aa r X i v : . [ m a t h . N T ] A p r UNIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVERNUMBER FIELDS
WATARU TAKEDA
Abstract.
We consider the upper bound of Piltz divisor problem over numberfields. Piltz divisor problem is known as a generalization of the Dirichlet divisorproblem. We deal with this problem over number fields and improve the errorterm of this function for many cases. Our proof uses the estimate of exponentialsums. We also show uniform results for ideal counting function and relatively r -prime lattice points as one of applications. Introduction
The behavior of arithmetic functions has long been studied and it is one of themost important research in analytic number theory. But many arithmetic functions f ( n ) fluctuate as n increases and it becomes difficult to deal with them. Thus manyauthors study partial sums P n ≤ x f ( n ) to obtain some information about arithmeticfunctions f ( n ). In this paper we consider Piltz divisor function I mK ( x ) over numberfield. Let K be a number field with extension degree [ K : Q ] = n and let O K beits ring of integers. Let D K be absolute value of the discriminant of K . Then Piltzdivisor function I mK ( x ) counts the number of m -tuples of ideals ( a , a , . . . , a m ) suchthat product of their ideal norm Na · · · Na m ≤ x . It is known that(1.1) I mK ( x ) ∼ Res s =1 (cid:18) ζ K ( s ) m x s s (cid:19) . We denote ∆ mK ( x ) be the error term of I mK ( x ), that is, I mK ( x ) − Res s =1 (cid:0) ζ K ( s ) m x s s (cid:1) .The case of m = 1 this function is the ordinary ideal counting function over K .For simplicity we substitute I K ( x ) and ∆ K ( x ) for I K ( x ) and ∆ K ( x ) respectively.There are many results about I K ( x ) from 1900’s. In the case K = Q , integer idealsof Z and positive integers are in one-to-one correspondence, so I Q ( x ) = [ x ], where[ · ] is the Gauss symbol. For the general case, the best estimate of ∆ K ( x ) hithertois the following theorem: Theorem 1.2.
The following estimates hold. For all ε > Mathematics Subject Classification.
Key words and phrases. ideal counting function, exponential sum, Piltz divisor problem. n = [ K : Q ] ∆ K ( x )2 O (cid:16) x (log x ) (cid:17) Huxley. [Hu00]3 O (cid:16) x + ε (cid:17) M¨uller. [M¨u88]4 O (cid:16) x + ε (cid:17) Bordell`es. [Bo15]5 ≤ n ≤ O (cid:16) x − n +1 + ε (cid:17) Bordell`es. [Bo15]11 ≤ n O (cid:16) x − n +6 + ε (cid:17) Lao. [La10]There are also many results about I m Q from 1800’s. In 1849 Dirichlet shows that I Q ( x ) = x log x + (2 γ − x + O (cid:16) x (cid:17) , where γ is the Euler constant, defined by the equation γ = lim n →∞ n X k =1 k − log n ! . The O -term is improved by many researchers many times, the best estimate hithertois x + ε [BW17].As we have mentioned above, there exists many results about other divisor prob-lems but it seems that there are not many results about piltz divisor problem overnumber fields. In 1993, Nowak shows the following theorem: Theorem 1.3 (Nowak [No93]) . When n = [ K : Q ] ≥ , then we get ∆ mK ( x ) = O K (cid:16) x − mn + mn (5 mn +2) (log x ) m − − m − n +2 (cid:17) for ≤ mn ≤ ,O K (cid:16) x − mn + m n (log x ) m − − m − mn (cid:17) for mn ≥ . For the estimate of lower bound, Girstmair, K¨uhleitner, M¨uller and Nowak ob-tain the following Ω-results:
Theorem 1.4 (Girstmair, K¨uhleitner, M¨uller and Nowak [GKMN05]) . For anyfixed number field K with n = [ K : Q ] ≥ mK ( x ) = Ω (cid:16) x − mn (log x ) − mn (log log x ) κ (log log log x ) − λ (cid:17) , where κ and λ are constants depending on K . To be more precise, let K gal be theGalois closure of K/ Q , G = Gal (cid:0) K gal / Q (cid:1) its Galois group and H = Gal (cid:0) K gal /K (cid:1) the subgroup of G corresponding to K . Then κ = mn + 12 mn n X ν =1 δ ν ν mnmn +1 − ! and λ = mn + 14 mn R + mn − mn , where δ ν = |{ τ ∈ G | |{ σ ∈ G | τ ∈ σHσ − }| = ν | H |}|| G | and R is the number of ≤ ν ≤ n with δ ν > . We know the following conditional result:If we assume the Lindel¨of hypothsis for Dedekind zeta function, it holds that forall ε >
0, for all K and for all m (1.6) ∆ mK ( x ) = O ε (cid:16) x + ε D εK (cid:17) . NIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS 3
In this paper we estimate the error term of ∆ mK ( x ) by using exponential sums. In[No93] and [GKMN05], they use other approaches, so we expect new developmentfor the Piltz divisor problem over number field. As a results, we improve theestimate of upper bound of ∆ mK ( x ) for many K and many m .In Section 2, we show some auxiliary theorems to consider the upper boundof the error term ∆ mK ( x ). First we give a review of the convexity bound for theDedekind zeta function and generalized Atkinson’s Lemma [At41]. Next we showproposition 2.6, which reduces an ideal counting problem to an exponential sumsproblem. This proposition plays a crucial role in our computing ∆ mK ( x ).In Section 3, we prove the following theorem about the error term ∆ mK ( x ) byusing estimate of exponential sums. Theorem 1.7.
For every ε > the following estimates hold. When mn ≥ , then ∆ mK ( x ) = O n,m,ε (cid:18) x mn − mn +1 + ε D m mn +1 + εK (cid:19) . This theorem gives improvement of upper bound of ∆ mK ( x ) for mn ≥ r -prime lattice points over number fields asa corollary of the first application.In Section 5, we consider a conjecture about estimates for Piltz divisor functionsover number field. It is proposed that for all number fields K and for all m thebest upper bound of the error term is better than that on the assumption of theLindel¨of Hypothesis (1.6). If mn ≤ Auxiliary Theorem
In this section, we show some important lemmas for our argument. Let s = σ + it and n = [ K : Q ]. We use the convexity bound of Dedekind zeta function to obtainan upper bound of the error term of Piltz divisor function ∆ mK ( x ).It is well-known fact that Dedekind zeta function satisfies the following functionalequation:(2.1) ζ K (1 − s ) = D s − K n (1 − s ) π − ns Γ( s ) n (cid:16) cos πs (cid:17) r + r (cid:16) sin πs (cid:17) r ζ K ( s ) , where r is the number of real embeddings of K and r is the number of pairs ofcomplex embeddings,The Phragmen-Lindel¨of principle and (2.1) give the well-known convexity boundof the Dedekind zeta function [Ra59]: For any ε > n = [ K : Q ](2.2) ζ K ( σ + it ) = O n,ε (cid:16) | t | n − nσ + ε D − σ + εK (cid:17) if σ ≤ ,O n,ε (cid:16) | t | n (1 − σ )2 + ε D − σ + εK (cid:17) if 0 ≤ σ ≤ ,O n,ε ( | t | ε D εK ) if 1 ≤ σ as | t | n D K → ∞ , where K runs through number fields with [ K : Q ] = n . In theprevious papers, we also use this convexity bound (2.2) to estimate the distributionof ideals. In the following sections, we show some estimate for ∆ mK ( x ) in the similarway to our previous papers. WATARU TAKEDA
Lemma 2.3 states the growth of the product of Gamma function and trigono-metric functions in the functional equation (2.1) of Dedekind zeta function.
Lemma 2.3.
Let τ ∈ { cos, sin } and n be a positive integer Γ( s ) n − s (cid:16) cos πs (cid:17) r + r (cid:16) sin πs (cid:17) r = Cn − ns Γ (cid:18) ns − n + 12 (cid:19) τ (cid:16) nπs (cid:17) + O n (cid:0) | t | − nσ − n (cid:1) , where C is a constant and s = σ + it .Proof. This lemma is shown from the Stirling formula and estimate for trigonomet-ric function. (cid:3)
Next we introduce the generalized Atkinson’s lemma. This lemma is quite usefulfor calculating integrals of the Dedekind zeta function.
Lemma 2.4 (Atkinson [At41]) . Let y > , < A ≤ B and τ ∈ { cos , sin } , andwe define I = 12 πi Z A + iBA − iB Γ( s ) τ (cid:16) πs (cid:17) y − s ds. If y ≤ B , then I = τ ( y ) + O y − min (cid:18) log By (cid:19) − , B ! + y − A B A − + y − ! . If y > B , then I = O (cid:18) y − A (cid:18) B A − min (cid:18)(cid:16) log yB (cid:17) − , B (cid:19) + A A − (cid:19)(cid:19) . Finally we introduce the following lemma to reduce the ideal counting problemto an exponential sum problem.
Lemma 2.5 (Bordell`es [Bo15]) . Let ≤ L ≤ R be a real number and f be anarithmetical function satisfying f ( m ) = O ( m ε ) , and let e ( x ) = exp (2 πix ) and F = f ∗ µ , where ∗ is the Dirichlet product symbol. For a ∈ R − { } , b, x ∈ R andfor every ε > the following estimate holds. X m ≤ R f ( m ) m a τ (cid:0) πxm b (cid:1) = O n,ε L − a + R ε max L
Proposition 2.6.
Let F K = I mK ∗ µ . For every ε > the following estimate holds. ∆ mK ( x )= O n,m,ε L − α + x mn − mn D n K R ε max L ≤ S ≤ R S − mn +12 mn ×× max S I mK ( x ) = X l ≤ x d mK ( l ) . Thus Perron’s formula plays a crucial role in this proof.We consider the integral 12 πi Z C ζ K ( s ) m x s s ds, where C is the contour C ∪ C ∪ C ∪ C shown in the following Figure 1. ✲✻ ✲ ✻✛ O ℜ ( s ) ℑ ( s ) r ❄ iT − iT − ε εC C C C Figure 1.
In a way similar to the well-known proof of Perron’s formula, we estimate(2.8) 12 πi Z C ζ K ( s ) m x s s ds = I mK ( x ) + O ε (cid:18) x ε T (cid:19) . We can select the large T , so that the O -term in the right hand side is sufficientlysmall. For estimating the left hand side by using estimate (2.2), we divide it intothe integrals over C , C and C . WATARU TAKEDA
First we consider the integrals over C and C as (cid:12)(cid:12)(cid:12)(cid:12) πi Z C ∪ C ζ K ( s ) m x s s ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z ε − ε | ζ K ( σ + iT ) | m x σ T dσ + 12 π Z ε − ε | ζ K ( σ − iT ) | m x σ T dσ.
It holds by the convexity bound of Dedekind zeta function (2.2) that their sum isestimated as (cid:12)(cid:12)(cid:12)(cid:12) πi Z C ∪ C ζ K ( s ) m x s s ds (cid:12)(cid:12)(cid:12)(cid:12) = O n,m,ε (cid:18)Z ε − ε ( T mn D mK ) − σ + ε x σ T dσ (cid:19) (2.9) = O n,m,ε (cid:18) x ε D εK T − ε + T mn − ε D m + εK x − ε (cid:19) . By the Cauchy residue theorem, (2.8) and (2.9) we obtain(2.10) ∆ mK ( x ) = Z C ζ K ( s ) m x s s ds + O n,m,ε (cid:18) x ε D εK T − ε + T mn − ε D m + εK x − ε (cid:19) . Thus it suffices to consider the integral over C as12 πi Z C ζ K ( s ) m x s s ds = 12 πi Z − ε + iT − ε − iT ζ K ( s ) m x s s ds. Changing the variable s to 1 − s , we have12 πi Z C ζ K ( s ) m x s s ds = 12 πi Z ε + iT ε − iT ζ K (1 − s ) m x − s − s ds. From this functional equation (2.1), it holds that12 πi Z C ζ K ( s ) m x s s ds = 12 πi Z ε + iT ε − iT (cid:18) D s − K n (1 − s ) π − ns Γ( s ) n (cid:16) cos πs (cid:17) r + r (cid:16) sin πs (cid:17) r ζ K ( s ) (cid:19) m x − s − s ds. By lemma 2.3 the integral over C can be expressed as12 πi Z C ζ K ( s ) x s s ds = Cx πi Z ε + iT ε − iT D − m K (cid:18) (2 n ) mn π mn xD mK (cid:19) − s Γ (cid:18) mns − mn + 12 (cid:19) τ (cid:16) mnπs (cid:17) ζ K ( s ) ds + O n,m,ε (cid:16) D m + εK T mn − ε x − ε (cid:17) . NIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS 7
Changing the variable mns − mn +12 to s , we have12 πi Z C ζ K ( s ) x s s ds = Cx mn − mn D n K πi Z mn − + mnε + mniT mn − + mnε − mniT mnπ (cid:18) xD mK (cid:19) mn ! − s Γ( s ) τ (cid:18) πs mn + 1) π (cid:19) × ζ K (cid:18) smn + mn + 12 mn (cid:19) ds + O n,m,ε (cid:16) D m + εK T mn − ε x − ε (cid:17) . From (2.7) the function ζ K ( s ) m can be expressed as a Dirichlet series. It is abso-lutely and uniformly convergent on compact subsets on ℜ ( s ) >
1. Therefore wecan interchange the order of summation and integral. Thus we obtain Z mnπ (cid:18) xD mK (cid:19) mn ! − s Γ( s ) τ (cid:18) πs mn + 1) π (cid:19) ζ K (cid:18) smn + mn + 12 mn (cid:19) ds = ∞ X l =1 d mK ( l ) l mn +12 mn Z mnπ (cid:18) lxD mK (cid:19) mn ! − s Γ( s ) τ (cid:18) πs mn + 1) π (cid:19) ds, where the integration is on the vertical line from mn − + mnε − mniT to mn − + mnε + mniT . Properties of trigonometric function lead to τ (cid:18) πs mn + 1) π (cid:19) = ± (cid:26) τ (cid:0) πs (cid:1) if mn is odd , √ (cid:0) τ (cid:0) πs (cid:1) ± τ (cid:0) πs (cid:1)(cid:1) if mn is even,where { τ, τ } = { sin , cos } . Hence it holds that12 πi Z C ζ K ( s ) m x s s ds = Cx mn − mn D n K πi ∞ X l =1 d mK ( l ) l mn +12 mn Z mn − + mnε + mniT mn − + mnε − mniT mnπ (cid:18) lxD mK (cid:19) mn ! − s Γ( s ) τ (cid:16) πs (cid:17) ds + O n,m,ε (cid:16) D m + εK T mn − ε x − ε (cid:17) . WATARU TAKEDA
Now we apply lemma 2.4 to this integral with y = 2 mnπ (cid:16) lxD mK (cid:17) mn , A = mn − + mnε, B = mnT and T = 2 π (cid:16) xRD mK (cid:17) mn , this becomes12 πi Z C ζ K ( s ) m x s s ds = Cx mn − mn D n K πi X l ≤ R d mK ( l ) l mn +12 mn τ mnπ (cid:18) lxD mK (cid:19) mn ! + O n,m,ε x mn − mn D n K X l ≤ R d mK ( l ) l mn +22 mn min ((cid:18) log Rl (cid:19) − , (cid:18) RxD mK (cid:19) mn ) + O n,m,ε x mn − mn D n K X l ≤ R d mK ( l ) l mn +22 mn (cid:18) Rl (cid:19) mn − mn + 1 ! + O n,m,ε x mn − mn D n K R mn − mn + ε X l>R d mK ( l ) l ε min ((cid:18) log lR (cid:19) − , (cid:18) RxD mK (cid:19) mn )! + O n,m,ε (cid:16) x mn − mn + ε D n + εK R mn − mn + ε (cid:17) . We evaluate three O -terms as follows.First we consider the first O -term. One can estimate (cid:0) log Rl (cid:1) − = O (cid:16) RR − l (cid:17) , so weobtain O n,m,ε x mn − mn D n K X l ≤ R d mK ( l ) l mn +22 mn min ((cid:18) log Rl (cid:19) − , (cid:18) RxD mK (cid:19) mn ) = O n,m,ε x mn − mn D n K X l ≤ [ R ] − d mK ( l ) l mn +22 mn (cid:18) log Rl (cid:19) − + x mn − mn D n K X [ R ] ≤ l ≤ R d mK ( l ) l mn +22 mn (cid:18) RxD mK (cid:19) mn = O n,m,ε x mn − mn D n K X l ≤ [ R ] − d mK ( l ) l mn +22 mn RR − l + x mn − mn D n K R mn X [ R ] ≤ l ≤ R d mK ( l ) l mn +22 mn = O n,m,ε (cid:16) x mn − mn D n K R mn − mn + ε + x mn − mn D n K R − mn +12 mn (cid:17) . Next we calculate the second O -term. O n,m,ε x mn − mn D n K X l ≤ R d mK ( l ) l mn +22 mn (cid:18) Rl (cid:19) mn − mn + 1 ! = O n,m,ε x mn − mn D n K R mn − mn X l ≤ R d mK ( l ) l . NIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS 9
Since it is well-known that d mK ( l ) = O ( l ε ), we get O n,m,ε x mn − mn D n K X l ≤ R d mK ( l ) l mn +22 mn (cid:18) Rl (cid:19) mn − mn + 1 ! = O n,m,ε x mn − mn D n K R mn − mn Z R t ε t dt ! = O n,m,ε (cid:16) x mn − mn D n K R mn − mn + ε (cid:17) . Finally we estimate the third O -term in a similar way to calculate the first O -term.One can estimate (cid:0) log lR (cid:1) − = O (cid:16) Rl − R (cid:17) , so we obtain O n,m,ε x mn − mn D n K R mn − mn + ε X l>R d mK ( l ) l ε min ((cid:18) log lR (cid:19) − , (cid:18) RxD mK (cid:19) mn )! = O n,m,ε x mn − mn D n K R mn − mn + ε X R Let S K ( x, S ) be the sum in the O -term, that is,max L ≤ S ≤ R S − mn +12 mn max S Estimate of counting function In the last section, we show that the error term of Piltz divisor function ∆ mK ( x )can be expressed as a exponential sum. Let X > ≤ M Lemma 3.2 (Wu [Wu98]) . Let α, β ∈ R such that αβ ( α − β − = 0 , and | a m | ≤ and | b n | ≤ and L = log ( XM N + 2) . Then L − S = O (cid:18) ( XM N ) + ( X M N ) + ( XM N ) + M N + ( X − M N ) + X − M N (cid:19) . Next Bordell`es also shows this lemma by using estimate for triple exponentialsums by Robert and Sargos. Lemma 3.3 (Bordell`es [Bo15]) . Let α, β ∈ R such that αβ ( α − β − = 0 , and | a m | ≤ and | b n | ≤ . If X = O ( M ) then ( M N ) − ε S = O (cid:16) ( XM N ) + N ( X − M ) + ( X − M N ) + M N + X − M N (cid:17) . The following Srinivasan’s result is important for our estimating ∆ mK ( x ). Lemma 3.4 (Srinivasan [Sr62]) . Let N and P be positive integers and u n ≥ , v p > , A n and B p denote constants for ≤ n ≤ N and ≤ p ≤ P . Then thereexists q with properties Q ≤ q ≤ Q and N X n =1 A n q u n + P X p =1 B p q − v p = O N X n =1 P X p =1 un + vp q A v p n B u n p + N X n =1 A n Q u n + P X p =1 B p Q − v p ! . The constant involved in O -symbol is less than N + P . Srinivasan remarks that the inequality in lemma 3.4 corresponds to the ‘bestpossible’ choice of q in the range Q ≤ q ≤ Q [Sr62]. We apply lemma 3.4 toimprove the error term ∆ mK ( x ). NIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS 11 Theorem 3.5. For every ε > the following estimates hold. When mn ≥ , then ∆ mK ( x ) = O n,m,ε (cid:18) x mn − mn +1 + ε D m mn +1 + εK (cid:19) as x tends to infinity.Proof. We note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X M 0. Let 0 ≤ α ≤ , weconsider four cases: Case 1. S α ≪ N ≪ S Case 2. S ≪ N ≪ S − α Case 3. S − α ≪ N Case 4. N ≪ S α When S α ≪ N ≪ S , we apply lemma 3.2 and this gives S − ε x mn − mn D n K S K ( x, S )(3.6) = O n,m,ε x mn − mn D n K R mn − mn + x mn − mn D n K R mn − mn + x mn − mn D n K R mn − mn + x mn − mn D n K R mn − mn − α + x mn − mn D n K R mn − mn + x mn − mn D n K R mn − mn . When S ≪ N ≪ S − α we use lemma 3.2 again reversing the role of M and N .We obtain the same estimate for the case that S α ≪ N ≪ S .For the case 3, we use lemma 3.3 S − ε x mn − mn D n K S K ( x, S )(3.7) = O n,m,ε x mn − mn D n K R mn − mn + α + x mn − mn D n K R mn − mn + α + x mn − mn D n K R mn − mn − α + x mn − mn D n K R mn − mn + α + x mn − mn D n K R mn − mn . If x mn (1 − α ) − D − mmn (1 − α ) − K ≪ S , the condition of Lemma 3.3 X = O ( N ) is satisfied.Therefore it suffices to choose L = x mn (1 − α ) − D − mmn (1 − α ) − K . For the case 4, we useLemma 3.3 again reversing the role of M and N . We obtain the same estimate for the case that N ≪ S α . Combining (3.6) and (3.7) with proposition 2.6, we obtain(3.8)∆ mK ( x ) = O n,m,ε x mn − mn D n K R mn − mn + ε + x mn − mn D n K R mn − mn + ε + x mn − mn D n K R mn − mn + ε + x mn − mn D n K R mn − mn − α + ε + x mn − mn D n K R mn − mn + ε + x mn − mn D n K R mn − mn + ε + x mn − mn D n K R mn − mn + α + ε + x mn − mn D n K R mn − mn + α + ε + x mn − mn D n K R mn − mn + α + ε + x mn − mn D n K R mn − mn + ε + x mn − mn + ε D n + εK R − mn + ε + x − αmn (1 − α ) − D − m (1 − α ) mn (1 − α ) − K . By lemma 3.4 with x mn (1 − α ) − D − mmn (1 − α ) − K ≤ R ≤ xD there exists R such that theerror term of estimate (3.8) is much less than x mn mn +7 + ε D m mn +7 + εK + x mn +35 mn +24 + ε D m mn +24 + εK + x mn − mn +13 + ε D m mn +13 + εK + x (1 − α ) mn + α − − α ) mn +1 + ε D (1 − α ) m (1 − α ) mn +1 + εK + x mn − mn +20 + ε D m mn +4 + εK + x mn − mn + ε D n + εK + x (2 α +1) mn − α (2 α +1) mn +5 + ε D (2 α +1) m (2 α +1) mn +5 + εK + x ( α +5) mn − α − α +5) mn +4 + ε D ( α +5) m ( α +5) mn +4 + εK + x (2 α +9) mn − α − α +9) mn +9 + ε D (2 α +9) m (2 α +9) mn +9 + εK + x mn − mn +1 + ε D m mn +1 + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x + ε + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x mn (1 − α ) − α mn (1 − α ) − + ε D m − mα mn (1 − α ) − + εK + x − αmn (1 − α ) − D − m (1 − α ) mn (1 − α ) − K . When mn ≥ α = mn +37 mn − , then we have∆ mK ( x ) = O n,m,ε (cid:18) x mn − mn +1 + ε D m mn +1 + εK (cid:19) . This proves the theorem. (cid:3) For mn ≥ K with [ K : Q ] = 4 then we improve the estimate for∆ K ( x ) as follows: Corollary 3.9. For any number field K with [ K : Q ] = 4 , ∆ K ( x ) = O K,ε (cid:16) x + ε (cid:17) . This result is better than Bordell`es’ result.4. Application In this section we introduce some applications of our theorems. First we obtainuniform estimate for ideal counting function I K ( x ). From the proof of theorem 3.5,we obtain the following theorem. NIFORM BOUNDS OF PILTZ DIVISOR PROBLEM OVER NUMBER FIELDS 13 Theorem 4.1. For all ε > for any fixed ≤ β ≤ n +5 − ε and C > thefollowings hold. If K runs through number fields with [ K : Q ] ≤ n and D K ≤ Cx β then ∆ K ( x ) = O C,n,ε (cid:16) x n − β n +1 + ε (cid:17) . The condition D K ≤ Cx β is caused by the relation between the principalterm and the error term. It is well known that I K ( x ) is very important to es-timate the distribution of relatively r -prime lattice points. We regard an ℓ -tuple ofideals ( a , a , . . . , a ℓ ) of O K as a lattice point in K ℓ . We say that a lattice point( a , a , . . . , a ℓ ) is relatively r -prime for a positive integer r , if there exists no primeideal p such that a , a , . . . , a ℓ ⊂ p r . Let V rℓ ( x, K ) denote the number of relatively r -prime lattice points ( a , a , . . . , a ℓ ) such that their ideal norm Na i ≤ x .B. D. Sittinger shows that V rℓ ( x, K ) ∼ ρ ℓK ζ K ( rℓ ) x ℓ , where ρ K is the residue of ζ K as s = 1 [Si10]. It is well known that(4.2) ρ K = 2 r (2 π ) r h K R K w K √ D K , where h K is the class number of K , R K is the regulator of K and w K is the numberof roots of unity in O ∗ K .After that we show some results for the error term: E rℓ ( x, K ) = V rℓ ( x, K ) − ρ ℓK ζ K ( rℓ ) x ℓ . In [Ta17] and [TK17] we consider the relation between relatively r -prime problemand other mathematical problems. If we assume the Lindel¨of Hypothesis for ζ K ( s ),then it holds that for all ε > E rℓ ( x, K ) = O ε (cid:16) x r ( + ε ) (cid:17) if rℓ = 2 ,O ε (cid:16) x ℓ − + ε (cid:17) otherwiseFrom easy calculation, we obtain the following corollary. Corollary 4.4. For all ε > and for any fixed ≤ β ≤ n +5 − ε and C > thefollowings hold. If K runs through number fields with [ K : Q ] ≤ n and D K ≤ Cx β ,then E rℓ ( x, K ) = O C,n,ε (cid:16) x n − r (2 n +1) + n +1 β + ε (cid:17) if rℓ = 2 ,O C,n,ε (cid:16) x ℓ − n +1 + n +5 − (2 n +1) ℓ n +1) β + ε (cid:17) otherwise. For the proof of this corollary, please see the proof of Theorem 4.1 of [TK17].5. Conjecture Theorem 4.1 states good uniform upper bounds. It is proposed that for allnumber fields K the best uniform upper bound of the error term is better thanthat on the assumption of the Lindel¨of Hypothesis (1.6). Conjecture 5.1. If K runs through number fields with D K < x , then ∆ mK ( x ) = o (cid:16) x (cid:17) . If K runs through cubic extension fields with D K ≤ Cx − ε , then this conjectureholds from theorem 4.1.From estimate (1.5), this conjecture may give the best estimate for uniformupper bound of ∆ mK ( x ). As we remarked above (Theorem 1.2) this conjecture isvery difficult even when K is fixed and m = 1. References [At41] F. V. Atkinson. A divisor problem. The Quarterly Journal of Mathematics, 1: 193–200.1941.[Bo15] O. Bordell`es, On the ideal theorem for number fields, Functiones et Approximatio 53(1):31–45. 2015.[BW17] J. Bourgain and N. Watt, Mean square of zeta function, circle problem and divisor prob-lem revisited, Preprint 2017, 23 pp. https://arxiv.org/abs/1709.04340.[GKMN05] K. Girstmair, M. K¨uhleitner, W. M¨uller, and W. G. Nowak, The Piltz divisor problemin number fields: An improved lower bound by Soundararajan’s method, Acta Arithmetica,117: 187-206. 2005.[Hu00] M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, NumberTheory for the Millennium, II: 275-290, 2002.[La10] H. Lao. On the distribution of integral ideals and Hecke Gr¨ossencharacters. Chinese Annalsof Mathematics, Series B, 31(3): 385–392. 2010.[M¨u88] W. M¨uller. On the distribution of ideals in cubic number fields. Monatshefte f¨ur Mathe-matik, 106(3): 211–219. 1988.[No93] W. G. Nowak. On the distribution of integral ideals in algebraic number theory fields,Math. Nachr. 161: 59-74. 1993.[Ra59] H. Rademacher, On the Phragm´en–Lindel¨of theorem and some applications, Math. Z. 72:192–204. 1959.[Si10] B. D. Sittinger. The probability that random algebraic integers are relatively r-prime. Jour-nal of Number Theory, 130(1): 164-171. 2010.[Sr62] B. R. Srinivasan. On Van der Corput’s and Nieland’s results on the Dirichlet’s divisorproblem and the circle problem. Proc Natl Inst Sci India, Part A. 28: 732–742. 1962.[Ta17] W. Takeda. Visible lattice points and the Extended Lindel¨of Hypothesis. Journal of Num-ber Theory, 180: 297–309. 2017.[TK17] W. Takeda and S. Koyama. Estimates of lattice points in the discriminant aspect overabelian extension fields. Forum Mathematicum, 30(3), 767–773. 2018.[Wu98] J. Wu, On the average number of unitary factors of finite abelian groups, Acta Arith-metica. 84:17-29. 1998. Department of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan E-mail address ::