On the Restricted Divisor Function in Arithmetic Progressions
aa r X i v : . [ m a t h . N T ] A ug On the Restricted Divisor Function inArithmetic Progressions
Igor E. Shparlinski
Department of Computing, Macquarie UniversitySydney, NSW 2109, [email protected] 25, 2018
Abstract
We obtain several asymptotic estimates for the sums of the re-stricted divisor function τ M,N ( k ) = { m M, n N : mn = k } over short arithmetic progressions, which improve some results ofJ. Truelsen. Such estimates are motivated by the links with the paircorrelation problem for fractional parts of the quadratic function αk , k = 1 , , . . . with a real α . Subject Classification (2010)
Keywords divisor function, congruences, character sums
There is a long history of studying the distribution of the divisor functionover short arithmetic progressions, see [2, 3, 5, 6, 7] and references therein.Recently, Truelsen [14] has introduced the restricted divisor function τ M,N ( k ) = { m M, n N : mn = k } pair correlation problem for fractional partsof the quadratic function αk , k = 1 , , . . . , with a real α , see also [10, 12]for various results and conjectures concerning this problem. In particular,it is conjectured in [14, Conjecture 1.2] that for any fixed ε, δ, c , c >
0, ifpositive integers
N, M, R and q satisfy N > q / ε c N M c N R > N δ , then, uniformly over all integers a with gcd( a, q ) = 1, we have R X r =1 X k ≡ ar (mod q ) τ M,N ( k ) ∼ M N Rq . (1)It is also shown in [14] that the asymptotic formula (1) yields explicit exam-ples of real α for which distribution of spacings between the fractional partsof αk is Poissonian .Towards the conjecture (1), several asymptotic formulas and estimatesare derived in [14].In particular, as in [14], for positive integer q , M , N and a divisor d | q ,we consider the sums∆ q ( d ; M, N ) = q X a =1gcd( a,q )= d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ≡ a (mod q ) τ M,N ( k ) − M Nq Φ( q, d ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2)where Φ( q, d ) = X e | d e X f | q/e f µ (cid:18) qef (cid:19) = X e | d eϕ ( q/e ) , (3)see [14, Equation (1.5)] and µ ( k ) is the M¨obius function. Also as in [14], forpositive integer q , M , N and R , we consider the sumsΓ q ( M, N, R ) = q − X a =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X r R X k ≡ ar (mod q ) τ M,N ( k ) − M N Rq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)Here, in Section 3.1, we show that a result of [13] almost instantly impliesthe estimate of [14, Theorem 1.8] on the sums ∆ q ( d ; M, N ), and in fact, ina slightly stronger form. Furthermore, using a different technique of multi-plicative character sums, in Section 3.2 we obtain a new estimates on thesums Γ q ( M, N, R ), which for some parameter ranges improves that of [14,Theorem 1.9]. We present our argument only in the case of prime q butcombining it with elementary (but somewhat cluttered) sieving it can alsobe used for arbitrary q . 2 Preliminaries
Throughout the paper, any implied constants in symbols O , ≪ and ≫ mayoccasionally depend on the positive parameters ε and δ and are absoluteotherwise. We recall that the notations U = O ( V ), U ≪ V and V ≫ U are all equivalent to the statement that | U | cV holds with some constant c > ϕ ( s, K ) = X k K gcd( k,s )=1 ϕ ( s, K ) = ϕ ( s ) s K + O ( s o (1) ) , (5)see [14, Equation (3.1)], that follows from the inclusion-exclusion principleand the well-known bound on the divisor and Euler functions τ ( s ) = s o (1) and ϕ ( s ) = s o (1) , see [9, Theorem 317] and [9, Theorem 328], respectively. Let Φ s be the set of all ϕ ( s ) multiplicative characters modulo s . We also use χ to denote the principal character and Φ ∗ s = Φ s \ { χ } to denote the set of nonprincipal multiplicative characters modulo s .For an integer Z and χ ∈ Φ s we define the sums S s ( Z ; χ ) = Z X z =1 χ ( z ) . (6)The following result is a combination of the P´olya-Vinogradov (for ν = 1)and Burgess (for ν >
2) bounds, see [11, Theorems 12.5 and 12.6].3 emma 1.
For a prime s and positive integers Z s , the bound max χ ∈ Φ ∗ s | S s ( Z ; χ ) | Z − /ν s ( ν +1) / ν + o (1) holds with an arbitrary fixed integer ν > . We combine Lemma 1 with a bound on the fourth moment of the sums S s ( Z, t ; χ ). First we recall the following estimate from [1] (for prime s ) and [7](for arbitrary s ), see also [4, 8], which we present in the following slightlyrelaxed form. Lemma 2.
For positive integers Z s , the bound X χ ∈ Φ ∗ s | S s ( Z ; χ ) | s o (1) Z holds. τ M,N ( k ) and congruences We note that sums of the restricted divisor function over an arithmetic pro-gression can be expressed via the number of solutions to a certain congruence.For example, X k ≡ a (mod q ) τ M,N ( k ) = T q ( M, N ; a ) , (7)where T q ( M, N ; a ) is number of solutions to the congruence mn ≡ a (mod q ) , m M, n N. (8)This interpretation underlines our approach.To estimate the function T s ( M, N ; a ) it is more convenient to work withthe quantity T ∗ s ( X, Y ; a ) which is defined as number of solutions to the con-gruence xy ≡ a (mod s ) , x X, gcd( x, s ) = 1 , y Y. One of our main tool is the following special case of [13, Theorem 1],combined with (5).
Lemma 3.
For positive integers s , X Y , we have s X a =1 (cid:12)(cid:12)(cid:12)(cid:12) T ∗ s ( X, Y ; a ) − ϕ ( s ) s XY (cid:12)(cid:12)(cid:12)(cid:12) XY s o (1) .
4e also define R s ( X, Y, Z ; a ) as number of solutions to the congruence xy ≡ az (mod s ) , with 1 x X, y Y, z Z. Lemma 4.
For a prime s and positive integers X, Y, Z < s we have s − X a =1 (cid:12)(cid:12)(cid:12)(cid:12) R s ( X, Y, Z ; a ) − XY Zs − (cid:12)(cid:12)(cid:12)(cid:12) XY ZU − /ν s ( ν +1) / ν + o (1) where U = min { X, Y, Z } and ν > is arbitrary fixed positive integer.Proof. We note that for every a with gcd( a, s ) = 1, we obtain R s ( X, Y, Z ; a ) = 1 s − X X x =1 Y X y =1 Z X z =1 X χ ∈ Φ s χ (cid:0) a − xyz − (cid:1) . Recalling the definition (6), changing the order of summation, using that χ (cid:0) z − (cid:1) = χ ( z ) , if gcd( z, s ) = 1 where χ is the complex conjugated character, we derive R s ( X, Y, Z ; a ) = 1 s − X χ ∈ Φ s χ ( a ) S s ( X ; χ ) S s ( Y ; χ ) S s ( Z ; χ ) . We now separate the contribution from the principal character χ = χ , get-ting R s ( X, Y, Z ; a ) − XY Zs − s − X χ ∈ Φ s ∗ χ ( a ) S s ( X ; χ ) S s ( Y ; χ ) S s ( Z ; χ ) . Using the orthogonality of characters, we easily derive s X a =1 (cid:12)(cid:12)(cid:12)(cid:12) R s ( X, Y, Z ; a ) − XY Zs − (cid:12)(cid:12)(cid:12)(cid:12) = 1 ϕ ( s ) X χ ∈ Φ s ∗ | S s ( X, u ; χ ) | | S s ( Y, v ; χ ) | | S s ( Z ; w, χ ) | = 1 ϕ ( s ) X χ ∈ Φ s ∗ | S s ( e X, u ; χ ) | | S s ( e Y , v ; χ ) | | S s ( e Z ; w, χ ) | , e X, e Y , e Z ) is any of ( X, Y, Z ). We now apply Lemma 1to the last sum and then use the Cauchy inequality, arriving to s X a =1 (cid:12)(cid:12)(cid:12)(cid:12) R s ( X, Y, Z ; a ) − XY Zs − (cid:12)(cid:12)(cid:12)(cid:12) e Z − /ν s ( ν +1) / ν + o (1) s − s X χ ∈ Φ s ∗ | S s ( e X ; χ ) | s X χ ∈ Φ s ∗ | S s ( e Y ; χ ) | . We now choose a permutation ( e X, e Y , e Z ) with e Z = U = min { X, Y, Z } . UsingLemma 2, we obtain the desired result. τ M,N ( k ) over Some Familiesof Progressions Here we estimate the sums ∆ q ( d ; M, N ) given by (2). and show how Lemma 3implies a stronger and more general form of the estimate [14, Theorem 1.8]which asserts that if M ≪ N ≪ M then∆ q ( d ; M, N ) q N max { / , − δ } + o (1) , (9)uniformly over q N − δ and d | q Theorem 5.
For arbitrary positive integers q , M and N and a divisor d | q we have ∆ q ( d ; M, N ) M N q o (1) Proof.
Without loss of generality we can assume that M > N .For each divisor e | d , we collect together the solutions to (8) withgcd( m, q ) = e , getting T q ( M, N ; a ) = X e | d T ∗ q/e ( ⌊ M/e ⌋ , N ; a/e ) . where T ∗ s ( X, Y ; a ) is defined in Section 2.3.6ecalling (7) and (3), we obtain∆ q ( d ; M, N ) = q X a =1gcd( a,q )= d (cid:12)(cid:12)(cid:12)(cid:12) T q ( M, N ; a ) − M Nq Φ( q, d ) (cid:12)(cid:12)(cid:12)(cid:12) = q X a =1gcd( a,q )= d X e | d (cid:12)(cid:12)(cid:12)(cid:12) T ∗ q/e ( ⌊ M/e ⌋ , N ; a/e ) − M N eq ϕ ( q/e ) (cid:12)(cid:12)(cid:12)(cid:12) . Thus, using the Cauchy inequality, we obtain∆ q ( d ; M, N ) q o (1) X e | d q X a =1gcd( a,q )= d (cid:12)(cid:12)(cid:12)(cid:12) T ∗ q/e ( ⌊ M/e ⌋ , N ; a/e ) − M N eq ϕ ( q/e ) (cid:12)(cid:12)(cid:12)(cid:12) . (10)We now note that M N eq ϕ ( q/e ) = ( M/e ) N ( q/e ) ϕ ( q/e )= ⌊ M/e ⌋ N ( q/e ) ϕ ( q/e ) + O ( N e/q ) = ⌊ M/e ⌋ N ( q/e ) ϕ ( q/e ) + O ( N d/q ) . We now see from (10) that∆ q ( d ; M,N ) q o (1) X e | d q X a =1gcd( a,q )= d (cid:12)(cid:12)(cid:12)(cid:12) T ∗ q/e ( ⌊ M/e ⌋ , N ; a/e ) − ⌊ M/e ⌋ N ( q/e ) ϕ ( q/e ) (cid:12)(cid:12)(cid:12)(cid:12) + N d q − o (1) X e | d q X a =1gcd( a,q )= d q o (1) X e | d q X a =1gcd( a,q )= d (cid:12)(cid:12)(cid:12)(cid:12) T ∗ q/e ( ⌊ M/e ⌋ , N ; a/e ) − ⌊ M/e ⌋ N ( q/e ) ϕ ( q/e ) (cid:12)(cid:12)(cid:12)(cid:12) + N dq − o (1) . Writing a = ce , we derive∆ q ( d ; M, N ) q o (1) X e | d q/e X c =1 (cid:12)(cid:12)(cid:12)(cid:12) T ∗ q/e ( ⌊ M/e ⌋ , N ; c ) − ⌊ M/e ⌋ N ( q/e ) ϕ ( q/e ) (cid:12)(cid:12)(cid:12)(cid:12) + N dq − o (1) . q ( d ; M, N ) M N q o (1) + N dq − o (1) ( M + N ) N q o (1) . Since M > N , this concludes the proof.Note that the bound of Theorem 5 is more general than (9) as it worksfor M and N of essentially different sizes and does not need the restriction.In particular, if M ≪ N ≪ M , then this bound takes form N q o (1) , whichimproves (9) for δ > /
2, that is, for N > q / ε for any fixed ε > Note that in [14] the bound (9) has been used to prove several other results.Theorem 5 can be used to get corresponding generalisations and improve-ments of these bounds. For example, bounds of ∆ q ( d ; M, N ) are used in [14,Theorem 1.9] to derive the estimate on the sums Γ q ( M, N, R ) given by (4).In particular, by [14, Theorem 1.9] we haveΓ q ( M, N, R ) N R (cid:0) R − + N max {− / , − δ } (cid:1) q − o (1) , (11)provided M ≪ N ≪ M , R q N − δ (note that the condition of [14,Theorem 1.9] that R > N η for some positive η > q ( d ; M, N ), using Theorem 5 one now obtains a simi-lar generalisation and improvement for Γ q ( M, N, R ). One can probably usesimilar arguments to sharpen [14, Theorem 4.5] as well.Furthermore, we now present a different approach, based on Lemma 4,which allows us to obtain estimates on Γ q ( M, N, R ) that are sometimesstronger that those of [14, Theorem 1.9] or following from Theorem 5. Wedemonstrate this approach only in the case of prime modulus q . In the gen-eral case, one can use it as well, but it involves rather cluttered expressionsarising from the inclusion-exclusion principle. Theorem 6.
For a prime q and positive integers M, N, R < q , the bound Γ q ( M, N, R ) M N RL − /ν q ( ν +1) / ν + o (1) holds, where L = min { M, N, R } and ν > is arbitrary fixed positive integer. roof. As in the proof of Theorem 5 we see thatΓ q ( M, N, R ) = q − X a =1 (cid:12)(cid:12)(cid:12)(cid:12) R q ( M, N, R ; a ) − M N Rq (cid:12)(cid:12)(cid:12)(cid:12) , and using Lemma 4, we conclude the proof.For example, if q is prime then for M, N = q / o (1) and R = q / o (1) ,applying Theorem 6 with ν = 2 we obtainΓ q ( M, N, R ) q / o (1) while (11) gives only Γ q ( M, N, R ) q / o (1) for the above choice of parameters. One can certainly easily produce manyother examples of the parameters ( M, N, R ) for which Theorem 6 is strongerthan (11).As we have said the argument used in the proof of Lemma 4 an thus ofTheorem 6 can also be applied in the case of composite q . However we recallthat the Burgess bound for character sums modulo a composite q has somelimitations on the possible choices of ν , see [11, Theorem 12.6] for details. Acknowledgement
The author is grateful to Jimi Truelsen and to the referee for very usefulcomments. During the preparation of this paper, the author was supportedin part by ARC grant DP1092835.