Primes in arithmetic progressions to large moduli

Transcription

1 CHAPTER 3 Primes in arithmetic progressions to large moduli In this section we prove the celebrated theorem of Bombieri and Vinogradov Theorem 3. (Bombieri-Vinogradov). For any A, thereeistsb = B(A) such that for any ma (; q, a) (q) () (a,q)= log A for Q = /2 / log B. Here (q) () = (n,q)= (n). Remark 3.. The term (q) () is the epected main term for the distribution of in arithmetic progressions of modulus q and coprime to q; we can also replace this term by the seemingly more natural term at the cost of an error of size O(log q/). positive power of () ' () Observe that for Q afied log Q log. Therefore the Bombieri-Vinogradov theorem states that the maimal error term on the distribution of primes in arithmetic progressions of modulus q E(,; q) = ma E(,; q, a) = ma (; q, a) (a,q)= (a,q)= (q) () is on average over q apple Q is O(/ log A ) and is therefore negligible compared to the average of the main term; put in another way for any A E(,; q) () log A for almost all q apple Q = /2 log B for some B = B(A). Observe that the GRH would give that for for any q apple E(,; q) Q /2 log 2 = / log B 2. 45

2 46 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI therefore ecepted for the dependency of B wrt to A the Bombieri-Vinogradov theorem does as good as the GRH for the distribution of primes in arithmetic progressions on average over the modulus.. Reduction to the large sieve inequality We return to the special case of the von Mangolt function: (; q, a) () = (a) (n) (n) + O( log q ) 6= apple (n) (n) + O( log q ). 6= the last term accounting for the contribution in the second term of the n not coprime with q. The total contribution of these lasts terms is bounded by log q apple log Q q q log2 Q. here we have used the following Lemma 3.. For Q, onehas log Q. Proof. This follows from the analytic properties of the L-function associated to the multiplicative function q 7! q/: indeed for <s > one has q q s = Y ( + ( p ) p s ( p s ) )= (s)h(s) p with q H(s) = Y p ( + O(p (s+) + p 2s )) is holomorphic for <s >/2. Therefore (s)h(s) is meromorphic in <s > /2 with a most a simple pole at s = (in fact this is a pole as more computation show that H() 6= 0). We need therefore to evaluate (n) (n). 6= We will also reduce this summation over primitive characters: given (modq) let (mod q )betheprimitiveinducing,wehave (n) (n) = (n) (n) = (n) (n) + O(log q) (n,q)=

3 . REDUCTION TO THE LARGE SIEVE INEQUALITY 47 by bounding trivially the contribution of the n which are coprime to q but not coprime to q (and are therefore powers of primes dividing q). Writing q = q q 0 we have 6= (n) (n) = '(q q 0 ) q q 0 appleq q >? (mod q ) (n) (n) + O(log q) Here P? mean that we average over primitive characters of modulus q. The second term is bounded by O(Q log Q) while for the first, we bound it using that '(q q 0 ) '(q )'(q 0 ) so that this term is bounded by <q appleq '(q ) by Lemma 3..? (mod q ) (n) (n) ( '(q 0 ) ) q 0 appleq/q log Q? '(q ) <q appleq (mod q ).. Applying Siegel s Theorem. We need to evaluate? (n) (n) < (n) (n). and for this we split the q-summation into two ranges: the small and the large moduli, = < < + Q < where Q = log C for some fied C to be choosen later. For the small range we use the Siegel-Walfisz theorem: since q>and each primitive being non-trivial, one has for any A and therefore (3.) <?? (n) (n) A (n) (n) A log A log A which will be admissible as long as we take A su ciently large compared to C. It is to bound the large moduli range? (3.2) (n) (n) Q < C

4 48 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI that we need the so called multiplicative large sieve inequality. 2. Large Sieve inequalities The above computations have reduce the pro of the Bombieri-Vinogradov theorem to the problem of evaluating on average of q apple Q and (primitive) the absolute values of linear forms `(, ; ) = (n) (n). The multiplicative large sieve inequality provide similar bounds for the average square of these linear forms for general arithmetic function (in place of just the van Mangolt function ): 2.. The multiplicative large sieve inequality. For this additive version of the large sieve we deduce a multiplicative version Theorem 3.2. For any M and ( m ) mapplem and any Q we have q? m (m) 2 (Q 2 + M) m 2. mapplem mapplem Here P? mean that we average over primitive characters of modulus q. Before embarking for the proof we deduce some corollaries Corollary 3.. For any ( n ) n and any Q,Q,N, wehave Q apple? napplen n (n) 2 log Q (Q 2 + N) n 2. Q napplen Here P? mean that we average over primitive characters of modulus q. Proof. We decompose the sum into a sum of O(log Q) dyadic intervals Q apple = Q 0 for each such sum we have Q 0 <qapple2q 0 apple 2Q 0 Q 0 <qapple2q 0 ; qapple2q 0 q and we apply the multiplicative large sieve inequality Multiplicative large sieve inequalities for convolutions. We deduce from this result a bound for the average value of linear forms of nontrivial Dirichlet convolution:

5 2. LARGE SIEVE INEQUALITIES 49 Corollary 3.2. For any sequences of comple numbers ( m ) mapplem, ( n ) napplen and any Q,Q one has Q apple? m n (mn) mapplem napplen log Q(Q + M /2 + N /2 + (MN)/2 Q )k k 2 k k 2. Remark 3.2. Observe that for Q > this bound is useless (with respect to the additional summation condition q Q )ifn = because then M /2 (MN) /2 /Q. What we will show is that the von Mangolt function ( (n)) can be decomposed, up to admissible terms into a sum of functions of non-trivial convolution ( m ) mapplem? ( n ) napplen for MN and M,N > so that one can apply Corollary 3.2. Proof. We decompose the q-sum into O(log Q) terms over dyadic intervals as above apple q Q 0 Q < Q appleq 0 appleq q Q 0 and use the factorization m n (mn) = mapplem q Q 0 mapplem napplen by Cauchy-Schwarz q m n m (m) napplen n (n) ; (M /2 + Q 0 )(N /2 + Q 0 )k k 2 k k 2 and we conclude with the bound Q 0 (M /2 + Q 0 )(N /2 + Q 0 ) log Q(Q + M /2 + N /2 + (MN)/2 ). Q Q appleq 0 appleq 2.3. Proof of theorem 3.2. We will reduce the proof of this inequality involving multiplicative characters modulo q to an analoguous one involving additive character modulo q: for mod q is primitive we have (n) = (a)e q (na) and therefore q a primitive n (n) 2 napplen

6 50 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI We have = apple = primitive a,a 0 a,a 0 a napplen a napplen n (a)e( an q ) 2 n (a)e( an q ) 2 (a) (a 0 ) n,n 0 n n 0e q (an a 0 n 0 ) (a) (a 0 )= (aa 0,q)= a=a 0 and therefore the above sum equals = n n 0e q (a(n n 0 )) n,n 0 To conclude it will su a (a,q)= = a (a,q)= n ce to prove that n e( an q ) 2 Theorem 3.3. We have for any ( n ) napplen 2 C N? (3.3) n e( an q ) 2 (Q 2 + N) n 2 napplen napplen a Here P? mean that we average over the congruence classes a which are coprime to q The duality principle. Let M, N be two finite sets and consider a matri :=( (m, n)) (m,n)2m N 2 C M N. this matri defines a linear map where : =( m ) m2m 2 C M 7! =( n ) n2n = ( ) 2 C N, n = m2m m (m, n). Equipping C M and C N with their usual structure of Hilbert spaces k k 2 =( m2m we have for any vector ( m ) 2 C M m 2 ) /2, k k 2 =( n2n k ( )k 2 2 applek k 2 2k k 2 2 n 2 ) /2

7 2. LARGE SIEVE INEQUALITIES 5 where k k 2 denote the operator norm of : ie. n2n k ( )k 2 k k 2 =sup <. 6=0 k k 2 In other terms for any 2 C M we have m (m, n) 2 applek k 2 2 m 2. m Let be the transpose matri m2m := ( (m, n)) (n,m)2n M 2 C N M, this matri defines the transpose linear map : =( n ) n2n 2 C N 7! =( m ) m2m = ( ) 2 C M, where m = n2n (m, n) n. The duality principle is the well known statement Theorem (Duality principle). One has k k 2 = k k 2. In other terms for any 2 C N,onehas n (m, n) 2 applek k 2 2 n 2 = k k 2 2 n 2. n m2m n2n n2n Proof. We have k ( )k 2 2 = n (m, n) 2 = n n 0 (m, n) (m, n0 ) m2m n m n,n 0 = n m (m, n), m = n 0 (m, n0 ). n m n 0 By Cauchy-Schwarz this is bounded by k k 2 ( n m m (m, n) 2 ) /2 = k k 2 k ( )k 2 applek k 2 k k 2 k k 2 but k k 2 2 = m n 0 n 0 (m, n0 ) 2 = m n n (m, n) 2 = k ( )k 2 2 and therefore k ( )k 2 2 applek k 2 k k 2 k ( )k 2 and hence for any, k ( )k 2 applek k 2 k k 2 or in other terms k k 2 applek k 2 ; the equality follows by symetry.

8 52 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI 2.5. The additive large sieve inequality. To prove theorem 3.3, we apply the duality principle to the following situation: and M = Q = {(a, q), q apple Q, (a, q) =}, N = {,,N} Theorem 3.3 states precisely that ((a, q),n)=e( an q ). k k 2 2 N + Q 2. By the duality principle this is equivalent to showing that k k 2 2 N + Q 2, or in other terms, that for any =( (a,q) ) (a,q)2q, one has? (a,q) e( an q ) 2 (N + Q 2 )k k 2 2. napplen a We will evaluate this last sum by computing the square and performing the n-summation; however before doing this we perform a smoothing trick: Let ' be a smooth, even, compactly supported function, and taking value on [, ]. We have? (a,q) e( an q ) 2 apple '( n N )? (a,q) e( an q ) 2 napplen a n2z a (3.4) =? q,q 0 appleq a a 0 0 By Poisson s formula the n-sum equals (a,q) (a 0,q 0 ) '( n N )e((a q N b'(n(n + a q n2z Observe that by construction the function n a 0 q 0 )) a 0 q 0 )n). 7! b' N,Z () := b'(n(n + )) n2z is periodic of period and therefor defines a smooth function on the additive group R/Z ' S. This implies that ' N,Z () =' N,Z (±kk) =' N,Z (kk) where kk =inf n2z n denote the distance between and the nearest integer: indeed either +kk or kk is a representative of the class (mod ) in R/Z and ' being even, b' is also even. Moreover since ' is compactly supported and smooth, its Fourier transform is rapidly decreasing and in particular b'() + 2.

9 From we we deduce that 3. HEATH-BROWN S IDENTITY 53 ' N,Z () Using this bound and the trivial bound +(Nkk) 2. (a,q) (a 0,q 0 ) apple (a,q) 2 + (a 0,q 0 ) 2 we obtain that (3.4) is bounded by (a,q) 2 (a,q) (a 0,q 0 ) N +(Nk a q a 0 q k). 2 0 Observe that when (a, q) 6= (a 0,q 0 ) the rational fractions a/q and a 0 /q 0 are distinct modulo and we have for any n 2 Z a 0 q 0 n = (a n)q0 a 0 q qq 0 qq 0 Q 2. a q Therefore k a a 0 q q 0 k Q 2 and for any other (a 00,q 00 ) 6= (a 0,q 0 ) one hasv (the triangle inequality for the distance function k.k on R/Z) (a 0,q 0 )6=(a,q) k a q a 0 q 0 k ka q a 00 q 00 k a0 q 0 a 00 q 00 k Q 2. Thereofore for any given (a, q) any interval in R of the shape [kq 2, (k + )Q 2 [, k 2 Z, contains at most one number of the shape k a a 0 q q k. It follows 0 that N +(Nk a a 0 q q k) apple N 2 +(knq 2 ) 2 N + Q2. 0 k 0 Therefore we have proved that? (a,q) e( an q ) 2 (N + Q 2 )k k 2 2. napplen a 3. Heath-Brown s identity In order to apply Corollary 3.2, we need to show that the vonmangolt function can be decomposed into a sum of airthmetic function which convolutions. We e ectuate this using an identity due to Heath-Brown but there are many other possibilities (for instance Vaughan s identify). Theorem 3.4 (Heath-Brown s identity). Let J an integer and >, onehasforanyn<2 J J (n) = ( ) j µ(m ) µ(m j ) log n, j j= m,,m j applez m m j n n j =n

10 54 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI where Z = /J. Proof. This identity is an immediate consequence of the following identity for Dirichlet series: let M Z (s) = napplez µ(n) n s be the truncation of the inverse of Riemann s zeta function M(s) = (s) = µ(n) n s. n In particular (since (s) (s) = or equivalently µ(d) = n= ) d n one has (s)m Z (s) =+ a Z (n) n s ; n>z in other terms the convolution of with the function µ. napplez takes value 0 between 2 and Z. It follows that for J thecoe cientsb Z,J (n) of the Dirichlet series ( (s)m Z (s)) J are zero for n apple Z J = and therefore given any Dirichlet series L(s) = n a(n) n s associated to some arithmetic function (a(n)) n one has L(s)( (s)m Z (s)) J = a b Z,J(n) n s. n>z J We apply this observation to L(s) = 0 (s) (s). By the binomial law, we have 0 (s) (s) ( (s)m Z(s)) J = 0 (s) J J (s) + ( ) j 0 (s) k j j= (s)m k Z(s). this gives Heath-Brown s identity for n<z J but we observe that since () =, the coe cient of the Dirichlet series on the lefthand side are in fact zero for all n<2z J. 4. Proof of the Bombieri-Vinogradov theorem The proof we present here is a bit of an overkill; for instance one can find in Kowalski-Iwaniec a very sleek and quite a bit shorter proof of the Bombieri-Vinogradov theorem. The purpose of this eposition is to propose alternative presentations which maybe useful in other contets.

11 4. PROOF OF THE BOMBIERI-VINOGRADOV THEOREM Eponent of distribution of arithmetic functions. Heath- Brown s identity states that on the interval [, 2[ the von Mangolt function (n) can be decomposed in a linear combination of functions of the shape (3.5) ( applez µ) (?j)? log? (?j ), j =,,J,Z = /J. It is therefore su cient to prove that any of the functions above one has ma E(,; q) (a,q)= log A where and E(,; q, a) = E(,; q) = ma E(,; q, a) (a,q)= n a (n) (n,q)= (n). In is therefore worthwhile this problem (ie. estimating the quality of the distribution of in arithmetic progressions on average) for general arithmetic functions. The case of arithmetic functions which are essentially bounded: functions for which there eists K 0 such that for any n (3.6) (n) (( + log n)d(n)) K We have therefore the following trivial bounds: for q apple Q apple and E(,; q) (log )O() E(,; q) (log ) O(). Definition 3.. Given 2 [0, ], an arithmetic function satisfying (3.6) has level of distribution if, for any A 0, thereeistsb = B(A) such that for Q apple / log B,onehas E(,; q) K,A log A. With this terminology we have Theorem 3. (Bombieri-Vinogradov). The von Mangolt function has level of distribution /2. The following simple result will be useful in the proof of the Bombieri- Vinogradov theorem: Lemma 3.. Let P be a polynomial, the function n 7! P (log n) has level of distribution.

12 56 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI Proof. Il is su cient to prove this for n 7! log k n which is continuous monotone, therefore for q apple n a log k (n) = Z a q 0 log k (qt+a)dt+o(log k )= q Z and therefore for Q apple E(log k,; q) log k Q log k log A a log k (t)dt+o(log k ) as long as Q apple /2 / log B with B k + A A Bombieri-Vinogradov theorem for factorable arithmetic functions. We will discuss now the problem of evaluating the eponent of distribution of essentially bounded arithmetic functions which admit factorizations =? as a convolution of arithmetic functions (the idea then will be to use Corollary 3.2). By Heath-Brown identity this is essentially the case of the von Mangolt function which a linear combination of such functions. For k 2, let be an arithmetic function of the shape (n) =?? k (n) = (n ) k (n k ) n n k =n where i are arithmetic functions satisfying (3.6); therefore also satisfies (3.6). We will give general su cient conditions to insure that has level of distribution / From hyperboloids to paralleloids. For this we will need to make first a technical reduction: writing a a convolution, we need to evaluate sums of the shape n..n k apple that is sums over integral point lying under the hyperboloid given by the equation.. k =. In view of Corollary 3.2, we would rather evaluate sums of the shape, with N..N k apple n applen,,n k applen k (this is sums over integral points contained in a paralleloid). We do this by approimating the region located under the hyperboloid by a union of su ciently small paralleloids and evaluate the error made because of this approimation.

13 4. PROOF OF THE BOMBIERI-VINOGRADOV THEOREM 57 Given < such that = log C for some fied constant C (to be choosen depending on A), we cover the cube [,] k by O(( log ) k )= log O(C) paralleloids of the shape ky ]N i,n i ( + )] i= where the N i are powers of + and restricting to each paralleloid, we bound the sum E(,; q, a) = (n) (n) n a (n,q)= by a sum of O(( log ) k ) sums of the shape E( N,; q) = ma (a,q)= n,,n k n n k a (n ) k (n k ) n,,n k (n n k,q)= (n ) k (n k ) where N runs over O(( log ) k ) k-tuples of the shape N =(N,,N k ), apple N N k apple and the n i are subject to the constraints n i 2]N i,n i ( + )], i = k and the additional constraint (3.7) n n k apple. Observe that in the sum E( N,; q) (3.7) is unnecessary if (3.8) ( + ) k Y i N i apple, in such a case we will then write E( N ; q) instead of E( N,; q). For all the other terms, the corresponding n = n n k satisfy ( + ) k apple n apple and the contribution of these terms to E( N,; q)

14 58 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI is bounded by (log ) O() (+ ) k apple n a d(n) K + (+ ) k apple (n,q)= d(n) K (log )O() log O() = log O() C. Up to an error term of size log O() C (less that / log A for C large enough) we are reducted to evaluate E( N ; q) where N =. ]N,N (+ )]?? k. ]Nk,N k (+ )] and N =(N,,N k ) satisfying (3.8). We may in fact assume that (3.9) /00 apple Y i N i k ; indeed the total contribution of terms with Q i N i apple 99/00 is bounded trivially by (log ) O() 99/00. We proceed as in the beginning of this chapter epressing the congruence condition n a in terms of characters and then in terms of nontrivial primitive characters of moduli q apple Q: writing (mod q ) for the primitive character underlying wehave E( N ; q) apple? '(q 0 ) '(q N(n) (n) ) q 0 appleq <q appleq/q 0 (mod q ) (n,q 0 )= = q 0 appleq '(q 0 ) <? (n,q 0 )= N(n) (n), upon changing notations. We separate the case of small and large moduli and are reduced to bound (here Q = (log ) D with D to be choosen large enough):? (n) N (n) < and Q <? (n,q 0 )= (n,q 0 )= (n) N (n). To each partition of the set {,,k} = I t I 0 into two subsets we associate a factorisation N =? =? i2i i? (? i 0 2I 0 i 0)

15 were, with We have 4. PROOF OF THE BOMBIERI-VINOGRADOV THEOREM 59 are supported on the intervals (n,q 0 )= [N I, ( + O( ))N I ], [N I 0, ( + O( ))N I 0] N I = Y i2i (n) N (n) = N i, N I 0 = Y mapplen I (m,q 0 )= i 0 2I 0 N i 0 (m) (m) napplen I 0 (n,q 0 )= (n) (n). We may and will use di erent partitions depending on which method we use Small moduli and the Siegel-Walfisz hypothesis. To deal with small moduli, we make the following assumption on : Hypothesis 3. (Siegel-Walfisz type bound). There eist an absolute constant E such that given q, q 0 and non-trivial and primitive, one has for any F, and any y 2 (n) (n) A (d(q 0 )q) E y Since nappley (n,q 0 )= log F y Under this assumption we have for any F? (n) N (n) A (d(q 0 ) log ) O() log F. N < I 0 q 0 appleq d(q 0 ) O() '(q 0 ) = (log ) O(), his bound is satisfactory as long as N I 0 for some fied >0 and F is choosen so that F (A + O()). In particular, suppose that all the i, =,,k satisfy Hypothesis 3.; taking = i with N i is maimal we have therefore N i k so that the above reasonning is valid with = k. Since we need to bound at most O(log O() ) such sums, under this assumption, we obtain upon taking that the contribution of the moduli q apple Q is / log A Large moduli and the large sieve inequality. For large moduli we apply Corollary 3.2 getting log (/2 + N /2 I + N /2 Q I + Q)k k 0 2 k k 2 with Q < k k 2 =( (n) ) /2 N /2 I log k, n N I

16 60 3. PRIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI and therefore Q < k k 2 =( n N I 0 (n) ) /2 N /2 I log k 0 /2 (ma(n I,N I 0) /2 + /2 log D + /2 log B ) log+2k Let us recall that the number of N is bounded by O(log O() ) therefore given A we will choose B,D A + O() we declare a tuple N good if there is a factorisation N =? such that and we obtain that N good Q apple ma(n I,N I 0) apple? log 2(A+O()) (n) N (n) log A. Suppose that N is not good, and let i be such that N i is maimal in the set {N j, j =,,k} (to fi ideas, let us assume that i = ); since N is not good and N is maimal, one has necessarily and therefore Setting N log 2(A+O()) Y N i 0 log 2(A+O()). i 0 6= =? i 0 6= i 0 and returning to the initial problem we have to bound ma (n 0 ) E(, [N, ( + )N ]; q, an 0 ) E( N,; q) apple (a,q)= n 0 log O() (n 0,)= (log ) O() E(, [N, ( + )N ]; q) (log ) O() E(, ( + )N ; q) + (log ) O() E(,N ; q) The latter sum is admissible if we show that the arithmetic function has level of distribution /2.

17 5. APPLICATION TO THE -FUNCTION 6 5. Application to the -function We use Heath-Brown identity with J = 2 and are reduced to proving that the convolutions =( applez µ) (?j)? log? (?j ), j =, 2, k =2, 4, Z = /2 have level of distribution /2. The following Proposition is left to the reader as an eercise: Proposition 3.. The Moebius function satisfies a Siegel-Walfisz type bound: there eist E 0 such that for any y, F, anyq, q 0 and primitive non-trivial, one has (n) (n) A (d(q 0 )q) E y log F y. nappley It is also obvious that the constant function and log satisfy a Siegel- Walfisz type bound, therefore we may apply the previous argument and it remains to bound the sum E( N,; q). N not good With the notation of the previous section we have that is either = applez µ or = log or = but since Z = /2 and is supported around (log ) O() the first case is not possible and therefore = or log. Since these have level of distribution we are done. More precisely, we have seen in the proof of Lemma 3. that for Q apple log Q log. E(,; q) In particular for = (log ) O() and Q apple /2 log B the later term is of size /2 (log ) O() / log A for any A.