A lower bound for the variance in arithmetic progressions of some multiplicative functions close to 1
aa r X i v : . [ m a t h . N T ] F e b A LOWER BOUND FOR THE VARIANCE IN ARITHMETICPROGRESSIONS OF SOME MULTIPLICATIVE FUNCTIONS CLOSE TO DANIELE MASTROSTEFANO
Abstract.
We investigate lower bounds for the variance in arithmetic progressions of certainmultiplicative functions “close” to . Specifically, we consider α N -fold divisor functions, when α N is a sequence of positive real numbers approaching in a suitable way or α N = 1 , and theindicator of y -smooth numbers, for suitably large parameters y .As a corollary, we will strengthen a previous author’s result on the first subject and obtainmatching lower bounds to some Barban–Davenport–Halberstam type theorems for y -smoothnumbers.Incidentally, we will also find a lower bound for the variance in arithmetic progressions of theprime factors counting functions ω ( n ) and Ω( n ) . Introduction
For any complex arithmetic function f : N → C and any positive integer N we indicate with V ( N, Q ; f ) := X q ≤ Q X h | q X a mod q ( a,q )= h (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Nn ≡ a mod q f ( n ) − ϕ ( q/h ) X n ≤ N ( n,q )= h f ( n ) (cid:12)(cid:12)(cid:12)(cid:12) the probabilistic variance of f in arithmetic progressions. Here ϕ ( · ) is the Euler totient function, < Q ≤ N is a real number and the symbol ( · , · ) stands for the greatest common divisor of twopositive integers.In a previous paper [6], the author found a lower bound for the quantity V ( N, Q ; f ) over a largeclass of multiplicative functions f , referred to as “generalized divisor functions”, which contains,as a particular instance, all the α –fold divisor functions d α ( n ) , for parameters α ∈ C \{{ }∪− N } .For all such values α , it was proved that:(1.1) V ( N, Q ; d α ) ≫ α,δ Q X n ≤ N | d α ( n ) | , uniformly on N / δ ≤ Q ≤ N , whenever δ > is sufficiently small and N is large enough withrespect to α and δ (see [6, Theorem 1.1]).When α = 1 , the lower bound (1.1) does not hold, as shown in [6, Proposition 1.10], wherethe estimate V ( N, Q ; d ) ≪ Q , for any Q ≥ , was proved, by an elementary direct inspectionof the variance. The first new result of this paper, consequence of some new computations ona certain related mean square integral of a complete exponential sum, demonstrates that suchan upper bound is sharp, at least in some ranges of Q . Theorem 1.1.
There exists an absolute constant c > such that for any cN / ≤ Q ≤ N and N large enough, we have V ( N, Q ; d ) ≫ Q . Mathematics Subject Classification.
Primary: 11N64. Secondary: 11B25, 11N37, 11L99.
Key words and phrases.
Multiplicative and additive functions in arithmetic progressions; partial sums withRamanujan sums; integrals of exponential sums over minor arcs.The author is funded by a Departmental Award and by an EPSRC Doctoral Training Partnership Award.The present work was carried out when the author was a second year PhD student at the University of Warwick.
For parameters α = α N := 1 + 1 /R ( N ) , where R ( N ) is a real non-vanishing function, themethod developed in [6], which makes strong use of the asymptotic expansion of the partialsum of divisor functions, also produced the following result (see [6, Theorem 1.11]). Proposition 1.2.
Let
A > and α N as above with | R ( N ) |≤ (log N ) A . Let δ > small enoughand N / δ ≤ Q ≤ N . Then there exists a constant B > such that if | R ( N ) |≥ B and N islarge with respect to δ and A , we have (1.2) V ( N, Q ; d α N ) ≫ A,δ QR ( N ) X n ≤ N d α N ( n ) ≫ QNR ( N ) exp (cid:18)(cid:18) R ( N ) (cid:19) log log NR ( N ) (cid:19) . In particular, we notice that the lower bound (1.2) is always of size
QNR ( N ) whenever | R ( N ) |≥ log log N . Remark 1.3.
By going through the proof of Proposition 1.2, it is not difficult to verify that thesame lower bound also holds when replacing the function d α N ( n ) with α ω ( n ) N or with α Ω( n ) N , where Ω( n ) and ω ( n ) stand for the prime divisors counting functions with or without multiplicity. In the following, we will indicate with ̟ ( n ) the function ω ( n ) or Ω( n ) , when a statementholds for both, and with d ̟α N ( n ) the function α ̟ ( n ) N .The main aim of this paper is to improve, by means of a different approach, the result ofProposition 1.2 to what we expect to be the best possible lower bound for the variance of d ̟α N ( n ) in arithmetic progressions. Theorem 1.4.
Let α N = 1 + 1 /R ( N ) , where R ( N ) is a non-zero real function. Assume N / δ ≤ Q ≤ N , with δ > sufficiently small. Then there exists a constant C = C ( δ ) > such that if C log log N ≤ | R ( N ) |≤ N δ/ and N is large in terms of δ , we have (1.3) V ( N, Q ; d ̟α N ) ≫ δ QNR ( N ) log (cid:18) log N log(2 N/Q ) (cid:19) + Q . Compared to (1.2), the lower bound (1.3) improves the exponent of R ( N ) , shows the presenceof the extra factor Q , which dominates on certain ranges of R ( N ) , and | R ( N ) | is allowed togrow much bigger than an arbitrarily large power of log N .In order to exploit more the extra cancellation we have, compared to (1.1), when α is closeto , we will input the Taylor expansion of the function d ̟α N ( n ) = (1 + 1 /R ( N )) ̟ ( n ) into ournew computations. Since the function ̟ ( n ) is, for the majority of positive integers n ≤ N , ofsize roughly log log N (see e.g. (2.1) below), this justifies the condition | R ( N ) |≥ C log log N inthe hypotheses of Theorem 1.4.Regarding the additive function ̟ ( n ) we will prove the following result. Theorem 1.5.
Assume N / δ ≤ Q ≤ N , with δ > sufficiently small. Then we have V ( N, Q ; ̟ ) ≫ δ Q (log log N ) + QN log (cid:18) log N log(2 N/Q ) (cid:19) , if N is large enough in terms of δ . Remark 1.6.
The proof of Theorem 1.5 contains some aspects and computations preliminaryto that of Theorem 1.4. This is why we decided to insert such result here.
The sequence of functions d ̟α N ( n ) is only one instance of a wide class of multiplicative functions“close” to . Another interesting representative of such class is the characteristic function of the y –smooth numbers, for parameters y near N . These are defined as those numbers made onlyby prime factors smaller than y . For y -smooth numbers we will prove the following theorem. HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Theorem 1.7.
Let N / δ ≤ Q ≤ N , with δ > sufficiently small. Let u := (log N ) / (log y ) .There exists a large constant C > such that the following holds. If C log N ≤ u ≤ and N is large enough in terms of δ , we have V ( N, Q ; y − smooth ) ≫ δ QN log u + Q . Remark 1.8.
We observe that Harper’s result [4, Theorem 2] gives a tight corresponding upperbound for the variance above, when Q = N/ (log N ) A , with A > and √ N ≤ y ≤ N − δ , say. We will show our new theorems by first reducing ourselves to study certain L –integrals ofthe exponential sums with coefficients , ̟ ( n ) , d ̟α N ( n ) or y − smooth ( n ) , through an applicationof a technique introduced in a seminal work of Harper and Soundararajan. For them we willdetermine their size, which constitute an interesting result on its own and, since we believethat for the aforementioned functions their variance in arithmetic progressions should be wellapproximated by such integrals, will also give us a strong indication of the fact that our theoremsshould be sharp. 2. Preliminary notions and results
Throughout the rest of this paper the letter p will be reserved for a prime number. Otherletters might still indicate a prime number but in each case it will be specified.2.1. Some basic facts about certain arithmetic functions.
It is a classical result goingback to Hardy and Ramanujan (see also Diaconis’ paper [1]) that the partial sum of the ̟ –function satisfies the following asymptotic expansion: X n ≤ x ̟ ( n ) = x log log x + B ̟ x + O (cid:18) x log x (cid:19) ( x ≥ , (2.1)where B ̟ is a constant depending on the function ̟ . In particular, we deduce that the meanvalue of ̟ ( n ) , over the integers n ≤ x , is roughly log log x . Regarding its variance, we canappeal to the Turán-Kubilius’ inequality (see e.g. [8, Ch. III, Theorem 3.1]), which states that X n ≤ x ( ̟ ( n ) − log log n ) ≪ x log log x ( x ≥ . (2.2)In particular, (2.1) and (2.2) together give X n ≤ x ̟ ( n ) ≪ x (log log x ) ( x ≥ . (2.3)Finally, we remind of the following bound on the maximal size of ̟ ( n ) (see e.g. [8, Ch. I, Eq.5.9]): ̟ ( n ) ≤ (log x ) / (log 2) (1 ≤ n ≤ x ) . (2.4)We will make use of the following result on the partial sum of some non-negative multiplicativefunctions. Lemma 2.1.
For any non-negative multiplicative function g ( n ) uniformly bounded on the primenumbers by a positive real constant B and such that the sum S = P q ( g ( q ) log q ) /q over all theprime powers q = p k , with k ≥ , converges, one has X n ≤ x g ( n ) ≪ B,S x log x X n ≤ x g ( n ) n ( x ≥ D. MASTROSTEFANO and ≪ B,S X n ≤ x g ( n ) n Y p ≤ x (cid:18) g ( p ) p (cid:19) − ≪ B,S x ≥ . Proof.
The first conclusion is [8, Ch. III, Theorem 3.5] and the second one is a special caseof [3, Lemma 20] of Elliott and Kish. (cid:3)
In particular, we evidence the following immediate consequence for the partial sum of certaintypes of divisor functions (see e.g. [8, Ch. III, Theorem 3.7]).
Corollary 2.2.
Let < y < . Then, uniformly for ≤ y ≤ y and x ≥ , one has X n ≤ x y ̟ ( n ) ≪ x (log x ) y − . Preliminaries about the variance in arithmetic progressions.
As usual, we definethe so called set of major arcs M = M ( K, Q , Q ) consisting of those θ ∈ R / Z having anapproximation | θ − a/q |≤ K/ ( qQ ) with moduli q ≤ KQ and reduced residue classes ( a, q ) = 1 .Let instead m = m ( K, Q , Q ) , the minor arcs, denote the complement of the major arcs in R / Z .Thus, the union of minor arcs is defined as the set of those real numbers in [0 , approximableby rational fractions with large denominator, as large as depending on K, Q and Q .As explained in [6], to produce lower bounds for the variance of complex sequences in arith-metic progressions we rely on an application of Harper and Soundararajan’s method introducedin [5], which points out a direct link between the variance and the L -norm of some exponentialsums over unions of minor arcs. This is the content of [5, Proposition 1], which we next report. Proposition 2.3.
Let f ( n ) be any complex sequence. Let N be a large positive integer, K ≥ be a parameter and K, Q and Q be such that (2.5) K p N log N ≤ Q ≤ N and N log NQ ≤ Q ≤ QK . Then we have V ( N, Q ; f ) ≥ Q (cid:16) O (cid:16) log KK (cid:17)(cid:17) Z m |S f ( θ ) | dθ + O (cid:16) N KQ X n ≤ N | f ( n ) | (cid:17) (2.6) + O (cid:18) X q ≤ Q q X d | qd>Q ϕ ( d ) (cid:12)(cid:12)(cid:12) X n ≤ N f ( n ) c d ( n ) (cid:12)(cid:12)(cid:12) (cid:19) , where c d ( n ) are the Ramanujan sums defined as c d ( n ) = X a =1 ,...,d ( a,d )=1 e ( an/d ) and S f ( θ ) := P n ≤ N f ( n ) e ( nθ ) with e ( t ) = e πit , for any t ∈ R . Remark 2.4.
One has the following representation for the Ramanujan sums as a sum overdivisors (see e.g. [7, Theorem 4.1]): (2.7) c d ( n ) = X k | ( n,d ) kµ ( d/k ) , where µ ( n ) is the Möbius function. To handle the L -integrals over minor arcs as in (2.6) for the function f ( n ) = y − smooth ( n ) we will appeal to [5, Proposition 3], which we next report adapted to our context. HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Proposition 2.5.
Keep notations as above and assume KQ < R := N / − δ/ . Then we have (2.8) Z m |S f ( θ ) | dθ ≥ (cid:18) Z m |S f ( θ ) G ( θ ) | dθ (cid:19) (cid:18) Z m |G ( θ ) | dθ (cid:19) − , where G ( θ ) = X n ≤ N (cid:18) X r | nr ≤ R g ( r ) (cid:19) e ( nθ ) , for any complex arithmetic function g ( r ) .If moreover there exists a constant κ > for which | g ( n ) |≤ d κ ( n ) , for any n ≤ N , we alsohave (2.9) Z m |S f ( θ ) G ( θ ) | dθ ≥ X KQ To estimate the L –integrals over minor arcs as in (2.6) for the functions f ( n ) = d ̟α N ( n ) and f ( n ) = ̟ ( n ) , we will instead need to invoke [5, Proposition 2], which we next report in a morecompact form. For ease of readability, we say that a real smooth function φ ( t ) belongs to the“Fourier class” of functions F if: • φ ( t ) is compactly supported in [0 , ; • ≤ φ ( t ) ≤ , for all ≤ t ≤ ; • R φ ( t ) dt ≥ / ; • | ˆ φ ( ξ ) |≪ A (1+ | ξ | ) − A , for any A > , where ˆ φ ( ξ ) := R + ∞−∞ φ ( t ) e ( − ξt ) dt denotes the Fouriertransform of φ ( t ) . Proposition 2.6. Keep notations as above and assume KQ < R ≤ Q/ K . Then we have (2.10) Z m |S f ( θ ) | dθ ≥ (cid:12)(cid:12)(cid:12)(cid:12) Z m S f ( θ ) G ( θ ) dθ (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) Z m |G ( θ ) | dθ (cid:19) − , where G ( θ ) = X n ≤ N (cid:18) X r | nr ≤ R g ( r ) (cid:19) φ (cid:16) nN (cid:17) e ( nθ ) , for any complex arithmetic function g ( r ) and real function φ ( t ) .Let M := max r ≤ R | g ( r ) | . If φ ( t ) ∈ F , then we also have Z m S f ( θ ) G ( θ ) dθ = X n ≤ N f ( n ) (cid:18) X r | nr ≤ R g ( r ) (cid:19) φ (cid:18) nN (cid:19) (2.11) − N X q ≤ KQ Z K/qQ − K/qQ (cid:18) X n ≤ N f ( n ) c q ( n ) e ( nβ ) (cid:19)(cid:18) X r ≤ Rq | r g ( r ) r (cid:19) ˆ φ ( βN ) dβ + O (cid:18) M KR √ Q log N √ Q s X n ≤ N f ( n ) (cid:19) . D. MASTROSTEFANO It turns out that to lower bound the integral R m |S y − smooth ( θ ) | dθ is sufficient to only look atcertain minor arcs, i.e. at those centred on fractions with denominators q ≤ R . This makesthe application of the Cauchy–Schwarz’s inequality in the form (2.8) efficient, which in turnsimplifies our task by means of (2.9). On the other hand, the contribution to the integrals R m |S d ̟αN ( θ ) | dθ and R m |S ̟ ( θ ) | dθ comes from all of the minor arcs, even from those centred onfractions with possibly very large denominators. This forces us to use the Cauchy–Schwarz’sinequality as in (2.10) and then to asymptotically estimate R m S f ( θ ) G ( θ ) dθ by means of (2.11).3. The L -integral of some exponential sums over minor arcs As already discussed, Harper and Soundararajan showed that, to lower bound the variance V ( N, Q ; f ) of complex arithmetic functions f ( n ) in arithmetic progressions, we can switch ourattention to integrals of exponential sums over unions of minor arcs, such as R m |S f ( θ ) | dθ , forwhich we seek for a sharp lower bound. This is accomplished by an application of Proposition2.3. Our aim is to employ such strategy in the case f ( n ) = 1 , f ( n ) = ̟ ( n ) , f ( n ) = d ̟α N ( n ) and f ( n ) = y − smooth ( n ) and with the choice of minor arcs m = m ( K, Q, Q ) given by K a largepositive constant, N / δ ≤ Q ≤ N , for any suitably small δ > , and Q satisfying (2.5). Thiswill indeed be the underlying choice of minor arcs in the next propositions.Regarding the constant function , we have the following result. Proposition 3.1. For any N large enough with respect to δ , we have Z m |S ( θ ) | dθ ≫ Q. (3.1)Regarding the additive function ̟ ( n ) , we will prove the next proposition. Proposition 3.2. Suppose KQ < N / − δ/ . If N is sufficiently large in terms of δ , we have (3.2) Z m |S ̟ ( θ ) | dθ ≫ δ Q (log log N ) + N log (cid:18) log N log(2 N/Q ) (cid:19) . Regarding the multiplicative function d ̟α N ( n ) , the result is the following. Proposition 3.3. Suppose KQ < N / − δ/ . There exists a large constant C = C ( δ ) > suchthat if C log log N < | R ( N ) |≤ N δ/ and N is large enough in terms of δ , we have (3.3) Z m |S d ̟αN ( θ ) | dθ ≫ δ NR ( N ) log (cid:18) log N log(2 N/Q ) (cid:19) + Q. Remark 3.4. From the proof of [6, Theorem 1.11] it can be easily evinced that Z m |S d ̟αN ( θ ) | dθ ≫ δ NR ( N ) exp (cid:18)(cid:18) R ( N ) (cid:19) log log NR ( N ) (cid:19) , whenever B < | R ( N ) |≤ log log N , for a suitable large constant B ≥ . Regarding the indicator of y –smooth numbers, we will show the following lower bound. Proposition 3.5. Assume that KQ ≤ N / − δ (log N ) . Let u := (log N ) / (log y ) . There existsa large constant C > such that the following holds. If C log N ≤ u ≤ and N is large enough in terms of δ , we have Z m |S y − smooth ( θ ) | dθ ≫ δ N log u + Q. (3.4) HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO In order to show that Q -times our lower bounds (3.1), (3.2), (3.3) and (3.4) provides us withthe expected best possible approximation for the related variances, we will produce correspond-ing sharp upper bounds for them, which in some cases will also turn out to be useful to deducethe aforementioned lower bounds themselves. Proposition 3.6. With notations as in Propositions 3.1, 3.2, 3.3 and 3.5, we have thata) (3.1) is sharp;b) (3.2) is sharp;c) the estimate (3.3) is sharp when | R ( N ) | > (log log N ) / ;d) (3.4) is sharp. Remark 3.7. It should be possible to produce a sharp upper bound for the integral in (3.3) inthe whole range | R ( N ) | > C log log N (see Remark 4.1 below). To work out the size of the L -integral over minor arcs of the exponential sum with coefficients f ( n ) = d ̟α N ( n ) , we will split f into a sum f = f d + f r of a deterministic part f d , constant, and apseudorandom one f r . By triangle inequality we will separate their contribution to the integralsto then analyse them individually. To deal with R m |S f d ( θ ) | dθ we will unfold the definition ofminor arcs and insert classical estimates for the size of a complete exponential sum. Regarding R m |S f r ( θ ) | dθ instead, when | R ( N ) | > (log log N ) / , we will reduce the problem to estimate the L -integral over minor arcs of the exponential sum with coefficients ̟ ( n ) . To this aim, we willwrite ̟ ( n ) = Σ + Σ , where Σ is a sum over prime numbers smaller than a power of N/Q and Σ the remaining part, and again use triangle inequality. To estimate R m |S Σ ( θ ) | dθ we will useParseval’s identity and an application of Turán–Kubilius’ inequality. Regarding R m |S Σ ( θ ) | dθ instead we will expand out the square inside the integral and unfold the definition of minor arcsto then conclude by counting the number of primes which are solution to certain systems ofcongruences. 4. Proof of Proposition 3.6 We set the parameter K to be a large constant, N / δ ≤ Q ≤ N , with N sufficiently large interms of δ , and Q satisfying (2.5). We keep these notations throughout the rest of this section.4.1. The case of the constant function . We use the well-known bound |S ( θ ) |≪ min (cid:26) N, || θ || (cid:27) , (4.1)where || θ || indicates the distance of θ from the nearest integer. Since θ = a/q + δ , with | δ |≤ K/qQ and q > KQ , we have that either || θ || = | θ | or || θ || = 1 − | θ | . Hence, by symmetry, we find that Z m |S ( θ ) | dθ ≪ X KQ We first observe that for any two complex numbers w, z we have | w + z | ≤ | w | + | z | ) . (4.2) D. MASTROSTEFANO By writing y − smooth ( n ) = 1 − ∃ p | n : p>y ( n ) and using (4.2) to separate their contribution to theintegral, we get Z m |S y − smooth ( θ ) | dθ ≪ Q + X n ≤ N ∃ p | n : p>y ≤ Q + N X y
The case of divisor functions close to . Let α N = 1 + 1 /R ( N ) , where R ( N ) is anon-vanishing real function with | R ( N ) | > C log log N , for a constant C > to determine lateron. By (4.2), one has(4.3) Z m |S d ̟αN ( θ ) | dθ ≤ Z m |S d ̟αN − ( θ ) | dθ + 2 Z m |S ( θ ) | dθ and we split the exponential sum with coefficients d ̟α N ( n ) − according to whether ̟ ( n ) ≤ A log log N or ̟ ( n ) > A log log N , with A > large to be chosen later. We do this only when | R ( N ) |≤ (log N ) / (log 2) . We separate their contribution to the integral by (4.2). The secondone is bounded by Parseval’s identity by X n ≤ N̟ ( n ) >A log log N ( α ̟ ( n ) N − ≤ X n ≤ N̟ ( n ) >A log log N ( α ̟ ( n ) N + 1) ≤ N ) A log(5 / X n ≤ N (cid:18)(cid:18) (cid:19) ̟ ( n ) + 1 (cid:19)(cid:18) (cid:19) ̟ ( n ) ≪ N (log N ) , say, by Corollary 2.2 and choosing A large enough.Let(4.4) Err ( N ) := (cid:26) N (log N ) if | R ( N ) |≤ (log N ) / (log 2);0 otherwise . From the above considerations and Proposition 3.6 a), we deduce that Z m |S d ̟αN ( θ ) | dθ ≪ Z m (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N̟ ( n ) ≤ A log log N ( α ̟ ( n ) N − e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ + Q + Err ( N ) , where the restriction on the sum is there only when | R ( N ) |≤ (log N ) / (log 2) . The integral onthe right-hand side by (4.2) is ≪ R ( N ) Z m (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N̟ ( n ) ≤ A log log N ̟ ( n ) e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ + Z m |S T N ( θ ) | dθ, where we let T N ( n ) := (cid:18) α ̟ ( n ) N − − ̟ ( n ) R ( N ) (cid:19) ̟ ( n ) ≤ A log log N . The second integral above, again by (4.2), is ≪ M N Z m |S ( θ ) | dθ + Z m |S T N − M N ( θ ) | dθ, where M N := α log log NN − − log log NR ( N ) . HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO By Proposition 3.6 a), the first term above is ≪ Q (log log N ) /R ( N ) ≤ Q, if C is largeenough. On the other hand, the second one, by Parseval’s identity, by Taylor expanding α ̟ ( n ) N and α log log NN (which we can do thanks to the restriction in the sum and reminding of the maximalsize (2.4) of ̟ ( n ) ) and using the well-known identity a k − b k = ( a − b ) P k − j =0 a j b k − − j , whichholds for a couple of positive real numbers a, b and any positive integer k , can be estimated with ≪ (log log N ) R ( N ) X n ≤ N ( ̟ ( n ) ̟ ( n ) ≤ A log log N − log log N ) ≪ N (log log N ) R ( N ) , if A, C ( A ) and N are sufficiently large, by inserting and after removing the condition ̟ ( n ) ≤ A log log N on the sum, at a cost of an acceptable error term, and performing the mean squareestimate using (2.2). Overall, by gathering all of the above considerations, we have showed that(4.5) Z m |S d ̟αN ( θ ) | dθ ≪ R ( N ) Z m |S ̟ ( θ ) | dθ + Q + N (log log N ) R ( N ) + NR ( N ) log N , say, whenever | R ( N ) | > C log log N and C and N are sufficiently large.It is then clear that assuming the upper bound in Proposition 3.6 b) for R m |S ̟ ( θ ) | dθ and | R ( N ) | > (log log N ) / we get Proposition 3.6 c). Remark 4.1. If we had T N ( n ) = ̟ ( n ) / (2 R ( N ) ) , we believe that we would roughly find Z m |S T N ( θ ) | dθ ≈ N (log log N ) R ( N ) log (cid:18) log N log(2 N/Q ) (cid:19) . This would imply that the lower bound (3.3) for the integral R m |S d ̟αN ( θ ) | dθ is sharp in the wholerange | R ( N ) | > C log log N , with C large. In practice, by writing T N ( n ) as a truncated Taylorseries up to order k , plus a remainder term, we believe we would get to prove that (3.3) issharp in the range | R ( N ) | > (log log N ) /k , for any fixed positive integer k , by inspecting thestructure of the minor arcs. Even though this would constitute an improvement over the resultof Proposition 3.6 c), we will not commit ourselves to formally proving this here. The case of the ̟ function. To begin with, we write X n ≤ N ω ( n ) e ( nθ ) = X n ≤ N ω ( n ) e ( nθ ) + X n ≤ N ω ( n ) e ( nθ ) , where ω ( n ) is the number of prime factors of n smaller than p N/Q and ω ( n ) that of primedivisors contained in the interval ( p N/Q, N ] . By (4.2), one has Z m | S ω ( θ ) | dθ ≪ Z m | S ω ( θ ) | dθ + Z m | S ω ( θ ) | dθ. (4.6)A simple calculation shows that ω ( n ) has a mean value of size log((4 log N ) / (log(2 N/Q ))) . Hence, isolating this term inside the corresponding integral gives Z m | S ω ( θ ) | dθ ≪ Q (cid:18) log (cid:18) N log(2 N/Q ) (cid:19)(cid:19) + Z m (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N (cid:18) ω ( n ) − log (cid:18) N log(2 N/Q ) (cid:19)(cid:19) e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ ≪ Q (cid:18) log (cid:18) N log(2 N/Q ) (cid:19)(cid:19) + N log (cid:18) N log(2 N/Q ) (cid:19) , by Proposition 3.6 a), Parseval’s identity and an application of the general form of the Turán–Kubilius’ inequality, which gives an analogue for ω ( n ) of (2.2) (see e.g. [8, Ch. III, Theorem X n ≤ N Ω( n ) e ( nθ ) = X n ≤ N ω ( n ) e ( nθ ) + X n ≤ N (cid:18) X p k | nk ≥ (cid:19) e ( nθ ) we immediately get Z m |S Ω ( θ ) | dθ ≪ Z m |S ω ( θ ) | dθ + X n ≤ N (cid:18) X p k | nk ≥ (cid:19) and, by expanding the square out and swapping summations, we see that the above sum is X n ≤ N X p k | nk ≥ X p j | nj ≥ X p ≤√ N j log N log p k X k =2 X p ≤√ N j log N log p k X j =2 X n ≤ Nn ≡ p k ,p j ]) ≤ N X p ≤√ N j log N log p k X k =2 j log N log p k X j =2 p max { k,j } + N X p ≤√ N j log N log p k X k =2 p k X p ≤√ Np = p j log N log p k X j =2 p j ≪ N. For the rest of this section, we will focus on showing the following statement. Claim 4.2. Let K be a large constant, N / δ ≤ Q ≤ N , with N sufficiently large in terms of δ , and Q satisfying (2.5) . Then we have Z m |S ω ( θ ) | dθ ≪ N. Assuming the validity of Claim 4.2, and collecting the above observations together, it isimmediate to deduce Proposition 3.6 b). We now then move to the proof of Claim 4.2. Byexpanding the integral, we find(4.7) Z m |S ω ( θ ) | dθ = X KQ Np , qpa (cid:27)(cid:19) . (4.8)Note that the above minimum is always of size q/pa . So, the above reduces to be = 1 Q X KQ Note that we have been able to facilitate the estimate of (4.9) thanks to our choiceof parameter p N/Q in (4.6) . The partial sum of some arithmetic functions twisted with Ramanujan sums A key step to find a lower bound for the variance of a function f in arithmetic progressionsis to produce a lower bound for the L -integral over minor arcs of the exponential sum withcoefficients f ( n ) . For smooth numbers, this will be accomplished by means of Proposition 2.5.More specifically, (2.9) allows us to reduce the problem to asymptotically estimate the partialsum of f ( n ) = y − smooth ( n ) twisted with the Ramanujan sums c q ( n ) . This will indeed constitutea crucial point in our argument and next we are going to state and prove the relative result. Lemma 5.1. Let C be a sufficiently large positive constant and consider √ N ≤ y ≤ N/C .Then for any prime number log N < q ≤ √ N and N large enough, we have (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Np | n ⇒ p ≤ y c q ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≫ N log (cid:18) log N log(max { N/q, y } ) (cid:19) and for any squarefree positive integer < q ≤ √ N with all the prime factors larger than N/y ,we have (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ Np | n ⇒ p ≤ y c q ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≫ N log u, where u := (log N ) / (log y ) .Proof. By [8, Ch. III, Theorem 5.8] we know that Ψ (cid:18) Nd , y (cid:19) := X n ≤ N/dp | n ⇒ p ≤ y (cid:26) ⌊ Nd ⌋ if d > N/y ; Nd (1 − log( log( N/d )log y )) + O ( Nd log y ) if d ≤ N/y. For any prime number q the identity (2.7) reduces to c q ( n ) = − q q | n . It is then immediateto verify the following equality: X n ≤ Np | n ⇒ p ≤ y c q ( n ) = − Ψ( N, y ) + q Ψ (cid:18) Nq , y (cid:19) , from which it is straightforward to deduce the first estimate of the lemma.By (2.7), and letting σ ( q ) := P d | q d , we can always rewrite the sum in the statement as X d | q dµ (cid:16) qd (cid:17) Ψ (cid:18) Nd , y (cid:19) = N X d | qd>N/y µ (cid:16) qd (cid:17) + N X d | qd ≤ N/y µ (cid:16) qd (cid:17) (cid:18) − log (cid:18) log( N/d )log y (cid:19)(cid:19) + O (cid:18) N log N X d | qd ≤ N/y σ ( q ) (cid:19) . In the hypothesis that q > has all the prime factors larger than N/y , the sums over the divisorsof q smaller than or equal to N/y reduce only to the single term corresponding to d = 1 . Hence,we actually have X n ≤ Np | n ⇒ p ≤ y c q ( n ) = − N µ ( q ) + N µ ( q )(1 − log u ) + O (cid:18) N log N (cid:19) = − N µ ( q ) log u + O (cid:18) N log N (cid:19) , since σ ( q ) ≪ q log log q ≪ √ N log log N ≤ N/ log N (see [8, Ch. I, Theorem 5.7]), if N is large,which immediately leads to deduce the second estimate of the lemma. (cid:3) To prove the lower bound for the variance of ̟ ( n ) and of d ̟α N ( n ) in arithmetic progressionswe will instead invoke Proposition 2.6. To this aim, we need to study the partial sum of ̟ ( n ) twisted with the Ramanujan sums and weighted by the smooth weight φ ( n/N ) , with φ ( t ) belonging to the Fourier class F as in Proposition 2.6. HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Lemma 5.2. Let R := N / − δ/ , for δ > small, and suppose that N / δ ≤ Q ≤ cN/ log log N ,for a certain absolute constant c > . Then for any N large enough with respect to δ , we have (cid:12)(cid:12)(cid:12)(cid:12) X N/Q
To begin with, we note that for prime numbers p the identity (2.7) reduces to c p ( n ) = − p p | n . Hence, the sum over n in the statement is = − X n ≤ N ̟ ( n ) φ (cid:16) nN (cid:17) + p X n ≤ Np | n ̟ ( n ) φ (cid:16) nN (cid:17) = − X n ≤ N ̟ ( n ) φ (cid:16) nN (cid:17) + p X k ≤ N/p ( ̟ ( k ) + 1) φ (cid:18) kpN (cid:19) + O (cid:18) p X k ≤ N/p ( ̟ ( k ) + 2) (cid:19) , where we used that ̟ ( pk ) ≤ ̟ ( k ) + ̟ ( p ) = ̟ ( k ) + 1 . By (2.1) the above big-Oh error termcontributes at most ≪ N (log log N ) /p .By partial summation from (2.1), it is easy to show that X n ≤ N ̟ ( n ) φ (cid:16) nN (cid:17) = J N log log N + J N B ̟ + O (cid:18) N log log N log N (cid:19) , for any N large enough, where J := R φ ( t ) dt ∈ [1 / , . This, applied once with N and oncewith N/p , together with the previous observations, gives X n ≤ N ̟ ( n ) c p ( n ) φ (cid:16) nN (cid:17) = J N (cid:18) (cid:18) − log p log N (cid:19)(cid:19) + O (cid:18) N log log N log N + N log log Np (cid:19) . Therefore, we see that the double sum in the statement is = J N X N/Q
Lemma 5.3. Let R := N / − δ/ , for δ > small, and q < R be a prime number. Then we have X n ≤ N c q ( n ) e (cid:16) nuN (cid:17) ≪ q (1 + | u | ) , uniformly for all real numbers u .Proof. To begin with, we notice that for any prime number q , the following estimate holds: S ( t ) := X n ≤ t c q ( n ) = X n ≤ tq | n q − X n ≤ t ≪ q, (5.1)by (2.7), for any t ≥ . Hence, by partial summation we find X n ≤ N c q ( n ) e (cid:16) nuN (cid:17) = Z N e (cid:18) tuN (cid:19) dS ( t ) = S ( N ) e ( u ) − S (1) e (cid:16) uN (cid:17) − uN Z N S ( t ) e (cid:18) tuN (cid:19) dt, from which, by using (5.1), the thesis follows. (cid:3) The last result of this section, preliminary to the proof of the lower bound for R m |S d ̟αN ( θ ) | dθ contained in Proposition 3.3, concerns the partial sum of the divisor functions d ̟α N ( n ) twistedwith Ramanujan sums and weighted by φ ( n/N ) , with φ ( t ) belonging to the Fourier class F asin Proposition 2.6. Lemma 5.4. Let α N = 1+1 /R ( N ) , where R ( N ) is a non-zero real function, and R := N / − δ/ ,for δ > small. Assume N / δ ≤ Q < cN (log log N ) /R ( N ) , for a certain absolute constant c > . There exists a sufficiently large constant C = C ( δ ) > such that if C log log N ≤| R ( N ) |≤ (log log N ) and N is large enough with respect to δ , we have (cid:12)(cid:12)(cid:12)(cid:12) X N/Q
By adapting the proof of [6, Theorem 1.11], it is not difficult to show that X n ≤ t d ̟α N ( n ) = c ( α N , ̟ )Γ( α N ) t (log N ) α N − (cid:18) O (cid:18) log log N | R ( N ) | log N (cid:19)(cid:19) + O (cid:18) N log log N log N (cid:19) , (5.2)for any t ∈ [ N/ log N, N ] , if N is large enough, where Γ( z ) stands for the Gamma function and c ( α N , ̟ ) := Q p (cid:16) − p (cid:17) α N (cid:16) α N p − (cid:17) if ̟ ( n ) = ω ( n ); Q p (cid:16) − p (cid:17) α N (cid:16) − α N p (cid:17) − if ̟ ( n ) = Ω( n ) . It is easy to verify that c ( α N , ̟ ) = 1 + O (cid:18) | R ( N ) | (cid:19) = Γ( α N ) , (5.3)if N is large enough (see [7, Appendix C] for basic results on the Gamma function).By Corollary 2.2, we certainly have X n ≤ N/ log N d ̟α N ( n ) φ (cid:16) nN (cid:17) ≪ X n ≤ N/ log N (cid:18) | R ( N ) | (cid:19) ̟ ( n ) ≪ N log N (log N ) / | R ( N ) | ≪ N log N . This, together with partial summation from (5.2) applied to the remaining part of the sum,leads to X n ≤ N d ̟α N ( n ) φ (cid:16) nN (cid:17) = c ( α N , ̟ )Γ( α N ) J N e log log NR ( N ) + O (cid:18) N log log N log N (cid:19) , where J := R φ ( t ) dt ∈ [1 / , and we made use of (5.3) to simplify the error term.Applying this asymptotic estimate with length of the sum N/p in place of N , we find X n ≤ Np | n d ̟α N ( n ) φ (cid:16) nN (cid:17) = α N X k ≤ N/pp ∤ k d ̟α N ( k ) φ (cid:18) pkN (cid:19) + X k ≤ N/p d ̟α N ( kp ) φ (cid:18) kp N (cid:19) = α N X k ≤ N/p d ̟α N ( k ) φ (cid:18) pkN (cid:19) + O (cid:18) X k ≤ N/p d ̟ / | R ( N ) | ( k ) (cid:19) = c ( α N , ̟ )Γ( α N ) J N α N p e log log( N/p ) R ( N ) + O (cid:18) N log log Np log N + Np (cid:19) , where we used ̟ ( pk ) ≤ ̟ ( k ) + 1 and Corollary 2.2 to handle the error term contribution. HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO The collection of the above estimates, taking into account of the identity (2.7) for the Ra-manujan sums, makes the sum over n in the statement equals to c ( α N , ̟ )Γ( α N ) J N e log log NR ( N ) ( α N e log(1 − log p log N ) R ( N ) − 1) + O (cid:18) N log log N log N + Np (cid:19) . (5.4)By Taylor expansion and thanks to (5.3), one has α N e log(1 − log p log N ) R ( N ) − (cid:18) R ( N ) (cid:19)(cid:18) − log p log N ) R ( N ) + O (cid:18) R ( N ) (cid:19)(cid:19) − 1= 1 + log(1 − log p log N ) R ( N ) + O (cid:18) R ( N ) (cid:19) and c ( α N , ̟ )Γ( α N ) e log log NR ( N ) = (cid:18) O (cid:18) | R ( N ) | (cid:19)(cid:19)(cid:18) O (cid:18) log log N | R ( N ) | (cid:19)(cid:19) = 1 + O (cid:18) log log N | R ( N ) | (cid:19) . Inserting the above estimates into (5.4), we see that the double sum in the statement is = (cid:18) J NR ( N ) X N/Q
By restricting the integral in the statement over minor arcs of the form (1 /q − /KqQ, /q +1 /KqQ ) , for positive integers q in the range Q/ (2 M ) < q ≤ Q/M , where M is a large positiveconstant to be chosen later, we can lower bound it with X Q/ (2 M ) Let K be a large constant, Q and Q be real numbers satisfying (2.5).7.1. Large values of Q . By isolating the constant term Z := log log N and expanding thesquare out, we have Z m |S ̟ ( θ ) | dθ ≥ Z m |S ̟ − Z ( θ ) | dθ + Z m |S Z ( θ ) | dθ − Z m |S ̟ − Z ( θ ) S Z ( θ ) | dθ ≥ Z m |S Z ( θ ) | dθ − sZ m |S ̟ − Z ( θ ) | dθ Z m |S Z ( θ ) | dθ, by an application of Cauchy–Schwarz’s inequality. By completing the integral R m |S ̟ − Z ( θ ) | dθ tothe whole circle and using Parseval’s identity followed by an application of the upper bound (2.2)on the second centred moment of ̟ ( n ) , we find it is ≪ N log log N . Since from Propositions3.1 and 3.6 a) we know that R m |S ( θ ) | dθ ≍ Q , on a wide range of Q , we also in particular have Z m |S Z ( θ ) | dθ ≍ Q (log log N ) , whenever e.g. Q ≥ cN/ log log N , for any fixed constant c > . By then choosing c suitablylarge, we get the lower bound (3.2) on such range of Q . HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Small values of Q . Assume now N / δ ≤ Q < cN/ log log N , with c as in the previoussubsection, and KQ < R , where R := N / − δ/ , for a small δ > . Let g ( r ) be the characteristicfunction of the set of prime numbers smaller than R . We apply Proposition 2.6 with such setsof minor arcs and functions g ( r ) and f ( n ) = ̟ ( n ) . Remark 7.1. In order to successfully apply Proposition 2.6, as a rule of thumb, we might thinkof g ( r ) as an approximation of the Dirichlet convolution f ∗ µ ( r ) , where µ ( r ) is the Möbiusfunction. This motivates our choice of g , since for any n ≤ N we either have g ∗ n ) = ω ( n ) or g ∗ n ) = ω ( n ) − , with ω ( n ) ≈ log log N ≈ Ω( n ) , for most of the integers n ≤ N , by (2.1) . With the notations introduced in Proposition 2.6, we have Z m |G ( θ ) | dθ ≪ N log (cid:18) log N log(2 N/Q ) (cid:19) , (7.1)which follows from Proposition 3.6 b), on our range of Q .Next, by (2.11), with f ( n ) = ̟ ( n ) , the integral R m S f ( θ ) G ( θ ) dθ is = X n ≤ N ̟ ( n ) (cid:18) X p | np ≤ R (cid:19) φ (cid:16) nN (cid:17) (7.2) − N X q ≤ KQ Z K/qQ − K/qQ (cid:18) X n ≤ N ̟ ( n ) c q ( n ) e ( nβ ) (cid:19)(cid:18) q> , prime q + X p ≤ R q =1 p (cid:19) ˆ φ ( βN ) dβ + O ( N − δ ) , if N is large enough with respect to δ , where we trivially estimated the error term using thebound (2.4) on the maximal size of ̟ ( n ) and our hypotheses on Q , Q and R . The secondexpression in (7.2) equals − N X n ≤ N ̟ ( n ) X p ≤ R p Z K/Q − K/Q e ( nβ ) ˆ φ ( βN ) dβ (7.3) − N X ≤ q ≤ KQ q prime q X n ≤ N ̟ ( n ) c q ( n ) Z K/qQ − K/qQ e ( nβ ) ˆ φ ( βN ) dβ. (7.4)By changing variable and since φ ( t ) belongs to the Fourier class F as in Proposition 2.6, onehas N Z K/Q − K/Q e ( nβ ) ˆ φ ( βN ) dβ = φ (cid:16) nN (cid:17) + O (cid:18) Z + ∞ KN/Q ˆ φ ( u ) du + Z − KN/Q −∞ ˆ φ ( u ) du (cid:19) (7.5) = φ (cid:16) nN (cid:17) + O (cid:18) Q N (cid:19) , where we remind that Q < cN/ log log N . Thus, by the asymptotic expansion (2.1) for thepartial sum of ̟ ( n ) and Mertens’ theorem, (7.3) equals − X p ≤ R p X n ≤ N ̟ ( n ) φ (cid:16) nN (cid:17) + O (cid:18) Q log log N (cid:19) . (7.6)We now split the sum over q in (7.4) into two parts according to whether q ≤ N/Q or q > N/Q .In the second case, since ˆ φ ( ξ ) is bounded, we find N Z K/qQ − K/qQ e ( nβ ) ˆ φ ( βN ) dβ = Z KN/qQ − KN/qQ e (cid:16) nuN (cid:17) ˆ φ ( u ) du ≪ NqQ , (7.7) from which we deduce that the contribution in (7.4) from the primes q > N/Q is ≪ NQ X q> N/Qq prime q X n ≤ N ̟ ( n ) | c q ( n ) |≪ N log log NQ X q> N/Qq prime q ≪ N log log N log(2 N/Q ) . (7.8)On the other hand, for values of q ≤ N/Q , by changing variable and by definition of φ ( t ) , wecan rewrite the integral R KN/qQ − KN/qQ e ( nu/N ) ˆ φ ( u ) du as φ (cid:16) nN (cid:17) + Z + ∞ KN/qQ e (cid:16) nuN (cid:17) ˆ φ ( u ) du + Z − KN/qQ −∞ e (cid:16) nuN (cid:17) ˆ φ ( u ) du = φ (cid:16) nN (cid:17) + O (cid:18) qQN (cid:19) , (7.9)from which we may deduce that the contribution in (7.4) coming from those primes is − X ≤ q ≤ N/Qq prime q X n ≤ N ̟ ( n ) c q ( n ) φ (cid:16) nN (cid:17) + O (cid:18) N log log N log(2 N/Q ) (cid:19) . (7.10)Collecting together (7.6), (7.10) and previous observations and thanks to the identity (2.7) forthe Ramanujan sums, we see that (7.2) equals to X N/Q
For any δ small enough and N sufficiently large with respect to δ , we have log (cid:18) log R log(2 N/Q ) (cid:19) ≥ δ log (cid:18) log N log(2 N/Q ) (cid:19) . Proof. The aimed inequality is equivalent to (cid:18) − δ (cid:19)(cid:18) log N log(2 N/Q ) (cid:19) − δ ≥ , which is satisfied when in particular (cid:18) − δ (cid:19) ≥ (cid:18) − δ + O ( δ ) (cid:19) − δ and N is sufficiently large with respect to δ . The above in turn is equivalent to δ log 2 + O ( δ )1 + δ log 2 + O ( δ ) ≤ − δ. Since the left-hand side above equals to − δ/ log 2 + O ( δ ) , the thesis immediately follows if δ is taken small enough. (cid:3) HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Proof of Proposition 3.3 Let K be a large constant, Q and Q be real numbers satisfying (2.5). Moreover, let C log log N ≤ | R ( N ) |≤ N δ/ , with C as in Lemma 5.4.8.1. Large values of Q . By isolating the constant term and expanding the square out, wehave Z m |S d ̟αN ( θ ) | dθ ≥ Z m |S d ̟αN − ( θ ) | dθ + Z m |S ( θ ) | dθ − Z m |S d ̟αN − ( θ ) S ( θ ) | dθ ≥ Z m |S ( θ ) | dθ − sZ m |S d ̟αN − ( θ ) | dθ Z m |S ( θ ) | dθ, by an application of Cauchy–Schwarz’s inequality. The estimate of the integral R m |S d ̟αN − ( θ ) | dθ has already been performed in subsect. . , where we found (see Eq. (4.5)): Z m |S d ̟αN − ( θ ) | dθ ≪ R ( N ) Z m |S ̟ ( θ ) | dθ + N (log log N ) R ( N ) + NR ( N ) log N . By Propositions 3.1 and 3.6 a), which together give R m |S ( θ ) | dθ ≍ Q , by Proposition 3.6 b),which shows that Z m |S ̟ ( θ ) | dθ ≪ Q (log log N ) + N log (cid:18) log N log(2 N/Q ) (cid:19) , and by the above considerations, we may deduce the lower bound (3.3), at least when Q ≥ cN (log log N ) /R ( N ) , for c a suitable positive constant, by taking N large enough and possiblyreplacing C with a larger value.8.2. Small values of Q . Let us now assume N / δ ≤ Q < cN (log log N ) /R ( N ) and KQ Large values of Q . We always have Z m |S y − smooth ( θ ) | dθ ≥ Z m |S ( θ ) | dθ + Z m (cid:12)(cid:12)(cid:12)(cid:12) X n ≤ N ∃ p | n : p>y e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ − Z m (cid:12)(cid:12)(cid:12)(cid:12) S ( θ ) X n ≤ N ∃ p | n : p>y e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) dθ. By Parseval’s identity and Mertens’ theorem, the second integral on the right-hand side aboveis ≪ N log u , where u := (log N ) / (log y ) . This, together with the upper bound for R m |S ( θ ) | dθ given in Proposition 3.6 a) and Cauchy–Schwarz’s inequality, makes the third integral insteadof size ≪ √ QN log u . By using the lower bound (3.1) for the integral R m |S ( θ ) | dθ , for values DN log u ≤ Q ≤ N , with D > a large constant, we may deduce the lower bound (3.4) on suchrange of Q .9.2. Small values of Q . Let δ > small. Let K be a large constant, Q and Q be real numberssatisfying (2.5) and such that N / δ ≤ Q < DN log u, with D as in the previous subsection,and log N < Q ≤ Q max0 := N / − δ (log N ) /K . Let R := N / − δ/ . We keep these notationsthroughout the rest of this section. Remark 9.1. The choice of the maximal possible size of Q only reflects the fact that, todeduce the lower bound on the variance of the y –smooth numbers in arithmetic progressions asin Theorem 1.7, we will take Q = N (log N ) /Q in Proposition 2.3. Case y small. Let √ N ≤ y ≤ N − δ/ . Let g ( r ) be the indicator of the prime numbers r ∈ [ Q max0 , R ] . We apply Proposition 2.5 with functions f ( n ) = y − smooth ( n ) and g ( r ) as above. Remark 9.2. The choice of g here has been inspired by the fact that the Dirichlet convolution y − smooth ∗ µ ( n ) equals primes ∈ ( y,N ] ( n ) . With notations as in Proposition 2.5, by Parseval’s identity, we have Z m |G ( θ ) | dθ ≤ X n ≤ N (cid:18) X p | nQ max0
From the work in subsubsect. 9.2.1, it is clear that we cannot make use of thesame type of g even when y is very close to N . For, we would always have Z m |G ( θ ) | dθ ≪ N max X p ∈ Supp ( g ) ∩ [ KQ ,R ] p , (cid:18) X p ∈ Supp ( g ) ∩ [ KQ ,R ] p (cid:19) , whereas by (2.9) and Lemma 5.1 we would always also have Z m |S f ( θ ) G ( θ ) | dθ ≫ N log u X p ∈ Supp ( g ) ∩ [ KQ ,R ] p , which are not of comparable size, whenever u is close to . For such values of y , we then optedfor a multiplicative function g with the right logarithmic density, suggested to us from the secondpart of Lemma 5.1 and the following computations. By Parseval’s identity, we have Z m |G ( θ ) | dθ ≤ X n ≤ N (cid:18) X r | nr ≤ Rp | r ⇒ N/y
12 log 2 − X k =0 X k N / − δ/ 12 log 2 − X k =0 S ( k ) S ( k ) , by Cauchy–Schwarz’s inequality, where we have restated the condition on the support of q ,implicit in h ( q ) , as µ ( q ) = 1 and ( q, P ) = 1 . By the fundamental lemma of sieve theory (seee.g. [8, Ch. I, Theorem 4.4]), taking δ small enough, and Mertens’ theorem, we have S ( k ) ≫ δ (cid:18) k N / − δ/ ϕ ( P ) P (cid:19) ≫ (cid:18) k N / − δ/ log( N/y ) (cid:19) . On the other hand, by Lemma 2.1 and Mertens’ theorem, we get that S ( k ) is ≤ X q ≤ k +1 N / − δ/ ( q, P )=1 µ ( q )=1 k +1 N / − δ/ h ( q ) ≪ (2 k N / − δ/ ) log N Y N/y
12 log 2 − X k =0 N/y ) ≫ δ log N log( N/y ) ≥ u − and consequently that Z m |S f ( θ ) G ( θ ) | dθ ≫ δ N log u ( u − . This, in combination with the upper bound (9.1) for the integral R m |G ( θ ) | dθ and log u ≫ u − ,if δ small, concludes the proof of Proposition 3.5 via the application of Proposition 2.5.10. Deduction of Theorem 1.1 By Proposition 2.3, we have V ( N, Q ; d ) ≫ Q Z m |S ( θ ) | dθ + O (cid:16) N Q + X q ≤ Q q X d | qd>Q ϕ ( d ) (cid:12)(cid:12)(cid:12) X n ≤ N c d ( n ) (cid:12)(cid:12)(cid:12) (cid:19) , (10.1)by choosing K large and where Q and Q need to satisfy (2.5).The sum in the big-Oh error term has already been estimated in [6, Proposition 4.3], but herewe are going to produce a better bound for the function d ( n ) .First of all, by (2.7), we notice that X n ≤ N c d ( n ) = X n ≤ N X k | ( n,d ) kµ (cid:18) dk (cid:19) = X k | d kµ (cid:18) dk (cid:19) X n ≤ Nk | n X k | d kµ (cid:18) dk (cid:19) (cid:22) Nk (cid:23) = O ( σ ( d )) , where σ ( d ) := P k | d k and where we used the well-known identity P k | d µ ( k ) = 0 , for any d > .Therefore, we need to study the following sum: X q ≤ Q q X d | qd>Q σ ( d ) ϕ ( d ) = X Q In this final section we prove the lower bound for the variance of d ̟α N ( n ) in arithmetic pro-gressions as presented in Theorem 1.4. The proofs of Theorems 1.5 and 1.7 are similar, so theywill be omitted.By plugging the lower bound (3.3) for the integral R m |S d ̟αN ( θ ) | dθ into the lower bound ex-pression (2.6) for the variance of f ( n ) = d ̟α N ( n ) in arithmetic progressions, and choosing K large enough, we find V ( N, Q ; d ̟α N ) ≫ δ QNR ( N ) log (cid:18) log N log(2 N/Q ) (cid:19) + Q + O (cid:18) N (log N ) Q (cid:19) , (11.1)where to estimate the error term we used [6, Proposition 4.3] with κ = 2 , say, and Corollary2.2. Taking Q := N R ( N ) (log N ) /Q , which satisfies the hypotheses of Proposition 3.3, weget the thesis, if N is large enough with respect to δ . Acknowledgements I am deeply indebted to my supervisor Adam J. Harper for some discussions and insightfulcomments that notably improved the results presented here and simplified the exposition in thispaper. References [1] P. Diaconis. Asymptotic expansions for the mean and variance of the number of prime factors of a numbern. Technical Report No. 96, Department of Statistics, Stanford University (1976).[2] F. Dress, H. Iwaniec, G. Tenenbaum. Sur une somme liée à la fonction de Möbius . J. Reine Angew. Math.340, 53–58 (1983).[3] P. D. T. A. Elliott, J. Kish. Harmonic analysis on the positive rationals II: multiplicative functions and Maassforms . J. Math. Sci. Univ. Tokyo 23 (3), 615–658 (2016).[4] A. J. Harper. Bombieri–Vinogradov and Barban–Davenport–Halberstam type theorems for smooth numbers . https://arxiv.org/abs/1208.5992 (2012).[5] A. J. Harper, K. Soundararajan. Lower bounds for the variance of sequences in arithmetic progressions:primes and divisor functions . Q. J. Math. 68 (1), 97–123 (2017).[6] D. Mastrostefano. A lower bound for the variance of generalized divisor functions in arithmetic progressions . https://arxiv.org/abs/2004.05602 . Accepted for publication in The Ramanujan Journal (2021).[7] H. Montgomery, R. Vaughan. Multiplicative Number Theory I: Classical Theory . Cambridge University Press,Cambridge (2006).[8] G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory . Graduate Studies in Mathematics,vol. 163, AMS, New York (2015). University of Warwick, Mathematics Institute, Zeeman Building, Coventry, CV4 7AL, UK Email address ::
. For, if pa = 0 then q | p and p ≤ N/Q , whichcannot happen since q > KQ . Hence, | pa/q + pβ |≥ pa/ q . Indeed, for any N large enoughcompared to δ , we have p | β |≤ KNqQ ≤ q ≤ pa q . HE VARIANCE IN APS OF MULTIPLICATIVE FUNCTIONS CLOSE TO Putting together the above information, we see that (4.7) is ≪ Q X KQ
KQ and by (2.5) Q ≤ Q/K , we require K > M , say.Moreover, we remind that K , and thus M , are absolute constants here. By partial summationit is easy to verify that (cid:12)(cid:12)(cid:12)(cid:12) X ≤ n ≤ N e ( n/q ) e ( nθ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) e πi ( N +1) /q − e πi/q e πi/q − (cid:12)(cid:12)(cid:12)(cid:12) + O (cid:18) NQ (cid:19) . We deduce that Z m ( K,Q ,Q ) |S ( θ ) | dθ ≥ X Q/ (2 M )
. Tothis aim, we apply the van der Corput’s inequality (see e.g. [8, Ch. I, Theorem 6.5]) to thefunction f N ( t ) := N/t, for which f N ( t ) ∈ C ( I ) with f ′′ N ( t ) ≍ N M /Q , for t ∈ I . We thus get (cid:12)(cid:12)(cid:12)(cid:12) X q ∈ I ℜ ( e πif N ( q ) ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ QM , for any M / N / ≤ Q ≤ N , if we take N sufficiently large, from which the thesis follows, bytaking M large enough. 7. Proof of Proposition 3.2
(log log N ) , we replace d ̟α N ( n ) inside (8.7) with ̟ ( n ) /R ( N ) + E ( n ) . Afterwards, we estimate the error contribution coming from the constantfunction using partial summation from the bound (5.1) on the partial sum of c q ( n ) and triviallythat from E ( N ) thanks to our current assumption on | R ( N ) | and arguing as before. Finally,the main contribution coming from ̟ ( n ) /R ( N ) can be immediately handled by Lemma 5.2.Combining the estimate we get, by proceeding in this way, for (8.2) together with the bound(8.1) via an application of Proposition 2.6, we may deduce the lower bound (3.3) also on thisrange of | R ( N ) | and thus conclude the proof of Proposition 3.3.9. Proof of Proposition 3.5