5. THE LARGE SIEVE. log log x

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1 5. THE LARGE SIEVE The key new ingredient which gave rise to the B-VMVT was the large sieve. This had been invented by Linnik [94,94] in work on the least quadratic non residue np modulo a prime p. He was able to show that for any fixed positive number δ there are at most log log x primes p x such that np > p δ. To give some idea of the background and explain what is otherwise a rather obscure terminology, consider a set {a n : n N} of complex numbers a n with the property that for each prime p the support of a n lies in gp residue classes modulo p, and we may certainly suppose that gp always. We can think of the a n as being the characteristic function of a set which has had p gp residue classes removed for each p. Let Zq, r = card{n A : n r mod q} and Z = Z, 0. We might hope that for each prime p the support of a n is fairly uniformly distributed into the gp residue classes. Let Rp be the set of gp residue classes modulo p which contain the support of the a n and consider the variance Let V p = r Rp Sα = Zp, r Z gp N a n enα. By the orthogonality of the additive characters modulo p, for any r modulo p Zp, r = p and Parseval s identity r Rp p e ra/p Zp, r = N a n ean/p = p p Zp, r = p. p e ra/psa/p p Sa/p.

2 5. THE LARGE SIEVE We also have V p = = p = p r Rp Zp, r R r Rp p Sa/p Z /gp p Sa/p Z p gp pgp Zp, rz/gp + Z /gp Thus pv p + Z p gp gp Summing over a suitable set P of primes p gives p = Sa/p. p P pv p + Z p P p gp gp = p P p Sa/p. Since the variance is non-negative, but might be small in the ideal situation of uniform distribution we conclude that Z p P p gp gp p P p Sa/p. Suppose we can obtain an non-trivial upper bound for the right hand side, such as p Sa/p Y p P N a n. By the way, we know from the Cauchy-Schwarz inequality that such bounds exist and indeed this sum could be written in terms of a Hermitian matrix, so Y could be taken to be its largest eigenvalue. Suppose further that a n is the characteristic function of an interesting set. Then Z = Z = a n so we have Z Y/ p P p gp. gp Suppose we also write ωp = p gp so that ωp is the number of residue classes removed. Then the sum here is ωp p ωp. p P

3 5. THE LARGE SIEVE 3 Doesn t this look a bit like the sum in the Selberg sieve? It certainly looks like an upper bound sieve estimate. For ω small, say or it only saves a log log so is not very good. But if ω = p /, say, then the sum will save a power of N maybe. In other words it looks interesting when we are removing a large number of residue classes. Thus any non-trivial value for Y N, Q for which q Q q a,q= a S Y N, Q q n=m+ a n holds for any complex numbers a n, has become called The Large Sieve. That such Y N, Q exist is clear via the Cauchy-Schwarz inequality applied to Sa/q. More generally one can ask for values of Y 0 N, δ such that whenever x,..., x R are R real numbers with x r x s δ whenever r s we have R Sx r Y 0 N, δ n=m+ a n for any complex numbers a n. Such inequalities are called The Large Sieve now also. By the way, α is the metric on T = R/Z, that is α = min α n. n Z It is useful to observe that if a, q = b, r =, q Q, r Q and a/q b/r, then Q /qr a/q b/r and so one can take Y N, Q = Y 0 N, Q. The first modern version of the large sieve is due to Roth [965], who obtained Bombieri [965] then obtained Y N, Q N + Q log Q. Y N, Q = N + CQ, Gallagher [967] gave a quite short proof that Y N, Q = πn + Q is permissible, and then there was a lot of work by a number of authors improving the constants. Finally Montgomery and Vaughan [973,974], with an added wrinkle by Paul Cohen [977], and Selberg [99, but known by him before 977] gave proofs that the bound holds with Y 0 N, δ = N + δ,

4 4 5. THE LARGE SIEVE and it had already been shown by Bombieri and Davenport [968] that this is best possible even when applied to Y N, Q. For an overall account of this work see the survey article by Montgomery [978]. For the most refined of the versions of the bounds of the kind see Montgomery [968] and Montgomery and Vaughan [973]. In some sense they are the duals of the Selberg sieve as applied to an interval. If we are not that concerned about the logarithmic power we need only the very simplest bound. To start with we state a lemma which, in fact is a statement from linear algebra. It says that if M is an N R matrix, then the two Hermitian matrices MM and M M, where here and only here the asterisk denotes the complex conjugate transpose, have the same largest eigenvalue. By the way, quite a number of the underlying ideas in this area are related to, or suggested by, ideas from linear algebra. Lemma, Duality Lemma. Suppose that c nr, n =,..., N,r =,..., R are complex numbers and λ is a real number such that for all complex numbers z r we have Then N R c nr z r λ holds for all complex numbers w n. Proof. We have Hence, by Cauchy s inequality, R N c nr w n λ LHS = N R z r. N w n R N w m c mr m= m= m= c nr w n. N N LHS w m R c mr z r where On hypothesis this does not exceed N z r = c nr w n. N R w m λ z r. m=

5 5. THE LARGE SIEVE 5 By definition of z r this is N LHSλ w m. m= By the way I. M. Vinogradov makes repeated use of the Duality Lemma in many special cases in his work on exponential sums, but always obtained directly via the Cauchy-Schwarz inequality and without, apparently, being aware that it was a special case of a general theorem! Below is a very simple proof of Roth s bound for the large sieve which would serve to establish Bombieri s theorem with a slightly inflated logarithmic power. This is certainly adequate for most applications of Bombieri;s theorem. Theorem, A Large Sieve Inequality 0. Suppose that 0 < δ and the x r, r =..., R satisfy x r x s δ whenever r s. Then holds with R Sx r Y 0 N, δ n=m+ Y 0 N, δ = N + δ log 3 δ a n and holds with q Q q a,q= a S Y N, Q q n=m+ Y N, Q = N + Q log 3Q. a n Proof. By the Duality Lemma it suffices to bound n=m+ R b r enx r The diagonal terms r = s contribute and when r s the sum over n satisfies e nx r x s n=m+ = R s= N R b r b s R b r n=m+ e nx r x s. sin πx r x s x r x s. 3

6 6 5. THE LARGE SIEVE Hence the non-diagonal terms contribute at most = R R s=,s r R R b r b r + b s x r x s s=,s r x r x s. Given an r we can add integers to the x s with s r so that the the resulting x s lie in [x r, x r + ]. For convenience write x r = x r. Now the numbers x s are all spaced δ apart. Moreover if x and x + are the smallest and largest values of the x s, then x + δ x + x + R δ. Thus Rδ and R s=,s r This establishes the theorem. x r x s k /δ kδ δ log 3 δ. In fact, in our proof of Bombieri s theorem we will assume the following slightly stronger statement. Theorem 3, A Large Sieve Inequality. The inequality holds with R Sx r Y N, δ Y N, δ N + δ n=m+ a n and holds with q Q q a,q= a S Y N, Q q Y N, Q N + Q. n=m+ a n Proof. The reason for the log in the previous result is because the characteristic function of [M +, M + N] has jump discontinuities. The solution is to majorise it by a smooth upper bound. One quite simple way to do this is to insert a factor fn in which majorises the characteristic function of [M +, M + N] is fx = max 0, n N 0 M, N

7 5. THE LARGE SIEVE 7 where N 0 = N/. When x [M +, M + N] we have N/ N 0 x N 0 M N N 0 N/. Thus on multiplying out and interchanging the order of summation the innermost sum over n becomes a Fejèr kernel fne nx r x s = N e N 0 Mx r x s n and satisfies in place of 3. Thus we find that n=m+ R b r enx r N h= N N h e hx r x s = N e N 0 Mx r x s N e jx r x s j=0 = e N 0 Mx r x s sin πnx r x s N sin πx r x s min N, N x r x s R R b r s= min N, By the spacing hypothesis for the x r it follows that this is If Nδ > then this is R b r N + and if Nδ, then it is R b r N + Thus in every case we may take k= min N, Nkδ. R b r N + N δ N k N δ + k>n δ Y N, δ N + δ. N x r x s. Nkδ N + δ. Selberg s argument which leads to an optimal Y N, δ is a more sophisticated variant of this idea. For more details see Montgomery s expository article Montgomery [978]. A this stage anyone who is not familiar with Dirichlet characters, and especially primitive characters should read the appendix.

8 8 5. THE LARGE SIEVE Theorem 4, A Large Sieve for Characters. Suppose that Sχ = n=m+ a n χn. Then holds with q Q q ϕq χ mod q Sχ Y N, Q Y N, Q N + Q. n=m+ a n Proof. By Appendix Lemma?, with χ replaced by χ, Sχ = n=m+ a n τχ Hence, by the last part of that lemma, χ mod q q Sχ q χaena/q = τχ χ mod q q χasa/q q χasa/q and by Parseval s identity this is ϕq q q a,q= a S. q We now proceed to convert this into a form more suitable for our application. The first step is a simple application of the Cauchy-Schwarz inequality. Lemma 5. Suppose that a,..., a M, b,..., b N are complex numbers. Then q Q q ϕq χ M N a m b n χmn m= M + Q N + Q M N a m b n. m= Proof. At once from our version of the large sieve for characters and the Cauchy-Schwarz inequality.

9 5. THE LARGE SIEVE 9 The next step is to insert a maximal condition into this. There are various ways of doing this. The one chosen here is motivated by something anyone who has seen a proof of the prime number theorem will be familiar with. This is the use of a formula of the kind n x c n = c+ πi c i c n x s n s x ds which is valid when c > 0, x N and, for example, the series n c n converges. The effect of this formula is to replace the condition n x by a twisting factor n s. To simplify matters we use a real version of this, i.e. a version on the line c = 0. Lemma 6. Suppose that x, Then on the premises of Lemma 5, q M N sup a m b n χmn ϕq q Q χ y x m= mn y M log xmn N M + Q N + Q a m b n. Proof. Let C = sin α α dα. We only need to know that C exists and C > 0 which is trivial from the observation that the integral can be written as π n 0 sin α πn + α dα and the terms in the series oscillate in sign and their absolute values form a decreasing sequence tending to 0, so Leibnitz test may be applied. Let γ > 0 and define { 0 β < γ, δβ = 0 β > γ. Then iβα sin γα e dα = δβ Cα since pairing α and α shows that the integral is real, and m= cos βα sin γα = sinγ + βα + sinγ βα. 4 Thus changing variables gives the value when 0 β < γ and 0 when β > γ.

10 0 5. THE LARGE SIEVE By integration by parts, provided that Y > 0 and A > 0, one has A sin Y α α dα Y A and so through the relationship 4 again one has δβ = A A iβα sin γα e Cα dα + O Now we specialise γ = log y +, β = log mn so { mn y, δlog mn = 0 mn > y and We have Thus Hence M m= δlog mn = min m,n N mn y A A. A γ β iα sin γα mn Cα dα + O A log y + log mn. y log + log mn = min log y + y a m b n χmn = = A δlog mn = M N A m= M m= A A iα sin γα y mn Cα dα + O. A N a m b n χmnδlog mn a m m iα b n n iα sin γα χmn Cα dα + O, log y + y + y. y A M m= N a m b n. The error term here is more than acceptable if we take A = xmn, and when y x the integral is A M N a m m iα b n n iα χmn min log x, dα. α By Lemma 5, q Q A m= q ϕq χ M N a m m iα b n n iα χmn m= M N M + Q N + Q a m b n m=

11 5. THE LARGE SIEVE and A A min log x, dα log xmn. α Appendix on properties of Dirichlet characters This is an adumbration of parts of Chapters 4 and 9 of MNT. It is often useful to represent the characteristic function of a reduced residue class mod q as a linear combination of totally multiplicative functions χn each one supported on the reduced residue classes and having period q. These are the Dirichlet characters. In the fancy language of abstract algebra we are examining the structure of homomorphisms from the units modulo q to an isomorphism of this group on the unit circle in the complex plane. Fundamental is that the homomorphisms themselves form a group which is also isomorphic to the original group. Since χn has period q we may think of it as mapping from residue classes, and since χn 0 if and only if n, q =, we may think of χ as mapping from the multiplicative group of reduced residue classes to the multiplicative group C of non-zero complex numbers. As χ is totally multiplicative, χmn = χmχn for all m, n, we see that the map χ : Z/qZ C is a homomorphism. The method we use to describe these characters applies when Z/qZ is replaced by an arbitrary finite abelian group G, so we consider the slightly more general problem of finding all homomorphisms χ : G C from such a group G to C. We call these homomorphisms the characters of G, and let Ĝ denote the set of all characters of G. We let χ 0 denote the principal character, whose value is identically. We note that if χ Ĝ then χe = where e denotes the identity in G. Let n denote the order of G. If g G and χ Ĝ, then gn = e, and hence χg n =. Consequently χg n =, and so we see that all values taken by characters are n th roots of unity. In particular, this implies that Ĝ is finite, since there can be at most nn such maps. If χ and χ are two characters of G, then we can define a product character χ χ by χ χ g = χ gχ g. For χ Ĝ, let χ be the character χg. Then χ χ = χ 0, and we see that Ĝ is a finite abelian group with identity χ 0. The following lemmas prepare for a full description of Ĝ in Theorem 9 below. Lemma 7. Suppose that G is cyclic of order n, say G = a. Then there are exactly n characters of G, namely χka m = ekm/n for k n. Moreover, and { n if χ = χ0, χg = 0 otherwise, g G { n if g = e, χg = 0 otherwise. χ Ĝ 3 In this situation, Ĝ is cyclic, Ĝ = χ.

12 5. THE LARGE SIEVE Proof. Suppose that χ Ĝ. As we have observed, χa is a nth root of unity, say χa = ek/n for some k, k n. Hence χa m = χa m = ekm/n. Since the characters are now known explicitly, the remaining assertions are easily verified. Next we describe the characters of the direct product of two groups in terms of the characters of the factors. Lemma 8. Suppose that G and G are finite abelian groups, and that G = G G. If χi is a character of G i, i =,, and g G is written g = g, g, g i G i, then χg = χ g χ g is a character of G. Conversely, if χ Ĝ, then there exist unique χi G i such that χg = χ g χ g. The identities and 3 hold for G if they hold for both G and G. We see here that each χ Ĝ corresponds to a pair χ, χ Ĝ Ĝ. Thus G = Ĝ Ĝ. Proof. The first assertion is clear. As for the second, put χ g = χg, e, χ g = χe, g. Then χi Ĝi for i =,, and χ g χ g = χg. The χi are unique, for if g = g, e then χg = χg, e = χ g χ e = χ g, and similarly for χ. If χg = χ g χ g, then χg = χ g χ g, g G g G g G so that holds for G if it holds for G and for G. Similarly, if g = g, g, then χg = χ Ĝ χ Ĝ so that 3 holds for G if it holds for G and G. χ g χ Ĝ χ g, Theorem 9. Let G be a finite abelian group. Then Ĝ is isomorphic to G, and and 3 both hold. Proof. Any finite abelian group is isomorphic to a direct product of cyclic groups, say G = C n C n C nr. The result then follows immediately from the lemmas. Though G and Ĝ are isomorphic, the isomorphism is not canonical. That is, no particular one-to-one correspondence between the elements of G and those of Ĝ is naturally distinguished.

13 5. THE LARGE SIEVE 3 Corollary 0. The multiplicative group Z/qZ of reduced residue classes mod q has φq Dirichlet characters. If χ is such a character, then If n, q = then q n,q= { φq if χ = χ0, χn = 0 otherwise. { φq if n mod q, χn = 0 otherwise χ 4 5 where the sum is extended over the φq Dirichlet characters χ mod q. As we remarked at the outset, for our purposes it is convenient to define the Dirichlet characters mod q on all integers; we do this by setting χn = 0 when n, q >. Thus χ is a totally multiplicative function with period q that vanishes whenever n, q >, and any such function is a Dirichlet character mod q. In this book a character is understood to be a Dirichlet character unless the contrary is indicated. Corollary. If χi is a character mod q i for i =,, then χ nχ n is a character mod [q, q ]. If q = q q, q, q =, and χ is a character mod q, then there exist unique characters χi mod q, i =,, such that χn = χ nχ n for all n. Proof. The first assertion follows immediately from the observations that χ nχ n is totally multiplicative, that it vanishes if n, [q, q ] >, and that it has period [q, q ]. As for the second assertion, we may suppose that n, q =. By the Chinese Remainder Theorem we see that Z/qZ = Z/q Z Z/q Z if q, q =. Thus the result follows from Lemma. Our proof of Theorem 9 depends on Abel s Theorem that any finite abelian group is isomorphic to the direct product of cyclic groups, but we can prove Corollary 0 without appealing to this result, as follows. By the Chinese Remainder Theorem we see that Z/qZ = Z/p α Z. p α q If p is odd then the reduced residue classes mod p α form a cyclic group; in classical language we say there is a primitive root g. Thus if n, p = then there is a unique ν mod φp α such that g ν n mod p α. The number ν is called the index of n, and is denoted ν = ind g n. From Lemma it follows that the characters mod p α, p >, are given by k indg n χkn = e φp α 6

14 4 5. THE LARGE SIEVE for n, p =. We obtain φp α different characters by allowing k to assume integral values in the range k φp α. By Lemma 8 it follows that if q is odd then the general character mod q is given by χn = e p α q k ind g n φp α for n, q =, where it is understood that k = kp α is determined mod φp α and that g = gp α is a primitive root mod p α. The multiplicative structure of the reduced residues mod α is more complicated. For α = or α = the group is cyclic of order or, respectively, and 6 holds as before. For α 3 the group is not cyclic, but if n is odd then there exist unique µ mod and ν mod α such that n µ 5 ν mod α. In group-theoretic terms this means that Z/ α Z = C C α 7 when α 3. By Lemma 8 the characters in this case take the form jµ χn = e + kν α for odd n where j = 0 or and k α. Thus 7 holds if 8 q, but if 8 q then the general character takes the form jµ χn = e + kν α + p α q p> l ind g n φp α when n, q =. By definition, if fn is totally multiplicative, fn = 0 whenever n, q >, and fn has period q, then f is a Dirichlet character mod q. It is useful to note that the first condition can be relaxed. Theorem. If f is multiplicative, fn = 0 whenever n, q >, and f has period q, then f is a Dirichlet character modulo q. Proof. It suffices to show that f is totally multiplicative. If mn, q > then fmn = fmfn since 0 = 0. Suppose that mn, q =. Hence in particular m, q =, so that the map k n + kq mod m permutes the residue classes mod m. Thus there is a k for which n + kq mod m, and consequently m, n + kq =. Then and the proof is complete. fmn = fmn + kq by periodicity = fmfn + kq by multiplicativity = fmfn by periodicity, 8 9

15 5. THE LARGE SIEVE 5 Primitive characters Suppose that d q and that χ is a character mod d, and set { χ n n, q = ; χn = 0 otherwise. Then χn is multiplicative and has period q, so χn is a Dirichlet character mod q. In this situation we say that χ induces χ. If q is composed entirely of primes dividing d then χn = χ n for all n, but if there is a prime factor of q not found in d then χn does not have period d. Nevertheless, χ and χ are nearly the same. Our immediate task is to determine when one character induces another. Lemma 3. Let χ be a character mod q. We say that d is a quasiperiod of χ if χm = χn whenever m n mod d and mn, q =. The least quasiperiod of χ is a divisor of q. Proof. Let d be a quasiperiod of χ, and put g = d, q. We show that g is also a quasiperiod of χ. Suppose that m n mod g and that mn, q =. Since g is a linear combination of d and q, and m n is a multiple of g, it follows that there are integers x and y such that m n = dx + qy. Then χm = χm qy = χn + dx = χn. Thus g is a quasiperiod of χ. With more effort it can be shown that if d and d are quasiperiods of χ then d, d is also a quasiperiod, and hence the least quasiperiod divides all other quasiperiods, and in particular it divides q since q is a quasiperiod of χ. The least quasiperiod d of χ is called the conductor of χ. Suppose that d is the conductor of χ. If n, d = then n+kd, d =. Also, if r, d = then there exist values of k mod r for which n+kd, r =. Hence there exist integers k for which n+kd, q =. For such a k put χ n = χn+kd. Although there are many such k, there is only one value of χn+kd when n + kd, q =. We extend the definition of χ by setting χ n = 0 when n, d >. It is readily seen that χ is multiplicative and that χ has period d. Thus χ is a character modulo d. Moreover, if χ 0 is the principal character modulo q, then χn = χ nχ 0 n. Thus χ induces χ. Clearly χ has no quasiperiod smaller than d, for otherwise χ would have a smaller quasiperiod, contradicting the minimality of d. In addition, χ is the only character mod d that induces χ, for if there were another, say χ, then for any n with n, d = we would have χ n = χ n+kd = χn+kd = χ n+kd = χ n, on choosing k as above. A character χ modulo q is said to be primitive when q is the least quasiperiod of χ. Such χ are not induced by any character having a smaller conductor. We summarize our discussion as follows. Theorem 4. Let χ denote a Dirichlet character modulo q and let d be the conductor of χ. Then d q, and there is a unique primitive character χ modulo d that induces q. We now give two useful criteria for primitivity.

16 6 5. THE LARGE SIEVE Theorem 5. Let χ be a character modulo q. Then the following are equivalent: χ is primitive. If d q and d < q then there is a c such that c mod d, c, q =, χc. 3 If d q and d < q, then for every integer a, q χn = 0. n a mod d Proof.. Suppose that d q, d < q. Since χ is primitive, there exist integers m and n such that m n mod d, χm χn, χmn 0. Choose c so that c, q =, cm n mod q. Thus we have. 3. Let c be as in. As k runs through a complete residue system mod q/d, the numbers n = ac + kcd run through all residues mod q for which n a mod d. Thus the sum S in question is q/d S = χac + kcd = χcs. k= Since χc, it follows that S = Suppose that d q, d < q. Take a = in 3. Then χ = is one term in the sum, but the sum is 0, so there must be another term χn in the sum such that χn, χn 0. But n mod d, so d is not a quasiperiod of χ, and hence χ is primitive. Bounds for character sums We first record some useful facts about the simplest character sums and Gauss sums. Given a character χ modulo q, we define the Gauss sum by q τχ = χaea/q. This can be thought of as an inner product between additive and multiplicative characters, and is the principal medium for translation between additive and multiplicative characters. Lemma 6. Suppose that χ is a character modulo q and either n, q = or χ is primitive. Then q χaena/q = χnτχ. When χ is primitive, τχ = q. Proof. The case n, q = is trivial. The case n, q >, which we now assume, is not quite. Choose m and d so that m, d = and m/d = n/q. Then q χaean/q = d ehm/d h= q a h mod d χa.

17 5. THE LARGE SIEVE 7 By Theorem 5 3 the innermost sum is 0. To evaluate τχ we take the square of the modulus of both sides of the first part of the lemma, and sum over n to see that φq τχ = q q χaean/q = q b= q χaχb q ea bn/q. The innermost sum on the right is 0 unless a b mod q, in which case it is equal to q. Thus φq τχ = φqq, and hence τχ = q. One use for the above is Theorem 7, The Pólya [98] I. M. Vinogradov [98] inequality. Suppose that χ is a non-principal character modulo q. Then uniformly in x and y with x y. x<n y χn q log q Proof. Schur [99] and Vinogradov [99]. We first prove this when χ is primitive. By the orthogonality of the additive characters modulo q and Lemma 6 we have x<n y χn = = q x<n y m= q h= m= q χm q = q χhτχ q h= q e hm n/q h= q χmehm/q x<n y since the sum over m is 0 when h = q. The sum over n is sin πh/q h/q and so by the last part of Lemma, our sum is x<n y e hn/q e hn/q and the primitive case follows. q q h/q q h= h q/ h

18 8 5. THE LARGE SIEVE To deduce the imprimitive case, let χ be the primitive character which induces χ and let r be the conductor of χ. Then χn = χ n x<n y x<n y n,q/r= = m q/r µmχ m x/m<l<y/m dq/rr log r q log q. χ l References E. Bombieri [965] On the large sieve, Mathematika, 0 5. E. Bombieri and H. Davenport [968], On the large sieve method, Abh. Zahlentheorie Anal., pp. 9. H. Davenport [967], Multiplicative number theory, Markham, Chicago. H. Davenport [000], Multiplicative Number Theory, third edition, Springer-Verlag, Berlin. T. Estermann [95], Introduction to modern prime number theory, Cambridge University Press, Cambridge, Tract No. 4. P. X. Gallagher [967], The large sieve, Mathematika 4, 4 0. P. X. Gallagher [968], Bombieri s mean value theorem, Mathematika 5, 6. Yu. V. Linnik [94] The large sieve, C. R. Dokl. Acad. Sci. URSS, n. Ser. 30, Yu. V. Linnik [94] A remark on the least quadratic non-residue, C. R. Dokl. Acad. Sci. URSS, n. Ser. 36, 9 0. H. L. Montgomery [968], A note on the large sieve, J. Lond. Math. Soc. 43, H. L. Montgomery [978], The analytic principle of the large sieve, Bull. Am. Math. Soc. 84, H. L. Montgomery and R. C. Vaughan [973], The large sieve, Mathematika 0, H. L. Montgomery and R. C. Vaughan [974], Hilbert s inequality, J. Lond. Math. Soc. 8, H. L. Montgomery and R. C. Vaughan [006], Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge. G. Pólya [98], Über die Verteilung der quadratischen Reste und Nichtreste, Nachr. Akad. Wiss. Göttingen 98, 9. K. F. Roth [965], On the large sieves of Linnik and Renyi, Mathematika, 9. I. Schur [98], Einige Bemerkungen zu der vorstehenden Arbeit des Herrn G. Pólya: Über die Verteilung der quadratischen Reste und Nichtreste, Nachr. Akad. Wiss. Göttingen 98, A. Selberg [99], Collected papers. Volume II, Springer-Verlag, Berlin. C. L. Siegel [935], Über die Klassenzahl quadratischer Zahlkörper, Acta Arith., E. C. Titchmarsh [930], A divisor problem, Rend. Circ. Mat. Palermo 54, R. C. Vaughan [977], Sommes trigonométriques sur les nombres premiers,, C. R. Acad. Sci. Paris, Série A 85, R. C. Vaughan [980], An elementary method in prime number theory, Acta Arith. 37, 5. A. I. Vinogradov [965], On the density hypothesis for Dirichlet L series, Izv. Akad. Nauk SSSR, Ser. Mat. 9, A. I. Vinogradov [966], Corrections to the work of A.I. Vinogradov On the density hypothesis for Dirichlet L series, Izv. Akad. Nauk SSSR, Ser. Mat. 30, I. M. Vinogradov [98], Sur la distribution des résidus et des nonrésidus des puissances, J. Soc. Phys. Math. Univ. Permi 98, 8 8. I. M. Vinogradov [99], Über die Verteilung der quadratischen Reste und Nichtreste, J. Soc. Phys. Math. Univ. Permi 99, 4. I. M. Vinogradov [937], Some theorems concerning the theory of primes, Recueil Math. 44, A. Walfisz [936], Zur additiven Zahlentheorie. II, Math. Z. 40,