ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM

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1 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM AKSHAA VATWANI AND PENG-JIE WONG ABSTRACT. We fomulate the Titchmash diviso poblem in aithmetic pogessions by consideing the sum τp a p b mod whee a b ae non-zeo integes and τ denotes the diviso function. We obtain an asymptotic fomula fo this sum unifom in a cetain ange of the modulus which can be thought of as a Siegel-Walfisz type esult in this contet. As a futhe genealization we develop a numbe field analogue of this poblem. Accodingly we conside the above sum ove pimes p satisfying Chebotaev conditions and obtain some esults in this setting. Anothe vaiant of inteest is the study of the above sum with the diviso function τn eplaced by τ y n whee τ y n is the numbe of positive divisos of n which ae bounded below a paamete y. Then fo this modification of the Titchmash diviso poblem in aithmetic pogessions in the special case that b a mod we obtain a Bombiei- Vinogadov type esult fo the sum.. INTRODUCTION The Titchmash diviso poblem is concened with the asymptotic behaviou of the summatoy function of the numbe of divisos of shifted pimes. To fomulate this pecisely let a be a fied intege and τn denote the numbe of positive divisos of n. In 93 Titchmash [6] showed that τp a = O. He also gave the following eplicit asymptotic fomula fo this sum unde the genealized Riemann hypothesis fo Diichlet L-functions: τp a = + log + O. pp p p a p a Date: Mach Mathematics Subject Classification. Pimay N37 Seconday N25. Key wods and phases. Titchmash diviso poblem numbe fields Chebotaev conditions. This autho is cuently a PIMS Post-doctoal Fellow at the Univesity of Lethbidge.

2 2 AKSHAA VATWANI AND PENG-JIE WONG The above fomula was fist poved unconditionally by Linnik [4] via the dispesion method. Moeove by applying the celebated Bombiei- Vinogadov theoem Halbestam and Rodiguez independently gave anothe poof cf. [2]. Subsequently Fouvy [ Coollaie 2] as well as Bombiei Fiedlande and Iwaniec [3 Coollay ] have shown that fo any A >. τp a = c + c Li + O A whee c and c ae effectively computable constants depending on a the implied constant depends only on a and A and Li is the usual logaithmic integal. Vaious vaiants of the classical Titchmash diviso poblem have been studied in the liteatue. Fo instance in [] Akbay and Ghioca fomulated some genealizations of this poblem in the setting of numbe fields as well as fo abelian vaieties. Moe pecisely by intepeting the classical poblem as a question of counting pimes with cetain Atin symbols in cyclotomic etensions ove they asked whethe simila asymptotic behaviou can be deived if one eplaces cyclotomic etensions by othe Galois etensions aising fom abelian vaieties. Anothe diection in which this poblem can be genealized is by esticting the sum in. to un ove specific subsequences of the pime numbes. A natual subsequence that comes to mind is that of pimes lying in a cetain aithmetic pogession. Fo instance we may conside the sum of divisos of p a with p constained to be conguent to a modulo fo some fied N. Since p a always has as a diviso fo this subsequence of pimes one is led to conside the sum p a τ. p a mod Such sums and subsequent connections to Atin s conjectue on pimitive oots wee investigated by Feli in [7 8]. Viewing this as a paticula case of the Titchmash diviso poblem in aithmetic pogessions Bombiei- Vinogadov type aveage esults fo the eo tems wee poved by Feli [7] and Fioilli [9]. They did this by applying an equidistibution esult of Bombiei-Fiedlande-Iwaniec [3] fo pimes in a fied esidue class a modulo q on aveage ove q. Even moe geneally we may conside the subsequence of pimes p b modulo whee b = and b may be distinct fom a mod. Then the Titchmash diviso poblem in aithmetic pogessions is concened with the behaviou of the sum.2 τ p a p b mod

3 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM 3 as. In this pape we study this poblem and obtain an asymptotic fomula which holds unifomly fo in a specific ange see Theoem 4.2. We also fomulate an analogue of this poblem in the setting of numbe fields by consideing Chebotaev sets of pimes. We obtain esults fo some specific numbe fields which we state in Theoem 4.. This diection of genealization has not been consideed befoe and bings to the foefont the question of whethe equidistibution estimates of the type poved by Bombiei-Fiedlande-Iwaniec [3 4 5] can be etended to pimes satisfying Chebotaev conditions. This is a deep question which begs caeful investigation and we do not delve moe into it hee. Howeve it is clea that this question has potential applications to the genealized Titchmash diviso poblem in aithmetic pogessions consideed in this pape. Finally we also discuss in Section 5 a efinement of an equidistibution estimate of Fioilli [9 Poposition 5.]. This bound was applied by Fioilli to obtain a Bombiei-Vinogadov type esult fo the Titchmash diviso poblem in aithmetic pogessions cf. [9 Theoem 2.4]. Ou efinement allows us to etend this esult to the function τ y p a whee τ y m denotes the numbe of positive divisos d of m satisfying d y. We thus obtain an analogue of Theoem 2.4 of [9] fo a modified Titchmash diviso poblem involving a tuncated diviso function. This is given by Theoem 5.3 of ou pape. 2. NOTATION In this note we will let K/ be a fied Galois etension of numbe fields with Galois goup G and absolute disciminant d K. Fo evey unamified pime p σ p denotes the Atin symbol at p. We let C denote a union of conjugacy classes of G and let P = PK C = {p P p is unamified with σ p C} be a fied Chebotaev set whee as late P denotes the set of ational pimes. We will define the diviso function τ K n with espect to K as τ K n := de=n ded K = We note that if K = then τ n is the usual diviso function τn.. 3. PRELIMINARIES We fist elaboate on the notion of the level of distibution fo Chebotaev sets. Let P be a set of pimes. We use the standad notation and π P = #{p P p } π P q a = #{p P p p a mod q}.

4 4 AKSHAA VATWANI AND PENG-JIE WONG A Chebotaev set P = PK C is said to have a level of distibution θ if thee eists a natual numbe M such that 3. ma ma y aq= π Py q a π Py φq A A q θ B qm= holds fo any A > 0. Fo the case that K = o equivalently P = P the well-known Bombiei-Vinogadov theoem assets that the above estimate holds when 0 < θ and M = and the Elliot-Halbestam conjectue pedicts that 2 the above estimate holds fo all 0 < θ <. In [5] M. R. Muty and V. K. Muty gave the following vaiant of the Bombiei-Vinogadov theoem. Theoem 3. Theoem 7.3 [5]. Let d = min H ma [G : H]χ χ whee the minimum is ove all subgoups H of G satisfying H C and AC is tue fo H i.e. all L-functions attached to all abelian twists of any non-tivial chaacte of H ae entie; while the maimum uns ove ieducible chaactes of H. Let η be defined as { d η = 2 if d 4; 2 if d < 4. Then the aveage esult 3. holds fo M = d K and 0 < θ /η. Fo any finite goup G we let cdg = {χ χ IG}. It is clea that if G is abelian then cdg = {}. By Atin ecipocity the mean estimate 3. holds fo all 0 < θ if K/ is an abelian Galois etension. 2 In light of this we pesent below a lemma. Lemma 3.2. With the same notation as above if cdg = { 3} o cdg { 2 4} then the aveage esult 3. holds fo all 0 < θ 2 with M = d K. Poof. In each case we can choose p equal to 2 o 3 such that χ is a powe of p fo evey χ IG. By [3 Theoem 6.9] G admits an abelian nomal p-complement N. In paticula G/N is a p-goup and thus G is nealy nilpotent. By a esult of the second-named autho [7 Theoem.2] Langlands ecipocity holds fo K/. Now the assumption AC follows fom this and Rankin-Selbeg theoy due to Jacquet Piatetski- Shapio and Shalika. We also emak that S 4 is of automophic type and that cds 4 = { 2 3} cf. [7 Coollay 2.8]. Thus we know that all S 3 - A 4 - and S 4 -etensions have a level of distibution /2. We will also equie the following well-known esult in ou eposition. We efe the eade to Theoem 7.3. of [6] fo the same.

5 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM 5 Theoem 3.3 The Bun-Titchmash inequality. Let a and q be copime integes and a positive eal numbe such that q θ fo some θ <. Then fo any ɛ > 0 thee eists ɛ > 0 such that fo all > ɛ. π q a d dα= 2 + ɛ φq log2/q We also ecall below an elementay estimate. 3.2 φd = C α + O whee the constant C α is given by 3.3 C α = + pp p α p α. p 4. A NUMBER FIELD ANALOGUE As seen in the discussion following Lemma 3.2 thee is a family of non-abelian etensions with level of distibution /2. We give a vaiant of the Titchmash diviso poblem fo such etensions. The following esult fo the Titchmash diviso poblem ove pimes lying in a fied Chebotaev set P oughly assets that the divisos ae equidistibuted ove the conjugacy classes of G = GalK/. Theoem 4.. Let a Z be fied. Suppose that P = PK C has a level of distibution /2. Then we have τ K p a = C log G C ad K + O p P whee the constant C adk is given by the Eule poduct 3.3. Poof. Let Then we have p P δn = τ K p a = p P { if n is a squae; 0 othewise. = 2 d a dd K = 2 d p a d p a dd K = p a mod d p P δp a + O.

6 6 AKSHAA VATWANI AND PENG-JIE WONG Let us note that if d a then the inne sum above is at most so that we may impose the condition d a = on the oute sum taking into account an eo of O. Thus τ K p a = 2 π P d a + O a< p P We split the fist sum as π P d a + d θ B dad K = d a dad K = θ B d a dad K = π P d a whee θ 0 /2] is chosen such that Theoem 3. holds. Fo the fist ange of d applying Theoem 3. gives π P d a = π P d a C Li + C Li G φd G φd d θ B dad K = θ B d /2 dad K = d θ B dad K = = C G Li d θ B dad K = θ B d /2 dad K = φd + O A d θ B dad K = Fo the second sum as log2/d in this ange of d the Bun- Titchmash inequality yields π P d a φd 2 θ log +. Thus by the estimate 3.2 we have a< σ p C pd K = τ K p a = C G C ad K + O 2 θ + log. Now the assumption that K has a level of distibution θ = /2 completes the poof. Fom the above agument it is clea that the eo is negligible if θ = /2. As mentioned in [5] it is epected that one can always pick θ = /2. Howeve achieving such a desied level of distibution is a deep poblem that is still out of each. Fo the special case when K is the cyclotomic field ζ the Chebotaev condition σ p C can be witten as a conguence condition p

7 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM 7 b mod fo some esidue class b modulo. Moeove since the disciminant of the field ζ is given by d ζ = φ/2 φ p pφ/p the condition p d ζ = that is needed to ensue that the pime p is unamified educes to the condition p =. Thus we have 4. τ ζp a = τ ζp a σ p C pd ζ = p b mod fo some fied esidue class b copime to. In this special case the esult of Theoem 4. can be made unifom fo D fo any D > 0 as shown below. This is a Siegel-Walfisz type esult fo this poblem which can also be viewed as a vaiant of Theoem.2 of Feli [7] who consideed essentially the same sum fo the case b = a. When b a we obtain pooe eo tems since this situation does not allow us to invoke the equidistibution esult of Bombiei-Fiedlande-Iwaniec [3] which plays a key ole in the poof of Theoem.2 in [7]. Theoem 4.2. Let N with > a b Z b = and D > 0. Then we have the following esult unifomly in D : τ ζp a = C a log φ + O p b mod whee C a is given as in 3.3. Poof. As done in the poof of Theoem 4. we wite τ ζp a = 2 δp a p b mod p b mod = 2 d p a d p a d= d a d= p b mod p a mod d + O. Again if a is not copime to d then the inne sum in the fist tem above is so that the contibution to the sum fom such esidue classes a is. Thus we may assume a d = hencefoth which gives p b mod τ ζp a = 2 d a da= π d c + O whee c mod d is uniquely detemined by the conguences c a mod d and c b mod. The esidue class c is now no longe fied and vaies with d in the above sum. It is this subtle distinction that hindes us fom

8 8 AKSHAA VATWANI AND PENG-JIE WONG invoking stonge equidistibution esults that ae available when c is fied cf. [3]. Instead we now ely on the Bombiei-Vinogadov theoem. We wite π d c = π d c + π d c d a da= d a B da= a B <d a da= fo some B > 0 sufficiently lage to be chosen late. As agued in the poof of Theoem 4. using the Bun-Titchmash theoem gives that the second sum above is log /. Fo the fist sum we have d a B da= π d c = Li d a B da= φd + O d a B Li π d c φd. Since D we choose B = BD sufficiently lage so that the Bombiei-Vinogadov theoem can be applied to give that the eo tem above is of the ode of / A fo any A > 0. Putting eveything togethe and using 3.2 and 3.3 we obtain p b mod This gives as equied. τ ζp a = 2Li d a B da= φd + O log = 2 Li C a log B log + O φ log + O C a φ + O log 5. ANOTHER VARIANT OF THE TITCHMARSH DIVISOR PROBLEM In [3] Bombiei Fiedlande and Iwaniec poved the following esult. Theoem 5.. Bombiei-Fiedlande-Iwaniec [3] Let a 0 λ < /0 and R < λ. Fo any A > 0 thee eists B = BA such that povided R < / B we have R a= q qa= ψ q a Λa aaλ φq A. A moe pecise vaiant of this esult was obtained by Fioilli in [9]. In this section we obtain the following efinement of Fioilli s esult. In paticula by taking = in the following poposition one ecoves Poposition 5. of [9].

9 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM 9 Poposition 5.2. Fi positive eal numbes λ < and D. Let M = M 0 be an intege such that M D. Then fo any ɛ > 0 λ+ 2 +ɛ and R = R λ we have R/2< R a= q M qa= ψ q a Λa φq M.T. = O aadλ whee by setting to be the poduct of all pime divisos of we have M.T. = C a with C a = ζ2ζ3 ζ6 log M +C 2a C a p a p p 2 p + p s M sa= φs A s M + p p 2 p + and C 2 a given by C a γ log p p 2 p + + p 2 log p p p 2 p + p p a p whee γ is the Eule-Mascheoni constant. p p log p p 2 p + Poof. The poof is essentially the same as the poof of Poposition 5. of [9]. The diffeence is that we utilise the full stength of Theoem 5.. Following Fioilli we fist split the inne sum ove q as follows: = +. q M qa= q RL qa= RL <q qa= M <q qa= Choosing L = A+B+D+4 with B = BA as in Theoem 5. the fist tem is contolled by Theoem 5. to give ψ q a Λa φq aadλ. A R/2< R a= P <q qa= q RL qa= Fo the emaining sums we wite ψ q a Λa = φq P <q qa= P <q qa= a <n n a mod q φq Λn

10 0 AKSHAA VATWANI AND PENG-JIE WONG whee P 2L will be eithe M o RL. By Lemma 4.3 of [9] the second sum above equals C a ωa P log log P + O 3. Also by Lemma 5.3 of [0] we can fist emove the pime powes fo the fist tem on the ight and employ Hooley s vaiant of the diviso switching technique. Witing p = a + qs as done in [9] we get up to an eo ɛ 2 +ɛ + Λn = log p P <q qa= whee E a = a <n n a mod q s< P ap sa= = = s< P ap sa= s< P ap sa= s< P ap sa= Upon witing q = s we have R/2< R a= s P +a< p a mod s θ s a θ s φs P s P + a s a + E a s θ s a θ + a s a s P φs P E a R ma y s P a= sa= q P R τq ma y θy s a y φs θy q a y φq.. Fom the Cauchy-Schwaz inequality and the tivial estimate fo θ q a it follows that the above epession is q P R /2 τq 2 q + q P R ma y θy q a y /2 φq. Since λ+ 2 +ɛ with ɛ > 0 we can use the Bombiei-Vinogadov theoem to bound the second tem in paenthesis by A / A fo any A > 0. Elementay estimates yield that the fist tem in paenthesis is of the ode O. Thus we have E a A / A fo any

11 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM A > 0. Putting eveything togethe we see the main tem fo now is C a log RL RL + O 3 ωa log + s φs RL/ C a ωa M log log M + O 3 Fom Lemma 4.3 in [9] we have that φs equals C a RL s RL sa= log RL + C 2a C a RL + O s RL sa= + s M sa= s RL/ + O 3 ωa log RL s. φs M 3 log RL ωa RL Simplifying the epession fo the main tem above gives C a log M + C 2a C a φs as equied. s M sa= s M Let τ y n denote the numbe of positive divisos d of n with d y. Clealy when y n this is the usual diviso function. To conclude this section we will use the equidistibution estimate poved above to etend Theoem 2.4 of [9] to the Titchmash diviso poblem with the diviso function τn eplaced by τ y n whee y is a paamete depending on the modulus. The following theoem can be thought of as a Bombiei-Vinogadov type theoem fo a closely elated vaiant of the Titchmash diviso poblem in aithmetic pogessions. In paticula with M = and = in the esult below we ecove Theoem 2.4 of [9]. A vesion of Theoem 2.4 of [9] was also obtained independently by Feli in [7]. Theoem 5.3. Fi positive eal numbes λ < and D. Let M = M be an 0 intege such that M D. Then fo any ɛ > 0 λ+ 2 +ɛ R = R λ and letting y = /M we have Λm + aτ y m M.T. aadλ A R a /<m / a=

12 2 AKSHAA VATWANI AND PENG-JIE WONG whee τ y m denotes the numbe of divisos of m that ae smalle than y and the main tem M.T. is given by C a +C 2 a +C a log 2 e s. φs M R/2< R a= q M qa= s M sa= Poof. Using Poposition 5.2 togethe with Lemma 4.3 of [9] and the tiangle inequality we have ψ q a Λa M.T. aadλ. A+ By a dyadic inteval consideation the whole sum ove R is aadλ. Assuming a > 0 echanging the ode of summation gives us A 5. Λn = Λn q a<n M n a mod q qa= a<n n a mod q M q n a qa= When n equals some pime p the condition p = a+mq with m a positive intege implies that q must be copime to a. Hence we may dop the condition q a = in this case. The contibution of pime powes p k with k 2 to the above sum can be estimated as follows. We have Λn τ p k a a<n n a mod n=p k k 2 q M q n a qa= ɛa 2 k a<p k p k a mod 2 k ɛa 2 /2+ɛ /k+ɛ fo any ɛ > 0. Thus the condition q a = may be dopped in 5. with the esulting eo bounded by / A when summed ove. Once this copimality condition is dopped the inne sum in 5. is then eactly the numbe of divisos of n a that ae at most /M. This completes the poof fo a > 0. The case a < 0 can be handled similaly this is as done in the poof of Theoem 2.4 in [9]. ACKNOWLEDGMENTS We thank Pofessos Ami Akbay Habiba Kadii and Nathan Ng fo helpful comments on a pevious vesion of this pape.

13 ON GENERALIZATIONS OF THE TITCHMARSH DIVISOR PROBLEM 3 REFERENCES [] A. Akbay and D. Ghioca A geometic vaiant of Titchmash diviso poblem Int. J. Numbe Theoy no [2] J. Athu and L. Clozel Simple Algebas Base Change and the Advanced Theoy of the Tace Fomula Annals of Mathematics Studies Pinceton Univesity Pess 989. [3] E. Bombiei J. B. Fiedlande and H. Iwaniec Pimes in aithmetic pogessions to lage moduli Acta Math no [4] E. Bombiei J. B. Fiedlande and H. Iwaniec Pimes in aithmetic pogessions to lage moduli II Math. Ann no [5] E. Bombiei J. B. Fiedlande and H. Iwaniec Pimes in aithmetic pogessions to lage moduli III J. Ame. Math. Soc no [6] A. Cojocau M. Ram Muty An intoduction to sieve methods and thei applications. London Mathematical Society Student Tets 66. Cambidge Univesity Pess Cambidge [7] A. T. Feli Genealizing the Titchmash diviso poblem Int. J. Numbe Theoy 8 no [8] A. T. Feli Vaiations on Atin s pimitive oot conjectue Ph.D. thesis ueen s Univesity Kingston Ontaio 20. [9] D. Fioilli On a Theoem of Bombiei Fiedlande and Iwaniec Canad. J. Math [0] D. FioilliResidue classes containing an unepected numbe of pimes Duke Math. J. Vol 6 no [] Étienne Fouvy Su le poblème des diviseus de Titchmash J. Reine Agnew. Math [2] H. Halbestam Footnote to the Titchmash-Linnik diviso poblem Poc. Ame. Math. Soc [3] I. M. Isaacs Chaacte Theoy of Finite Goups Dove New Yok 994. [4] Ju. V. Linnik The dispesion method in binay additive poblems Tanslated by S. Schuu Ame. Math. Soc. Povidence Rhode Island 963. [5] M. Ram Muty and V. Kuma Muty A vaiant of the Bombiei-Vinogadov theoem Numbe theoy Monteal ue CMS Conf. Poc. 7 Ame. Math. Soc. Povidence RI 987. [6] E. C. Titchmash A diviso poblem Rend. di Palemo [7] P.-J. Wong Langlands ecipocity fo cetain Galois etensions Jounal of Numbe Theoy Vol DEPARTMENT OF PURE MATHEMATICS UNIVERSITY OF WATERLOO WATERLOO ONTARIO N2L 3G CANADA. addess: avatwani@uwateloo.ca DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF LETH- BRIDGE LETHBRIDGE ALBERTA TK 3M4 CANADA addess: pengjie.wong@uleth.ca