYNES AND CHRISTOPHER WHITE
Theorem implies that there is a constant κ > n , and for all a ∈ N with ( a, n ) = 1, the smallestprime which is a modulo n is less than n κ . After many subsequentimprovements, Heath-Brown showed in [7] that κ can be taken to be5 .
5, and this has recently been improved by Xylouris in [17] to 5 .
2. Forour results we will prove the following extensions of Linnik’s Theorem,which give information about small primes in arithmetic progressionswhich lie in intervals of prescribed lengths.
Theorem 1.5. If κ > . then for any ǫ > , for any a, n ∈ N with n sufficiently large (depending on ǫ ) and ( a, n ) = 1 , and for any x > n κ ,there is a prime in the interval [ x, (1 + ǫ ) x ) which is equal to a modulo n . Theorem 1.5 is a version of Bertrand’s Postulate for primes in arith-metic progressions. The next theorem is similar, but with smaller in-tervals.
Theorem 1.6. If κ > . then there is an ǫ ≥ such that, for any a, n ∈ N with n sufficiently large and ( a, n ) = 1 , and for any x > n κ ,there is a prime in the interval h x, x + xn ǫ (cid:17) which is equal to a modulo n . We remark that similar problems were considered in [2] and [15],however our results are stronger. For example, in [15, Theorem 6.3.1](see also [16, Theorem 4.1]) it is proven that for n sufficiently large,if ( a, n ) = 1 and if κ = 328 and θ = 655 / (cid:2) n κ , n κ + n κθ (cid:1) which is a mod n . It is not difficult to deduce from our Theorem 1.6that this can be improved to κ = 20 . θ = 24 /
25, and in fact, byour proof, better constants than this could also be obtained.If we assume the Generalized Riemann Hypothesis then it is rela-tively easy to show that the constant κ in Theorem 1.5 could be takento be any real number larger than 2, while the constant κ in Theorem1.6 could be taken to be any real number larger than 4. However wedo not make this assumption. ERIODIC POINTS AND PRIMES IN PROGRESSIONS 5
The layout for this paper is as follows. In Section 2 we define the F -method and prove Theorems 1.1 and 1.2, assuming the truth of The-orems 1.5 and 1.6. In Section 3 we prove the latter two theorems, andin Section 4 we prove Theorems 1.3 and 1.4.Our notation and conventions are: ϕ denotes the Euler phi function, µ the M¨obius function, Λ the von Mangoldt function. We use d ( n )to denote the number of positive divisors of an integer n , ω ( n ) todenote the number of distinct prime divisors of n , and ( m, n ) to denotegreatest common divisor of m, n ∈ Z . All summations are assumed tobe restricted to positive integers. If p is prime and a ∈ N then thenotation p a k n means that p a | n but p a +1 ∤ n . If f and g are complexvalued functions with domain X then the notation f ( x ) ≪ g ( x ), or f ( x ) = O ( g ( x )), means that there exists a constant c > | f ( x ) | ≤ c | g ( x ) | for all x ∈ X . If X = N , Z , or R then the notation f ( x ) = o ( g ( x )) means that f ( x ) /g ( x ) → x → ∞ . We say thata function f : N → C is multiplicative if f ( mn ) = f ( m ) f ( n ), for all m, n ∈ N with ( m, n ) = 1.2. The F -method and proofs of Theorems 1.1 and 1.2 Following Ward’s construction, we consider sequences of pairs of in-tegers { ( p n , g n ) } ∞ n =1 satisfying the following properties:(i) Each p n is either prime or equal to 1, and p = 1,(ii) Each p n satisfies p n = 1 mod n ,(iii) If p n = 1 then g n = 0, otherwise g n is a primitive root mod p n .Formally, we define the F -method to be a function from the collection ofall such sequences to the collection of pairs ( G, T ) where G is a compactgroup and T is an automorphism of G . The group G associated to thesequence { ( p n , g n ) } is G = ∞ Y n =1 F p n , the direct product (with the product topology) of the additive groupsof the finite fields F p n , each taken with the discrete topology. When p n = 1 we adopt the convention that F p n is the group with one element.The automorphism T is defined by T ( x , x , . . . , x n , . . . ) = ( T ( x ) , T ( x ) , . . . , T n ( x n ) , . . . ) , where T n : F p n → F p n is the trivial automorphism when p n = 1 , and isotherwise defined by T n ( x n ) = g ( p n − /nn x n . ALAN HAYNES AND CHRISTOPHER WHITE
If a group-automorphism pair (
G, T ) lies in the image of the F -methodthen we say that it can be constructed (or simply that T can be con-structed) using the F -method.Now assume that ( G, T ) is the image under the F -method of thesequence { ( p n , g n ) } . For any m ∈ N , if x m ∈ F p m is not the additiveidentity then the least period of x m under the map T m is equal to m .It follows that, for each n ∈ N , F n ( T ) = (cid:8) x ∈ G : lcm { m : p m = 1 , x m ∈ F ∗ p m }| n (cid:9) = Y d | n p d . (2.1)Here we remark that the reason for allowing some of the integers p n to be 1 is to give us the flexibility to slow the growth of F n ( T ) alongcertain subsequences of n . We will return to this thought in the finalsection. Now we use equation (2.1) to prove the following slightly moregeneral version of Theorems 1.1 and 1.2, assuming for the moment thevalidity of Theorems 1.5 and 1.6. Theorem 2.1.
Given r : N → R , let s : N → R be defined by s ( n ) := X d | n µ ( d ) r ( n/d ) . (2.2) Then: (i)
If for all sufficiently large n with s ( n ) = 0 we have that s ( n ) > . · log n, then there exists a compact group automorphism T with F n ( T )exp( r ( n )) ≍ . (ii) If for all sufficiently large n we have that s ( n ) > . · log n, then there exists a compact group automorphism T with lim n →∞ log F n ( T ) r ( n ) = 1 . Proof.
To prove (i), choose ǫ >
0. By Theorem 1.6, for every sufficientlylarge n for which s ( n ) = 0, we can find a prime number p n = 1 mod n which lies in the interval (cid:20) exp( s ( n )) , exp( s ( n )) (cid:18) n ǫ (cid:19)(cid:19) . ERIODIC POINTS AND PRIMES IN PROGRESSIONS 7
Then using (2.1) we have thatlog F n ( T ) = X d | n s ( d ) + X d | n log (cid:18) O (cid:18) d ǫ (cid:19)(cid:19) = r ( n ) + X d | n log (cid:18) O (cid:18) d ǫ (cid:19)(cid:19) , (2.3)where the last equality follows by M¨obius inversion in (2.2). For theerror term we have that X d | n log (cid:18) O (cid:18) d ǫ (cid:19)(cid:19) ≪ X d | n d ǫ ≪ , so exponentiating both sides of (2.3) finishes the proof.The proof of (ii) is almost the same, except that we appeal to Theo-rem 1.5 to choose, for all sufficiently large n , a prime p n = 1 mod n inthe interval [exp( s ( n )) , s ( n ))) . Then for each n we have thatlog F n ( T ) = r ( n ) + X d | n log(1 + ξ d ) , where each ξ d is a real number in the interval [0 , d ( n ), and since r ( n ) = X d | n s ( d ) ≫ X d | n log d = d ( n ) log n , we have that log F n ( T ) r ( n ) = 1 + O (cid:18) n (cid:19) . (cid:3) Finally, to derive Theorems 1.1 and 1.2 from Theorem 2.1, supposethat κ > r ( n ) is multiplicative function which satisfies r ( p a ) − r ( p a − ) > κa log p, (2.4)for all primes p and for all a ∈ N . Then we have that s ( n ) = Y p a k n ( r ( p a ) − r ( p a − )) > Y p a k n κa log p. ALAN HAYNES AND CHRISTOPHER WHITE
Since κa log p > p and a , the product on the right hand sideis greater than X p a k n κa log p = κ log n. Substituting κ = 20 . κ = 13 . Proofs of Theorems 1.5 and 1.6
In this section we assume familiarity with the basic tools of analyticnumber theory, as presented for example in [1]. Our proofs of Theorems1.5 and 1.6 are based on deep but well known results which describethe distribution of zeros of Dirichlet L-functions in the critical strip.Our approach is modeled after the proofs of [3, Theorem 7] and [4,Theorem 2].To begin, let n, a ∈ N satisfy ( a, n ) = 1 and for x > ψ ( x, a, n ) = X m ≤ xm = a mod n Λ( m ) . If χ is a Dirichlet character to modulus n then define ψ ( x, χ ) = X m ≤ x χ ( m )Λ( m ) , and note by the orthogonality relations that ψ ( x, a, n ) = 1 ϕ ( n ) X χ mod n χ ( a ) ψ ( x, χ ) . (3.1)The approximate explicit formula for ψ ( x, χ ) (see [1, Section 19]) statesthat, for 2 ≤ T ≤ x , ψ ( x, χ ) = δ χ x − X | γ |≤ T x ρ ρ + O (cid:18) x (log nx ) T + x / log x (cid:19) , where δ χ is 1 if χ is the principal character and 0 otherwise. The sumon the right hand side here is a sum over all non-trivial zeros ρ = β + iγ of the Dirichlet L-function L ( s, χ ), which have | γ | ≤ T . Substitutingthis in (3.1) we have for 0 < h ≤ x that ψ ( x + h, a, n ) − ψ ( x, a, n )= hϕ ( n ) − ϕ ( n ) X χ mod n χ ( a ) X | γ |≤ T ( x + h ) ρ − x ρ ρ ERIODIC POINTS AND PRIMES IN PROGRESSIONS 9 + O (cid:18) x (log nx ) T + x / log x (cid:19) . (3.2)For the double sum here we have the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X χ mod n χ ( a ) X | γ |≤ T ( x + h ) ρ − x ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X χ mod n X | γ |≤ T (cid:12)(cid:12)(cid:12)(cid:12)Z x + hx y ρ − dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ h X χ mod n X | γ |≤ T x β − . The main goal of the rest of the proof will be to show that for ourchoices of x, h, we can choose T , depending on n , so that X χ mod n X | γ |≤ T x β − < . The other error terms involved will all be asymptotically negligible as n → ∞ , which will imply that ψ ( x + h, a, n ) − ψ ( x, a, n ) > , for all large enough n , and the conclusions of the theorems will easilyfollow. The proof is split into two cases, depending on whether or notthere is a Siegel zero corresponding to a Dirichlet character of modulus n . Case 1:
Suppose that n is sufficiently large and that 1 − β / log n is aSiegel zero to modulus n . We know from Siegel’s theorem that, for any ǫ > β ≫ ǫ n − ǫ . (3.3)All other zeros of Dirichlet L-functions to modulus n must satisfy β < − c (log β − )log( n (2 + | γ | )) , and the constant c may be taken to be any real number less than 2 / η = c (log β − )log( n (2 + T )) , (3.4)and denoting the sum over non-Siegel zeros with a prime, we have that X χ mod n X ′| γ |≤ T x β − = − Z − η x α − d α N n ( α, T )= N n (0 , T ) x + Z − η x α − N n ( α, T ) log x dα, (3.5) where N n ( α, T ) denotes the number of zeros of all L-functions of mod-ulus n with α < β < | γ | ≤ T . A zero-density estimate of Huxley[8] implies that, for any ǫ >
0, there exists an integer n = n ( ǫ ) suchthat, for all T ≥ N n ( α, T ) ≤ ( nT ) (12 / ǫ )(1 − α ) for all n ≥ n . For α ≤ /
12 it is better to use the standard estimate that, assuming n has been chosen large enough, N n ( α, T ) ≤ ( nT ) ǫ for all n ≥ n . For the rest of the proof we will assume that ǫ is a fixed small number,to be specified later, and that n ≥ n . Suppose that κ and δ are realnumbers satisfying δ > κ > (1 + δ )(12 / ǫ ) , (3.6)and set x = n κ and T = n δ −
2. Then we have that Z − η x α − N n ( α, T ) log x dα ≤ κ Z − η n (1 − α )((1+ δ )(12 / ǫ ) − κ ) log n dα ≤ κκ − (1 + δ )(12 / ǫ ) · n η ((1+ δ )(12 / ǫ ) − κ ) For the other term appearing in (3.5) we have that N n (0 , T ) x ≤ n (1+ δ )(1+ ǫ ) − κ , and this tends to 0 as n → ∞ . For the secondary error term appearingin (3.2) we have that x (log nx ) T + x / log x ≪ n κ − δ (log n ) + n κ/ , and this will be o ( hβ /ϕ ( n )) provided that n κ − δ (log n ) + n κ/ = o ( hβ ) as n → ∞ . (3.7)Assuming that h is chosen so that (3.7) is satisfied, we have that ψ ( x + h, a, n ) − ψ ( x, a, n ) ≥ hϕ ( n ) (cid:18) − κκ − (1 + δ )(12 / ǫ ) · n η ((1+ δ )(12 / ǫ ) − κ ) − n − κβ / log n + o ( β ) (cid:19) . ERIODIC POINTS AND PRIMES IN PROGRESSIONS 11
Now what we are going to show is that, for choices of x and h corre-sponding to each of our theorems, we can choose δ so that (3.6) and(3.7) are satisfied and such that1 − κκ − (1 + δ )(12 / ǫ ) · n η ((1+ δ )(12 / ǫ ) − κ ) − n − κβ / log n ∼ κβ , (3.8)as n → ∞ .To prove Theorem 1.5 (in the case where there is a Siegel zero) take h = ǫn κ . In this case we suppose that κ > /
5, let δ be any numbersatisfying κ > (1 + δ ) · / , (3.9)and then choose ǫ in (3.3) so that (3.7) holds. We will show that δ can be chosen so that (3.8) also holds. By taking n large enough, β can be assumed to be as close to 0 as necessary, and we therefore havethat 1 − n − κβ / log n ∼ κβ as n → ∞ . Now we have that n η ((1+ δ )(12 / ǫ ) − κ ) = β c ( κ/ (1+ δ ) − (12 / ǫ ))0 , and this will be o ( β ) as n → ∞ , provided that κ > (1 + δ )(12 / ǫ + c − ) . (3.10)Since δ can be taken as close to 1 as necessary and ǫ > n , provided that κ > /
5. Inthis case what we have shown is that ψ ( x + h, a, n ) − ψ ( x, a, n ) ≫ hβ ϕ ( n ) ≫ n κ − − ǫ . The contribution from higher powers of primes is bounded by X p ≤ x + h X m ≥ p m ≤ x + h log p ≪ ( x + h ) / log x ≪ n k/ log n, and this verifies the first part of the statement of Theorem 1.5, for all κ > /
5, in the case when there is a Siegel zero (the constant 13 . h = n κ − − ǫ . In this case we assume that κ > (3 + ǫ ) · /
5, let δ be any number satisfying (3.9), and then choose ǫ in (3.3) so that (3.7) holds. The analysis is the same as before exceptthat in equation (3.10) we are allowed to take any value of δ > ǫ , and this verifies the first part of Theorem 1.6, for all κ > (3 + ǫ ) · / Case 2:
If there are no Siegel zeros to modulus n then much of theproof is the same as before, except that we must choose η = c log( n (2 + T )) , where the constant c may be taken to be 0 . ψ ( x + h, a, n ) − ψ ( x, a, n ) ≥ hϕ ( n ) (cid:18) − κκ − (1 + δ )(12 / ǫ ) · n η ((1+ δ )(12 / ǫ ) − κ ) + o (1) (cid:19) , whenever (3.6) is satisfied and n κ − δ (log n ) + n κ/ = o ( h ) as n → ∞ . Now we show that, for choices of x and h corresponding to each of ourtheorems, we can choose δ so that the above conditions are satisfiedand such that κκ − (1 + δ )(12 / ǫ ) · n η ((1+ δ )(12 / ǫ ) − κ ) < . As before, for the proof of Theorem 1.5 we have the freedom tochoose δ > ǫ > κκ − / · n η (24 / − κ ) = κκ − / · exp( c (12 / − κ/ < . The function on the left hand side decreases as κ increases, and theinequality is satisfied for all κ > . δ > ǫ and we willbe able to obtain the required result provided that κκ − (3 + ǫ ) · / · exp( c (12 / − κ/ (3 + ǫ ))) < . When ǫ = 0 this inequality will be satisfied for all κ > .
1. Thereforefor any κ > .
1, it is possible to choose ǫ >
ERIODIC POINTS AND PRIMES IN PROGRESSIONS 13 Proofs of Theorems 1.3 and 1.4
In this final section we will explore the limits of the F -method inconstructing automorphisms with various logarithmic growth rates ofperiodic points. We begin by demonstrating why certain growth ratesare not possible. Assume that we naively apply the F -method with allof the integers p n greater than 1. Then, since each p n = 1 mod n ,log F n ( T ) = X d | n log p d > X d | n log d = d ( n ) log n . It is well known (see [6, Theorem 317]) that d ( n ) > exp (cid:18) log n n (cid:19) for infinitely many n ∈ N , and this establishes a lower bound (albeit on a very thin subsequence)for the growth of F n ( T ). Note that as n → ∞ the functionexp (cid:18) log n n (cid:19) grows more quickly than any power of log n .This example is taking advantage of the fact that there is a thin setof integers which have an unusually large number of divisors. We mayhope to taper the growth of F n ( T ) by cleverly employing the F -methodwith some of the p n equal to 1. However it is not difficult to prove thateven in this case there are arbitrarily large growth rates which are notattainable. Proof of Theorem 1.3.
Using notation as in the statement of the Theo-rem, and assuming without loss of generality that t ( n ) → ∞ as n → ∞ ,define r : N → R by r ( n ) = ( t ( n ) if n is prime ,t ( n ) else.If, for some T constructed using the F -method, log F n ( T ) were asymp-totic to r ( n ) then we would have, for all large enough integers n , that r ( n ) / < log F n ( T ) < r ( n ) . Suppose this is the case, let q and q be two large enough primes, andlet n = q q . Then, using formula (2.1),2 t ( n ) > log F n ( T ) = log F q ( T ) + log F q ( T ) + log p n > (1 / (cid:0) t ( q ) + t ( q ) (cid:1) > t ( n ) , which is impossible as n → ∞ . (cid:3) On the extreme of slowly growing numbers of periodic points, it iscertainly not possible using the F -method to obtain polynomial (orsub-polynomial) growth. To see why, suppose thatlim n →∞ log F n ( T ) k log n = 1for some k ∈ N . Then for all large enough n there is a number ξ n ∈ ( − ,
1) such that log F n ( T ) = (1 + ξ n (8 k ) − ) k log n. If q is a prime which is large enough and if a ∈ N then we havelog p q a +1 = log F q a +1 ( T ) − log F q a ( T )= k log q + (( a + 1) ξ q a +1 + aξ q a )8 log q. Setting a = 2 k −
1, we have the bound (cid:12)(cid:12)(cid:12)(cid:12) (( a + 1) ξ q a +1 + aξ q a )8 log q (cid:12)(cid:12)(cid:12)(cid:12) < ( k/
2) log q, which implies that q k/ < p q k < q k/ . However this is impossible, since we require that p q k = 1 mod q k .In contrast to the above argument, we conclude with our proof ofTheorem 1.4, in which we explain how to construct automorphismswith growth rates of periodic points only marginally asymptoticallyfaster than polynomial. Proof of Theorem 1.4.
With notation as in the statement of the Theo-rem, let r ( n ) = ι ( n ) log n. Taking κ = 13 . c > n ∈ N and for any x > cn κ ,there is a prime number in the interval [ x, x ) which is 1 modulo n .We apply the F -method to choose the integers p n as follows.If n is not a prime power then we set p n = 1. For each prime q andfor each a ∈ N we choose p q a so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a X i =1 log p q i − r ( q a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log( cq aκ ) . (4.1) ERIODIC POINTS AND PRIMES IN PROGRESSIONS 15
To verify that this is possible we argue for each prime q by inductionon a . First of all if r ( q ) ≤ log( cq κ ) , then we take p q = 1, otherwise we choose p q to be a prime equal to1 mod q which lies in the interval[exp( r ( q )) , r ( q ))) . In the second case we have that | log p q − r ( q ) | ≤ log 2 , so (4.1) is satisfied with a = 1. Now suppose the inequality holds forsome a ∈ N . Then if (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a X i =1 log p q i − r ( q a +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ log( cq ( a +1) κ ) , we take p q a +1 = 1. Otherwise (since r is non-decreasing and the in-equality was assumed to be true for a ) it must the case that δ a +1 := r ( q a +1 ) − a X i =1 log p q i > log( cq ( a +1) κ ) , and we may choose p q a +1 to be a prime equal to 1 mod q in the interval[ δ a +1 , δ a +1 ). Then as before we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a +1 X i =1 log p q i − r ( q a +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | δ a +1 − log p q a +1 | < log 2 , which verifies (4.1).Finally, suppose that n ∈ N and write its prime factorization as n = q a · · · q a k k . Then we have thatlog F n = X d | n log p d = k X i =1 a i X a =1 log p q ai = k X i =1 ( r ( q a i i ) + O (log( q a i i )))= k X i =1 r ( q a i i ) + O (log n ) . Letting L ( n ) := log log n , we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ι ( n ) log n − k X i =1 r ( q a i i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k X i =1 ( ι ( n ) − ι ( q a i i )) log( q a i i ) ≤ k X i =1 q aii > L ( n ) q a i i ) + k X i =1 q aii ≤L ( n ) ι ( n ) log( q a i i ) , (4.2)where we have used the inequalities | ι ( n ) − ι ( q a i i ) | < q a i i > L ( n ),and | ι ( n ) − ι ( q a i i ) | < ι ( n ) otherwise. The first sum on the right handside is bounded above by 2 log n . For the second we use the PrimeNumber Theorem (or even Chebyshev’s Theorem) to obtain the bound k X i =1 q aii ≤L ( n ) ι ( n ) log( q a i i ) ≪ ι ( n ) log log n. Therefore the right hand side of (4.2) is o ( ι ( n ) log n ), and the proof iscompleted. (cid:3) References [1] H. Davenport:
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