ON THE AVERAGE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

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1 ON THE AVERAGE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS KOSTADINKA LAPKOVA Abstract. We give an asymtotic formula for the ivisor sum c<n N τ n bn c for integers b < c of the same arity. Interestingly, the coefficient of the main term oes not een on the iscriminant as long as it is a full square. We also rovie effective uer bouns of the average ivisor sum for some of the reucible quaratic olynomials consiere above, with the same main term as in the asymtotic formula.. Introuction Let τn enote the number of ositive ivisors of the integer n an P x Z[x] be a olynomial. There are many results on estimating average sums of ivisors N. τ P n, n= one of which was obtaine by Erős [8], who showe that for an irreucible olynomial P x an for any N >, we have N log N P N n= τp n P N log N. Here the imlie constants can een both on the egree an the coefficients of the olynomial. When P x is a quaratic olynomial Hooley [4] an McKee [7], [8] obtaine asymtotic formulae for the sum.. When eg P x 3 no asymtotic formulae for. are known. A certain rogress in this irection was mae by Elsholtz an Tao in 7 of [7]. When the olynomial P x is reucible the behavior is a little bit ifferent. Ingham [5] consiere the aitive ivisor roblem an rove that for a fixe ositive integer q the following asymtotic hols τnτn + q π 2 σ qn log 2 N, n N as N, where σ a q = q a for a, q Z. Later Hooley [4] reicte that.2 τn 2 r 2 = ArN log 2 N + ON log N, n N but only recently Duek [5] rovie the exact value of the constant A, namely /ζ2 = /π 2. The first aim of this aer is to exten Duek s work an to fin the exact values of Ar for any integer r. Actually we fin the main term in the asymtotic formula for. for slightly more general olynomials P n = n bn c for integers b < c, such that b + c is even. For integers k 0 an > 0 we efine.3 ρ k := # { 0 x < : x 2 k mo }. Date: Mathematics Subject Classification. Primary N5; Seconary D09. Key wors an hrases. number of ivisors, quaratic olynomial, Dirichlet series.

2 2 K. LAPKOVA The main result we nee for the asymtotic estimate of the average ivisor sum. for reucible quaratic olynomials P x is the following Theorem, which is of interest of its own. Theorem. For any integer r we have the asymtotic formula as N. From Theorem we can euce λ N ρ r 2λ N log N, π2 Theorem 2. Let b < c be integers with the same arity. Then we have the asymtotic formula as N. c<n N τ n bn c π 2 N log2 N, Clearly the reucible quaratic olynomials of the tye n 2 r 2 consiere in.2 are covere by Theorem 2, this is the case c = b = r > 0. It is very interesting that the constants Ar in the formula.2 are uniform for r. This woul be no surrise if there is a relation of the function ρ r 2 with a certain class number, which woul not change if we factor the ositive iscriminant with a full square. Such a corresonence was escribe by McKee in [7], [8], [9], however, only for square-free iscriminants. Hooley [4] suggeste one ossible way to get to the values Ar an rove Theorem 2, namely to start from Ingham s work [5]. However we follow Duek s metho which relies on a Tauberian theorem. Let for a multilicative function λn we enote the Dirichlet series D λ s := n= λn/ns. In orer to fin the value A Duek uses information for the function ρ an then a Tauberian theorem for the Dirichlet series D ρ s. It turns out that all the necessary information for ρ n for any integer n can be extracte from section 4 of Hooley s aer [3]. There Hooley examines the Dirichlet series D ρn s. Actually his further investigations can also lea to a roof of our Theorem 2 with an exlicit error term. However, we woul use these further investigations, more recisely formula from [3], only in the secon art of the resent aer where we estimate exlicitly from above the average ivisor sum c<n N τ n bn c, much in the sirit of our earlier aer []. One of the motivations to consier also exlicit uer bouns for the sum of ivisors. for quaratic olynomials comes from their alication in Diohantine sets roblems. Let n 0 be an integer. A set of m ositive integers {a,..., a m } is calle a Dn m-tule if a i a j + n is a erfect square for all i, j with i < j m. The classical an most extensively stuie tye of such sets are the Diohantine sets D. In our aer [] we gave a similar exlicit uer boun for the sum. for an irreucible quaratic olynomial P x of certain tye, which allowe to imrove the maximal ossible number of D -quarules. We believe that in a similar way the uer bouns which will be given by Theorem 3 an Corollary 4 state below can be useful for estimating the number of D4, D or other Dk 2 - sets, which are investigate in a number of aers, e.g. [], [9], [0], [], [2]. We have the following theorem. Theorem 3. Let b < c be integers with the same arity an δ = b c 2 /4 factor as δ = 2 t Ω 2 for some even t 0 an o Ω. Assume that σ Ω 4/3. Let c = max, c + an

3 c n N ON THE AVERAGE NUMBER OF DIVISORS 3 X = fn. Then for any integer N c we have 3 τ n bn c < 2N π 2 log2 X + π 2 + CΩ log X + CΩ where CΩ = 2 Ω + 2X log X + CΩ π2 2τΩ/.749 σ Ω/ Remark. When a ositive c is close to N, i.e. we have relatively few summans, one can ajust the uer boun by subtracting the negative quantity 2 c c/ρ δ. In orer to estimate well enough this quantity, however, we nee to establish also a strong effective lower boun of the sum c ρ δ/, a task we o not ursue here. The first imortant feature of Theorem 3 is that uner the conition.4 σ Ω = Ω 4 3 we can rovie an exlicit uer boun with the same main term as in the asymtotic formula from Theorem 2, because X = N + O. Secon feature of Theorem 3 is that it rovies bouns for a larger family of quaratic reucible olynomials than the most stuie case u to now, this for P n = n 2, which satisfies conition.4. Inee, an immeiate observation is that when Ω = we have Ω / = < 4/3. This case inclues the olynomials P n = n2 4 s for integer s 0 an we have the following corollary. Corollary 4. For any integer N the following inequalities hol: i Let s be a nonnegative integer. Then ρ 4 sλ < 3 λ π 2 log2 N log N ii λ N N τn 2 < N n= π 2 log2 N log N Previous exlicit uer bouns for the average number of ivisors of P n = n 2 were obtaine by Elsholtz, Filiin an Fujita [] with A 2. Trugian [2] imrove this to A 2/π 2, Ciu [3] got A 9/π 2 an very recently Ciu an Trugian [4] also achieve the best leaing coefficient A /π 2 using ifferent metho than ours. The main goal of all these aers is to boun the maximal ossible number of Diohantine quintules. In another recent aer, estimating the number of D4-quintules, Bliznac an Filiin [] showe that A2 /π Asymtotic formula In [5] Duek uses the following Tauberian theorem Theorem 2.4. in [2]. a n Lemma. Let F s = n s be a Dirichlet series with non-negative coefficients converging for n= Rs >. Suose that F s extens analytically at all oints on Rs = aart from s =, an,

4 4 K. LAPKOVA that at s = we can write Hs F s = s α for some α R an some Hs holomorhic in the region Rs an non-zero there. Then a n γx log x α with where Γ is the usual Gamma function. n x γ := H Γ α, The key information which we nee in orer to exten the result of [5] is containe in the following lemma. We write α n when α n but α+ n. Lemma 2. Fix the integer arameter r an write simly ρ for the function ρ r 2. Let β, t 0 be such that β r 2 for > 2, 2 t r 2 an 2β := β, 2t := t. Then for the value of the function ρ α at rime owers we have the following cases. i 2r ii r, 2 iii = 2 ρ α = ρ α = 2, if α. { [ 2 α], if α β, 2 β, if α > β. 2 [ 2 α], if α t, ρ2 α 2 t, if α = t +, = 2 t +, if α = t + 2, 2 t +2, if α > t + 2. Proof. The lemma follows from section 4 of Hooley s aer [3], more recisely his cases a, an e. There the values of the function ρ n α are examine for a general integer arameter n. Note that the conition that n is square-free, which is imose in the theorems of [3], is not require in 4 [3]. Proof of Theorem. Let us fix the integer arameter r. Consier the function ρλ F s := D ρ s = λ s. Clearly ρλ λ, hence the Dirichlet series F s is absolutely convergent for Rs > 2 an we can write it as an Euler rouct 2. F s = + ρ s + ρ2 2s +... =: A s. From the following comutations it will become clear that F s is absolutely convergent for Rs >. Accoring to Lemma 2 for the factors A s we obtain the following cases. λ=

5 i 2r. Then ON THE AVERAGE NUMBER OF DIVISORS A s = + 2 s s +... = 2s s. ii r, 2. Then we get 2.3 A s = + s + 2s + β s = + s + β s β 2s iii = 2. In this case we have A 2 s = + 2 s s + 2 2t s 2 2 = + 2 s + 2 2t s 2 ts We use 2.2 an the fact that for Rs > so we can write ζ 2 s ζ2s = F s = ζ2 s ζ2s A 2s 2 s β β + + β 2s β s βs + β + s βs s. 2t 2t + 2t ts t+s 2 t+2s 2 t+3s + 2t + 2t t+2s 2 t+3s 2s s 2 = + 2 s r 2 + s s, + 2 s + 2 2s s A s s + s =: ζ2 s ζ2s Gs. + 2 s + 2 2s +... It is clear that Gs is holomorhic in the half-lane Rs, though it is not obvious that it is non-zero there. By Lemma 3 below it follows that this is inee true, because the finitely many factors A s for = 2 an 2 but r are non-zero for Rs. Then F s fulfills the conitions of Lemma, with α =, so we obtain 2.5 ρλ γn log N with λ N 2. γ = lim s s 2 F s = ζ2 G. By 2.3 in the case r, 2 we have A = = = β β β + β = +, which oes not een on β. β + β β β + + = β + β

6 K. LAPKOVA By 2.4 in the case = 2 we get A 2 = = t + 2 t t t t t = t t t = 3, which oes not een on t. Now from 2.7 an 2.8 lugge in the efinition of Gs it follows that G =. From 2. it follows that γ = /ζ2 an Theorem follows from 2.5. Lemma 3. If = 2 or r the functions A s have no zeros in the half-lane Rs. Proof. We will verify the claim in elementary way. First, we observe that from Lemma 2 it follows that for r an 5 we have ρ α α/2 for every integer α. Let us write Rs = σ, with σ. Then we easily obtain A s = α=0 ρ α αs α= α= ρ α αs α= α/2 σ /2 = ασ σ /2 > 0. ρ α ασ When = 3 an r, let us assume that A 3 s = 0. In this case from 2.3 β A 3 s = + 3 s 3 α 3 α=0 2αs + 3β + 3 s 3βs 3 s = + 3 β 2s 3 s 3 2s + 3 β 2s 3 s an the exression in the secon brackets shoul be zero, therefore 3 s 3 β 2s 2.9 = 3 2s 3 β 2s. Using triangle inequalities for the absolute values of both sies of 2.9 an the simle fact 3 σ /3 for σ, we see that the absolute value of the exression on the left-han sie of 2.9 is at least /2, while the absolute value of the exression on the right-han sie of 2.9 is less than /2 when β > - a contraiction. When β = 2.9 gives = 3 s + 3 2s which has no solutions for σ, again by a simle comarison of the absolute values. Thus the assumtion that A 3 s = 0 when σ is wrong. It remains to check the case = 2. Assume that A 2 s = 0 for σ. Then from 2.4 it follows that A 2 s = an necessarily we have + 2 s 2 2s 2 t +2s + 2 t +2s 2s + 2 s + 2 s s 2s 2 s + = 2 t +2s 2 t +2s. The left-han sie of 2.0 oes not een on t an we can see, again using triangle inequalities an the simle facts 2 σ /2 an 2 2σ /2, that its absolute value is at least /9. On the other han, the right-han sie of 2.0 equals /2 t +2s an its absolute value is at most /2 t + /5 for t 3. This gives /9 /5 - a contraiction. The cases t {0,, 2} can be ealt in a similar way, substituting the corresoning value of t in 2.0, an then arranging the exressions in an equation with absolute values on the two sies which cannot be equal. Thus the assumtion A 2 s = 0 for σ is wrong an this comletes the roof of the Lemma.

7 ON THE AVERAGE NUMBER OF DIVISORS 7 Proof of Theorem 2. Let us write fn := n bn c an 2. SN := τ fn = c<n N c<n N fn Note that fn is ositive for n > c an increasing. By Dirichlet hyerbola metho we have SN = c<n N 2 fn fn + O = 2 c<n N. fn fn 0 mo + ON. Recall that δ = b c 2 /4. We notice that fn = n bn c = n b + c/2 2 b c/2 2 = n b + c/2 2 δ an the conition fn 0 mo is equivalent to n b + c/2 2 δ mo. If we enote 2.2 Mx, := # { m x : fm 0 mo }, clearly we have Mx, = x ρ δ + O ρ δ. Then we rocee in the stanar way. For X = fn we have SN = 2 + O ρ δ + ON X n N X fn 0 mo = 2 MN, + O ρ δ + ON X X = 2N X ρ δ + O ρ δ + ON. X Since δ is a full square of an integer we can aly Theorem. As X = N + O it follows that ρ δ X log X N log N. Therefore we get X 2.3 SN = 2N X ρ δ Again using Theorem an Abel s summation we get X ρ δ = π 2 X log t X t + o t = 3 π 2 log2 N + o log 2 N. + O N log N. log t t t + O ρ δ X X The statement of Theorem 2 follows from lugging the latter asymtotic formula into 2.3.

8 8 K. LAPKOVA 3. Exlicit uer boun First we will reeat the argument of Hooley [3] an recreate his formula with the ientity 3.5. We resent the etails of the argument for the sake of clarity an to work out recisely the secific quantities for our secial case of a square-full iscriminant δ. In this section we will again sometimes omit δ in the notation for the function ρ δ, since δ is fixe. Recall 2. where we enote D ρ s = F s = A s. We again enote r 2 = δ = b c 2 /4 for fn = n bn c an β, t 0 are such that β r 2 for > 2, 2 t r 2 an 2β := β, 2t := t. In case ii when r, 2 we can continue the exression 2.3 in the following way: 3. A s = + s = + s + β s γ 2 β = + s γ 2 β γ γ 2s γ γ 2s β /γ 2 β 2s + β δ/γ 2 s s, βs s because δ = r 2 is a full square an β r 2, an.. is the Jacobi symbol. If 2r we notice that β = 0 an the last sum over γ from 3. equals exactly s, so we can rewrite in a similar way also the factors 2.2. For 2r we obtain 3.2 A s = + s γ 2 β γ γ 2s δ/γ 2 s. The ientity 2.4 for case iii when = 2 can be continue as 3.3 A 2 s = + 2 s [ t ] 2 α 2t αs 2 t+2s 2 s 2 2s + 2t +2 2 t+3s 2 2s α=0 = + 2 s Ks, where Ks := α=0 a α 2 αs an a irect calculation shows that 2 α/2 for 0 α t an 2 α, 3.4 a α = 0 for α t + an 2 α, 2 t + for α t + 2.

9 ON THE AVERAGE NUMBER OF DIVISORS 9 From 3., 3.2 an 3.3 an + s = ζs/ζ2s we obtain where u an we use that so we can finally write D ρ s = Ks ζs ζ2s = Ks ζs ζ2s = Ks ζs ζ2s δ/ 2u 3.5 D ρ s = where 3. h= ξ h h s We introuce the character Here := ζs ζ2s =3 = λ= =3 γ 2 β 2 δ,2= 2 δ,2= δ/ 2 ρλ λ s = χ l = 2s 2s γ γ 2s =3 =3 δ/γ 2 δ/ 2u δ/ 2 s s s,. Now recall that δ = 2 t Ω 2 with even t an o Ω, α=0 δ/ 2 { δ/ 2 l a α 2 αs 2s Ω h= s = ζs ζ2s, if 2 l, 0, otherwise. δ/ 2 2 t Ω 2 / 2 Ω 2 / 2 = = l l l ξ h h s, l= l,2= δ/ 2 l l s. is or 0, eening on whether the conition l, Ω/ = hols. This means that the character χ is actually the rincial character moulo 2Ω/, i.e. {, if l, 2Ω/ =, 3.7 χ l = 0, otherwise. Now we can write 3.8 We note that h= ξ h h s ζs ζ2s = = ζs ζ2s n= µ 2 n n s l= χ l l s. = D µ 2s an from 3.8 an the uniqueness of Dirichlet series exansion it follows that for every n we have ξ n = µ 2 lχ m. lm=n

10 0 K. LAPKOVA By 3.5 vali for Rs > 2, again using the uniqueness of Dirichlet series exansion, it follows that for the coefficients of the corresoning Dirichlet series we have the ientity a α ξ h. λ X ρλ = Ω 2 α 2 h X We summarize the latter results in the following lemma. Lemma 4. Let δ = 2 2t Ω 2 for integers t 0 an Ω, such that Ω, 2 =. Given the efinitions.3, 3.4, 3. an 3.7, for any X an Ω, we have the ientities a α ξ h an λ X ρ δ λ = Ω ξ n = lm=n 2 α 2 h X µ 2 lχ m. 3.. Proof of Theorem 3. Using the notation 2.2, an again the Dirichlet hyerbola metho, we have SN := τfn = 2 c n N 2 X fn maxc,b+ n N c n N fn 2 X n N fn c n N fn fn = 2 X MN,. We use that n b 2 > n bn c 2. One can achieve more recise uer boun for ositive c by more careful argument at this oint. We mae a cruer ste by summing over all n N instea of consiering only those n which satisfy simultaneously fn 2 an n c. From the efinition 2.2 it is clear that 3.9 MN, N ρ + ρ. Therefore 3.0 SN 2 X N ρ + ρ = 2N ρ + 2 ρ. X X In the sequel we will estimate exlicitly the sum Y ρ for any Y. From this we can easily extract also an exlicit uer boun of the sum Y ρ/ through Abel s summation. To achieve our goal we will use Lemma 4 an the Dirichlet convolution reresentation it rovies in a similar way as in our revious aer [], where Lemma 2. laye a key role roviing a comfortable Dirichlet convolution. In the current case, however, we o not have factoring in familiar multilicative functions irectly of the function ρ δ for a square-free δ, rather of another multilicative function ξ in the resentation of ρ δ for a square-full δ. From one sie this makes the argument more technical than in [], from the other sie the character sums we consier in the resent case are much simler because we eal just with the rincial character χ. From Lemma 4 it follows that 3. a α ξ h. λ X ρλ = Ω 2 α 2 X We concentrate first on estimating the innermost sum. h X/2 α 2

11 ON THE AVERAGE NUMBER OF DIVISORS 3... Estimation of ξ h. Let Y an is a fixe ivisor of Ω. By the Dirichlet convolution h Y reresentation in Lemma 4 it follows that 3.2 ξ h = µ 2 lχ m = µ 2 l χ m. h Y lm Y l Y m Y/l By 3.7 for a real Y, an writing temorarily q := 2Ω/ for the conuctor of the rincial character χ, we have χ m = = m Y m Y m Y m Y f q m Y/f 3.3 m,q> = [Y ] [ ] Y f f q f 2 Y f q f 2 f 2 Y f Let us use the notation θ a q := a = σ a q. q 2 Then the inequality 3.3 can be written as 3.4 χ m Y θ q + θ 0 q. m Y Plugging this in 3.2 we get ξ h h Y l Y 3.5 Y µ 2 l l θ q + θ 0 q = Y θ q l Y µ 2 l l + θ 0 q l Y. µ 2 l. At this ste we nee to have θ q = θ 2Ω/ 0 for every ivisor Ω. First we see that if k is o, then 3. θ 2k = 3 2 σ k. Inee, we notice that all ivisors of 2k which are at least 2 can be resente as f = 2 γ g for γ {0, } an g k, with γ = when g =, because k is o. Thus f = g + 2 g = 3 2 g. g k g k g k f 2k f 2 Then for every Ω we inee have θ 2Ω/ = σ Ω/ σ Ω 0, because we have assume the conition.4. Now we can use an uer boun ue to Ramaré Lemma 3.4 [20]. Lemma 5. Ramaré, [20] Let x be a real number. We have µ 2 n log x +.. n π2 n x

12 2 K. LAPKOVA Using also the trivial boun for the last sum in 3.5, we get ξ h Y θ q log Y +. + θ π2 0 qy h Y 3.7 = π 2 θ q Y log Y +. θ q + θ 0 q Y =: c qy log Y + c 2 qy Estimation of λ X ρ δ λ. Now we lug the latter boun in the first ientity from Lemma 4. Using that c q 0, we get a α λ X ρ δ λ = Ω Ω 2 α 2 X 2 α 2 X a α π 2 X log X Ω h X/2 α 2 ξ h c q X 2 α 2 log X 2 α 2 + c 2q X 2 α 2 θ 2Ω/ a α 2 α + X 2 α X/ 2 Ω By a irect calculation using 3.4, or by 2.8 an 3.3, we see that a α 3.8 K = 2 α = 2. Therefore we can boun the artial sums of a α /2 α by 2 an we obtain 3.9 ρ δ λ 2 π 2 C ΩX log X + C 2 ΩX, where λ X 3.20 C Ω := Ω an C 2 Ω := 2 Ω α=0 θ 2Ω/ = Ω c 22Ω/ = 2 Ω. f 2Ω/ f 2 f 2Ω/ f 2 c 22Ω/ f f + f 2Ω/ f 2 a α 2 α. 2 α X/ 2. For the constant C Ω we have the following crucial uer boun which woul guarantee the right main term. Lemma. Let Ω be an o integer satisfying σ Ω 4/3. Then the constant C Ω efine in 3.20 satisfies the inequalities Proof. From 3. it follows that 0 < C Ω 2. θ 2Ω/ = 3 2 σ Ω/.

13 ON THE AVERAGE NUMBER OF DIVISORS 3 Then C Ω = 2 3 = 2 2 g 3 2 g. Ω g Ω/ Ω Ω g Ω/ When we take out the contribution of = from the first sum an of = from the secon sum we get C Ω = g 3 2 g Ω g Ω Ω g Ω/ > g> > = g. Ω > Ω > If there are ivisors Ω which are greater than, then the innermost sum satisfies g Ω/ /g with contribution at least from g =. Then C Ω = 2 2. Ω Ω Ω > > > In articular, when Ω =, we have C Ω = /2. When Ω >, by the conition.4 we know that for every Ω we have θ 2Ω/ 0. Note that in this case there is at least one rime ivisor 3 such that Ω an we have the strict inequality 0 θ 2Ω < θ 2Ω/. Then by 3.20 we surely have C Ω > 0. g Ω/ From Lemma an 3.9 we arrive at ρ δ λ π 2 X log X + C 2ΩX. λ X From 3. an the analogous observation θ 0 2Ω/ = f 2Ω/,f 2 = 2σ 0Ω/ we check by a irect calculation that C 2 Ω = CΩ. Therefore 3.2 ρ δ λ X log X + CΩX. π2 λ X Estimation of λ X ρ δ λ/λ. Write P X := λ X ρ δλ. By 3.2 an Abel s summation formula we have 3.22 λ X ρ δ λ λ = P X X X X P u X log X + CΩX π2 π 2 log X + CΩ + π 2 X = 3 π 2 log2 X + u = P X u X + π 2 + CΩ + X log u u X P u u u 2 u u log u + CΩu π2 u 2 u + CΩ log X + CΩ. Now using 3.2 an 3.22, the inequality 3.0 turns into [ ] [ ] 3 SN 2N π 2 log2 X + π 2 + CΩ log X + CΩ + 2 X log X + CΩX, π2 X u u

14 4 K. LAPKOVA where X = fn. This roves Theorem Proof of Corollary 4. Instea of irectly alying Theorem 3 we will take use of the secific form of δ an fn. This way we can gain better minor terms coefficients, whereas the main coefficient remains the right one. When δ = 4 s, i.e. Ω =, then clearly we have only one ivisor of Ω an a single character to consier : χ = χ which is at even numbers an 0 at o. Then [ ] Y + χ m = Y From 3.2 an Lemma 5 we get m Y ξ h = µ 2 l χ l h Y l Y m Y/l l Y = Y µ 2 l + µ 2 l Y 2 l 2 2 l Y l Y = 3 π 2 Y log Y + 2.Y 2. Y µ 2 l Then by Lemma 4, the latter inequality an 3.8 we see that ρ 4 sλ = λ X 2 α X a α 3 π 2 X log X λ N h X/2 α ξ h 2 α X 2 α X a α 2 α X 2l + 2 log Y +. π2 + Y 2 3 X a α π 2 2 α log X 2 α + 2. X 2 2 α 2 α X a α 2 α < X log X + 2. X π2 From the last inequality an alying Abel s summation we obtain ρ 4 sλ 3 λ π 2 log N2 + π log N an the statement of i follows after a ecimal aroximation of the secon coefficient. For the roof of ii we note that N N τn 2 2 n=2 n=2 <n n 2 = 2 N = <n N n 2 Clearly M, = ρ an by 3.9 it follows that N τn 2 2N n=2 N = = 2 ρ N MN, M,. The secon statement of Corollary 4 follows from alying the inequality i to the innermost sum Examles. To rovie an exlicit uer boun for the average ivisor sum over any reucible quaratic olynomial fn = n bn c with δ = 2 t Ω 2 where Ω is o, t 0 is even, Theorem 3 requires the conition.4 for the ivisors of Ω. In Corollary 4 we showe an imrove such uer boun, which is vali when Ω = an fn = n 2. In this subsection we woul give some more examles when Theorem 3 hols. =.

15 I. Ω =, 3 is a rime. Inee, then ON THE AVERAGE NUMBER OF DIVISORS 5 σ Ω = Ω = = 4 3. II. Ω = k, 5 for integer k 2. Inee, if = 3 an k 2 conition.4 fails. If 5 we nee to have = k = k+ 4 3, Ω which is equivalent to / k+ 4/3 4/3, or further to 4/3 / k+ /3. The latter is true because 4/3 / k+ < 4/3 4/3 5 < /3. III. Ω = q, 5 < q an q 3 + / 3. Like in examle II. one sees that if = 3 the sum σ Ω > 4/3. Then one easily obtains the secon conition on q starting from the necessary inequality = + + q + q 4 3. Ω Thus when = 5 we can have any q. Funing. This work was suorte by a Hertha Firnberg grant of the Austrian Science Fun FWF [T84-N35]; an artially by the National Research, Develoment an Innovation Office NKFIH [K0483]. References [] M. Bliznac, A. Filiin, Uer boun for the number of D4-quintules, Bull. Aust. Math. Soc., to aear [2] A. C. Cojocaru, M. R. Murty, An introuction to sieve methos an their alications, Cambrige University Press, 200 [3] M. Ciu, Further remarks on Diohantine quintules,acta Arith , [4] M. Ciu, T. Trugian, Searching for Diohantine quintules, Acta Arith., ublishe online 8 May 20 [5] A. Duek, On the number of ivisors of n 2, Bull. Aust. Math. Soc , no. 02, [] C. Elsholtz, A. Filiin, Y. Fujita, On Diohantine quintules an D -quarules, Monatsh. Math , no. 2, [7] C. Elsholtz, T. Tao, Counting the number of solutions to the Erős-Straus equation on unit fractions, J. Aust. Math. Soc , no., [8] P. Erős, On the sum x k= fk, J. Lonon Math. Soc , 7 5 [9] A. Filiin, Extensions of some arametric families of D-triles, Int. J. Math. Math. Sci. 2007, Art. ID 3739, 2. [0] A. Filiin, An irregular D4-quarule cannot be extene to a quintule, Acta Arith , no. 2, 7 7

16 K. LAPKOVA [] A. Filiin, There are only finitely many D4-quintules, Rocky Mountain J. Math. 4 20, no., [2] Y. Fujita, Extensions of the D±k 2 -triles k 2, k 2 ±, 4k 2 ±, Perio. Math. Hungar , no., 8 98 [3] C. Hooley, On the reresentation of a number as the sum of a square an a rouct, Math. Z , [4] C. Hooley, On the number of ivisors of quaratic olynomials, Acta Math. 0 93, 97 4 [5] A. E. Ingham, Some asymtotic formulae in the theory of numbers, J. Lonon Math. Soc , [] K. Lakova, Exlicit uer boun for an average number of ivisors of quaratic olynomials, Arch. Math. Basel 0 20, no. 3, [7] J. McKee, On the average number of ivisors of quaratic olynomials, Math. Proc. Camb. Philos. Soc , [8] J. McKee, A note on the number of ivisors of quaratic olynomials. Sieve methos, exonential sums, an their alications in number theory Cariff, 995, , Lonon Math. Soc. Lecture Note Ser. 237, Cambrige Univ. Press, Cambrige, 997 [9] J. McKee, The average number of ivisors of an irreucible quaratic olynomial, Math. Proc. Cambrige Philos. Soc , no., 7 22 [20] O. Ramaré, An exlicit ensity estimate for Dirichlet L-series, Math. Com , no. 297, [2] T. Trugian, Bouns on the number of Diohantine quintules, J. Number Theory , Graz University of Technology, Institute of Analysis an Number Theory, Steyrergasse 30/II, 800 Graz, Austria aress: lakova@math.tugraz.at