On the divisor problem with congruence conditions
aa r X i v : . [ m a t h . N T ] A ug ON THE DIVISOR PROBLEM WITH CONGRUENCECONDITIONS
LIRUI JIA, WENGUANG ZHAI, AND TIANXIN CAI
Abstract.
Let d ( n ; r , q , r , q ) be the number of factorization n = n n satisfying n i ≡ r i (mod q i ) ( i = 1 ,
2) and ∆( x ; r , q , r , q ) be the errorterm of the summatory function of d ( n ; r , q , r , q ) with x ≥ ( q q ) ε , ≤ r i ≤ q i , and ( r i , q i ) = 1 ( i = 1 , x ; r , q , r , q ), and prove that for a sufficiently large constant C ,∆( q q x ; r , q , r , q ) changes sign in the interval [ T, T + C √ T ] for any large T . Meanwhile, we show that for a small constant c ′ , there exist infinitely manysubintervals of length c ′ √ T log − T in [ T, T ] where ± ∆( q q x ; r , q , r , q ) >c x always holds. Introduction
Dirichlet divisor problem.
Let d ( n ) be the Dirichlet divisor function, D ( x ) = P n ≤ x d ( n ) = P n n ≤ x D ( x ) = x log x + (2 γ − x + O ( √ x ) , where γ is the Euler constant.Let ∆( x ) = D ( x ) − x log x − (2 γ − x be the error term in the asymptotic formula for D ( x ). Dirichlet’s divisor problemconsists of determining the smallest α , for which ∆( x ) ≪ x α + ε holds for any ε >
0. Clearly, Dirichlet’s result implies that α ≤ . Since then, there are manyimprovements on this estimate. The best to-date is given by Huxley [5, 6], reads(1.1) ∆( x ) ≪ x log x. It is widely conjectured that α = is admissible and is the best possible.Since ∆( x ) exhibits considerable fluctuations, one natural way to study the upperbounds is to consider the moments.In 1904, Voronoi [17] showed that Z T ∆( x ) dx = T O ( T ) . Mathematics Subject Classification.
Key words and phrases.
Divisor problem, sign change, congruence conditions.The first author is supported by the National Natural Science Foundation of China (GrantNo. 11871295 and Grant No. 11571303), China Postdoctoral Science Foundation (Grant No.2018M631434). The second author is supported by the National Key Basic Research Programof China (Grant No. 2013CB834201). The third author is supported by the National NaturalScience Foundation of China (Grant No. 11571303).
Later, in 1922 Cram´er [1] proved the mean square formula Z T ∆( x ) dx = cT + O ( T + ε ) , ∀ ε > , where c is a positive constant. In 1983, Ivic [7] used the method of large values toprove that(1.2) Z T | ∆( x ) | A dx ≪ T A + ε , ∀ ε > ≤ A ≤ . The range of A can be extended to by the estimate(1.1). In 1992, Tsang [15] obtained the asymptotic formula(1.3) Z T ∆( x ) k dx = c k T k + O ( T k − δ k ) , for k = 3 , , with positive constants c , c , and δ = , δ = . Ivi´c and Sargos [8] improvedthe values δ , δ to δ ′ = , δ ′ = , respectively. Heath-Brown [3] in 1992 provedthat for any positive real number k < A , where A satisfies (1.2), the limit c k = lim X →∞ X − − k Z X ∆( x ) k dx exists. Then, there followed a series of investigations on explicit asymptotic formulaof the type (1.3) for larger values of k . In 2004, Zhai [18] established asymptoticformulas for 3 ≤ k ≤ x ) = − π √ x ∞ X n =1 d ( n ) √ n (cid:0) K (4 π √ nx ) + π Y (4 π √ nx ) (cid:1) , where K , Y are the Bessel functions, and the series on the right-hand side isboundedly convergent for x lying in each fixed closed interval.Heath-Brown and Tsang [4] studied the sign changes of ∆( x ). They proved thatfor a suitable constant C >
0, ∆( x ) changes sign on the interval [ T, T + C √ T ]for every sufficiently large T . Here the length √ T is almost best possible sincethey proved that in the interval [ T, T ] there are many subintervals of length ≫√ T log − T such that ∆( x ) does not change sign in any of these subintervals.1.2. The divisor problem with congruence conditions.
A divisor functionwith congruence conditions is defined by d ( n ; r , q , r , q ) = X n = n n n i ≡ r i (mod q i ) i =1 , , of which, the summatory function is D ( x ; r , q , r , q ) = X n n ≤ xn i ≡ r i (mod q i ) i =1 , . N THE DIVISOR PROBLEM WITH CONGRUENCE CONDITIONS 3
From Richert [13], we can find that for x ≥ q q , 1 ≤ r i ≤ q i ( i = 1 , D ( x ; r , q , r , q )= xq q log (cid:16) xq q (cid:17) − (cid:18) Γ ′ Γ (cid:16) r q (cid:17) + Γ ′ Γ (cid:16) r q (cid:17) + 1 (cid:19) xq q + ∆( x ; r , q , r , q ) . From Huxley’s estimates [5], it follows that(1.5) ∆( x ; r , q , r , q ) ≪ (cid:16) xq q (cid:17) (cid:16) log (cid:16) xq q (cid:17)(cid:17) uniformly in 1 ≤ r ≤ q ≤ x, ≤ r ≤ q ≤ x . It is conjectured that(1.6) ∆( x ; r , q , r , q ) ≪ (cid:16) xq q (cid:17) + ε uniformly in 1 ≤ r ≤ q ≤ x, ≤ r ≤ q ≤ x , ∀ ε >
0, which is an analogue of thewell-known conjecture that ∆( x ) ≪ x + ε .M¨uller and Nowak [12] studied the mean value of ∆( x ; r , q , r , q ). Theypointed out(1.7) Z T ∆( x ; r , q , r , q ) dx = (cid:0) r q − (cid:1)(cid:0) r q − (cid:1) T + O (cid:0) ( q q ) T (cid:1) , and(1.8) Z T ∆ ( x ; r , q , r , q ) dx = c ( q q ) T + o (cid:0) ( q q ) T (cid:1) , uniformly in 1 ≤ r i ≤ q i ≤ T ( i = 1 , T is a large number, and c is a constant.In [9], we show that(1.9) Z T | ∆( q q x ; r , q , r , q ) | A dx ≪ T A L A , for 0 ≤ A ≤ and T ≫ ( q q ) ε .Here we study ∆( x ; r , q , r , q ) further and give some more results about it. Notations . For a real number t , let [ t ] be the largest integer no greater than t , { t } = t − [ t ], ψ ( t ) = { t } − , k t k = min( { t } , 1 − { t } ), e ( t ) = e πit . C , R , Z , N denote the set of complex numbers, of real numbers, of integers, and of naturalnumbers, respectively; f ≍ g means that both f ≪ g and f ≫ g hold. Throughoutthis paper, ε denote sufficiently small positive constants, and L denotes log T .2. Main results
In this paper, we will first discuss the power moments of ∆( x ; r , q , r , q ) andget the following Theorem 2.1. If T ≫ ( q q ) ε is large enough. If A > satisfies Z T | ∆( q q x ; r , q , r , q ) | A dx ≪ T A + ε , then for any fixed integer ≤ k < A , we have (2.1) Z T ∆ k ( q q x ; r , q , r , q ) dx = C k Z T x k dx + o (cid:0) T k (cid:1) , where C k ≍ are explicit constants. LIRUI JIA, WENGUANG ZHAI, AND TIANXIN CAI
From (1.9), we can take A = , which means Corollary 2.1. If T, r i and q i ( i = 1 , satisfying the hypothesis of Theorem 2.1,then (2.1) holds for any fixed integer ≤ k ≤ . By using the estimates above, we can get the sign changes of ∆( x ; r , q , r , q )as following Theorem 2.2.
Let c > be a sufficiently small constant and c > be a suf-ficiently large constant, q ≥ , q ≥ , ≤ r i ≤ q i and ( r i , q i ) = 1 ( i = 1 , .For any real-valued function | f ( t ) | ≤ c t , the function ∆( q q t ; r , q , r , q ) + f ( t ) changes sign at least once in the interval [ T, T + c √ T ] for every sufficientlylarge T ≫ ( q q ) ε . In particular, there exist t , t ∈ [ T, T + c √ T ] such that ∆( q q t ; r , q , r , q ) ≥ c t and ∆( q q t ; r , q , r , q ) ≤ − c t . Theorem 2.3.
There exist three positive absolute constants c , c , c such that,for any large parameter T ≫ ( q q ) ε , and any choice of ± signs, there are atleast c √ T log T disjoint subintervals of length c √ T log − T in [ T, T ] , such that ± ∆( q q t ; r , q , r , q ) > c t , whenever t lies in any of these subintervals. More-over, we have the estimate meas (cid:8) t ∈ [ T, T ] : ± ∆( q q t ; r , q , r , q ) > c t (cid:9) ≫ T. We also study the Ω-result of the error term in the asymptotic formula (2.1) forodd k by using Theorem 2.3. Define F k (cid:0) q q x ; r , q , r , q (cid:1) := Z T ∆ k (cid:0) q q x ; r , q , r , q (cid:1) dx − C k T k . We have the following
Theorem 2.4.
For any T ≫ ( q q ) ε , the interval [ T, T ] contains a point X , forwhich F k (cid:0) q q X ; r , q , r , q (cid:1) ≫ X + k L − . Remark 2.1.
Although at the present moment we can only prove (2.1) for ≤ k ≤ , Theorem 2.4 holds for any odd k ≥ . proof of Theorem 2.1 In this section, we prove Theorem 2.1 by using the Voronoi-type formula for∆( x ; r , q , r , q ). Lemma 3.1. ( See [9] ) Let J = [ L +2 log q q − L log 2 ] , H ≥ be a parameter to be determined, and T ε In this section, we prove Theorem 2.2 following the approach of [4].Suppose | f ( t ) | ≤ c t . Let∆ ∗∗ ( t ) = √ πt − (cid:16) ∆( q q t ; r , q , r , q ) + f ( t ) (cid:17) , for t ≥ . Define K ζ ( u ) := (1 − | u | ) (cid:0) ζ sin(4 παu ) (cid:1) for | u | ≤ , with ζ = 1 or − 1, and α > Lemma 4.1. Suppose T ≫ ( q q ) ε is a large parameter. Then for each √ T ≤ t ≤√ T , we have Z − ∆ ∗∗ ( t + αu ) K ζ ( u ) du = − ζ (cid:18) πt − π (cid:16) r q + r q + 18 (cid:17)(cid:19) + O ( α − )+ O (cid:0) t − sup | u |≤ f (( t + αu ) ) (cid:1) + O (cid:0) t − L (cid:1) . Proof. Let J = [ L +2 log q q − L log 2 ], H ≥ T ε < y ≤ min( H , ( q q ) T ) L − . From (3.1), we have∆ ∗∗ ( t ) = R ∗ ( t ; y )+ R ∗ ( t ; y, H )+ R ∗ ( t ; y, H )+ √ πt − f ( t )(4.1) + O (cid:0) t − (cid:0) G ∗ ( t ; H )+ G ∗ ( t ; H ) (cid:1)(cid:1) + O (cid:0) t − L (cid:1) , LIRUI JIA, WENGUANG ZHAI, AND TIANXIN CAI where R ∗ ( t ; y ) = X n ≤ y n X n = hl cos (cid:18) π √ nt − π (cid:16) hr q + lr q + 18 (cid:17)(cid:19) ,R ∗ ( t ; y, H ) = X y Take H = T , y = T . Then clearly y > 1. Thus we get Z − R ∗ ( t + αu ) K ζ ( u ) du (4.4) = − ζ (cid:18) πt − π (cid:16) r q + r q + 18 (cid:17)(cid:19) + O (cid:16) X n> d ( n ) α n ( √ n − (cid:17) = − ζ (cid:18) πt − π (cid:16) r q + r q + 18 (cid:17)(cid:19) + O ( α − ) , by using P n> d ( n ) n ( √ n − ≪ 1. Noting that H = T , t ≍ T , by (4.2)-(4.4), we see Z − ∆ ∗∗ ( t + αu ) K ζ ( u ) du = − ζ (cid:18) πt − π (cid:16) r q + r q + 18 (cid:17)(cid:19) + O ( α − )+ O (cid:0) t − sup | u |≤ f (( t + αu ) ) (cid:1) + O (cid:0) t − H − T L (cid:1) + O (cid:0) t − L (cid:1) . Thus we complete the proof of Lemma 4.1 (cid:3) The mean value of ∆( q q x ; r , q , r , q ) in short intervals In this section, we need the following Lemma. Lemma 5.1. ( Hilbert’s inequality )( See e.g. [14] ) Let x < x < · · · < x n be asequence of real numbers. If there exists δ > , such that min s = r | x r − x s | ≥ δ r ≥ δ > ≤ r ≤ n ) , then there exists an absolute constant C , such that (cid:12)(cid:12)(cid:12)(cid:12) X s = r u r ¯ u s ( x r − x s ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ C X r δ r − | u r | , for arbitrary complex numbers u , u , · · · , u n . Suppose T ≫ ( q q ) ε is a large parameter, 1 ≤ h ≤ √ T . Denote ∆ ∗ ( q q x ) =∆( q q x ; r , q , r , q ). In this section we shall estimate the integral I ( T, h ) = Z T (cid:0) ∆ ∗ ( q q ( x + h ) − ∆ ∗ ( q q x ) (cid:1) dx, which would play an important role in the proof of Theorem 2.3. This type ofintegral was studied for the error term in the mean square of ζ ( + it ) by Good [2],for the error term in the Dirichlet divisor problem by Jutila [10] and for the errorterm in Weyl’s law for Heisenberg manifold by Tsang and Zhai [16]. Here we followsthe approach of Tsang and Zhai [16] and prove the following Lemma 5.2. The estimate I ( T, h ) ≪ T h log √ Th + T L holds uniformly for ≤ h ≤ √ T . N THE DIVISOR PROBLEM WITH CONGRUENCE CONDITIONS 9 Proof. Write(5.1) I ( T, h ) = Z + Z , where Z = Z 100 max( h ,T )1 (∆ ∗ ( q q ( x + h ) − ∆ ∗ ( q q x ) (cid:1) dx, Z = Z T 100 max( h ,T ) (∆ ∗ ( q q ( x + h ) − ∆ ∗ ( q q x ) (cid:1) dx. From Corollary 2.1, we see that(5.2) Z ≪ ( h + T ) ≪ T h . For R , first we estimate the integral(5.3) J ( U, h ) = Z UU (∆ ∗ ( q q ( x + h ) − ∆ ∗ ( q q x ) (cid:1) dx, 100 max( h , T ) ≤ U ≤ T. Let T = 2 U in (3.1). Then∆ ∗ ( q q x ) = R ( x ; y )+ R ( x ; y, H )+ R ( x ; y, H )+ G ( x ; H )+ G ( x ; H )+ O (cid:0) log U (cid:1) . Take H = U , y = min (cid:0) U h − , U log − U (cid:1) . From [11, Lemma 4.1 and eq.(4.11)],we see Z UU | G ( x ; H )+ G ( x ; H ) | dx ≪ U log U, Z UU | R ( x ; y, H )+ R ( x ; y, H ) | dx ≪ U y − log U. Therefor Z UU (cid:0) ∆ ∗ ( q q x ) − R ( x ; y ) (cid:1) dx ≪ U y − log U + U log U (5.4) ≪ U h log U + U log U. We now estimate R UU (cid:0) R ( x + h ; y ) − R ( x ; y ) (cid:1) dx . Set θ ( h, l ) = 2 π ( hr q + lr q ).From (3.2), we have(5.5) R ( x + h ; y ) − R ( x ; y ) = F ( x ) + F ( x ) , where F ( x ) = 1 √ π (cid:0) ( x + h ) − x (cid:1) X n ≤ y n X n = hl cos (cid:0) π p n ( x + h ) − θ ( h, l ) − π (cid:1)(cid:17) ,F ( x ) = x √ π X n ≤ y n X n = hl (cid:16) cos (cid:0) π p n ( x + h ) − θ ( h, l ) − π (cid:1) − cos (cid:0) π √ nx − θ ( h, l ) − π (cid:1)(cid:17) . From [11, Proof of Lemma 4.2], we get Z UU F ( x ) dx ≪ h U − Z UU R ( x + h ) dx ≪ h U − . (5.6)For the mean square of F ( x ), we see(5.7) F = F + F , where F ( x ) = x π X n ≤ y n × (cid:16) X n = hl cos (cid:0) π p n ( x + h ) − θ ( h, l ) − π (cid:1) − cos (cid:0) π √ nx − θ ( h, l ) − π (cid:1)(cid:17) ,F ( x ) = x π X n ,n ≤ yn = n n n ) X n = h l X n = h l × (cid:16) cos (cid:0) π p n ( x + h ) − θ ( h , l ) − π (cid:1) − cos (cid:0) π √ n x − θ ( h , l ) − π (cid:1)(cid:17) × (cid:16) cos (cid:0) π p n ( x + h ) − θ ( h , l ) − π (cid:1) − cos (cid:0) π √ n x − θ ( h , l ) − π (cid:1)(cid:17) = x π X n ,n ≤ yn = n n n ) X n = h l X n = h l X j =0 1 X j =0 ( − j + j × cos (cid:0) π p n ( x + j h ) − θ ( h , l ) − π (cid:1) cos (cid:0) π p n ( x + j h ) − θ ( h , l ) − π (cid:1) . Write F ( x ) =: F ( x ) + F ( x ) , (5.8)with F ( x ) = x π X j =0 1 X j =0 ( − j + j X n ,n ≤ yn = n n n ) X n = h l X n = h l × cos (cid:0) π p n ( x + j h ) − π p n ( x + j h ) − θ ( h − h , l − l ) (cid:1) ,F ( x ) = x π X j =0 1 X j =0 ( − j + j X n ,n ≤ yn = n n n ) X n = h l X n = h l × sin (cid:0) π p n ( x + j h ) + 4 π p n ( x + j h ) − θ ( h + h , l + l ) (cid:1) . Let g ± ( x ) = 4 π p n ( x + j h ) ± π p n ( x + j h ) − θ ( h ± h , l ± l ) . Using (1 + t ) = 1 + ∞ X v =1 d v t v (cid:0) | t | ≤ (cid:1) , N THE DIVISOR PROBLEM WITH CONGRUENCE CONDITIONS 11 with | d v | < 1, we see g ± ( x ) = 4 π √ x ( √ n ± √ n ) + 4 π ∞ X v =1 d v h v x v − ( √ n j v ± √ n j v ) − θ ( h ± h , l ± l ) . Noting that n , n ≤ y ≤ U h − , we have | g ′± ( x ) | ≫ √ x |√ n ± √ n | ( n = n ) . Then by the the first derivative test we get Z UU F ( x ) dx ≪ U X n ,n ≤ yn = n n n ) X n = h l X n = h l |√ n − √ n | = U X n ,n ≤ yn = n n n ) d ( n ) d ( n ) |√ n − √ n | , Z UU F ( x ) dx ≪ U X n ,n ≤ yn = n n n ) d ( n ) d ( n ) |√ n + √ n | . Noting P n ≤ N d ( n ) ≪ N log N , by using Lamma 5.1 and (5.8), we obtain Z UU F ( x ) dx ≪ U X n ,n ≤ yn = n n n ) d ( n ) d ( n ) |√ n − √ n | ≪ U log y. (5.9)By the elementary formulascos u − cos v = − (cid:0) u + v (cid:1) sin (cid:0) u − v (cid:1) , and sin( u − v ) = sin u cos v − cos u sin v, we have F ( x ) = 2 x π X n ≤ y n sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) (5.10) × (cid:16) X n = hl sin (cid:0) π p n ( x + h )+ 2 π √ nx − θ ( h, l ) − π (cid:1)(cid:17) , = : F + F + F , where F = 2 x π X n ≤ y n sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) × sin (cid:0) π p n ( x + h )+ 2 π √ nx (cid:1)(cid:16) X n = hl cos (cid:0) θ ( h, l )+ π (cid:1)(cid:17) , F = 2 x π X n ≤ y n sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) × cos (cid:0) π p n ( x + h )+ 2 π √ nx (cid:1)(cid:16) X n = hl sin (cid:0) θ ( h, l )+ π (cid:1)(cid:17) ,F = − x π X n ≤ y n sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) sin (cid:0) π p n ( x + h )+ 4 π √ nx (cid:1) × X n = hl sin (cid:0) θ ( h, l )+ π (cid:1) X n = h ′ l ′ cos (cid:0) θ ( h ′ , l ′ )+ π (cid:1) . It is easy to see that0 ≤ F + F ≤ x π X n ≤ y n sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) d ( n ) . By using Taylor’s expansion, we have for x ≥ h ,sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) = sin (cid:0) πh n x − + O ( h n x − ) (cid:1) = sin (cid:0) πh n x − (cid:1) + O ( h n x − ) . which suggests Z UU x sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) dx ≪ Z UU x min (cid:0) , h nx − (cid:1) + O ( h n x − ) dx ≪ (cid:26) U h n, n ≤ U h − ,U , n > U h − , in view of the fact h < U and n ≤ y < U . Hence, Z UU F + F dx ≪ h U X n ≤ Uh − d ( n ) n + U X n>Uh − d ( n ) n ≪ U h log √ Uh , (5.11)where we used the well-known estimate P n ≤ N d ( n ) ≪ N log N .By the first derivative test, we have L n ( t ) := Z tU x sin (cid:0) π p n ( x + h )+ 4 π √ nx (cid:1) dx ≪ U n − , U ≤ t ≤ U. Using the integration by parts, we obtain Z UU x sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) sin (cid:0) π p n ( x + h )+ 4 π √ nx (cid:1) dx = Z UU sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) dL n ( x )= L n (2 U ) sin (cid:0) π p n (2 U + h ) − π √ nU (cid:1) − Z UU L n ( x ) × sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) cos (cid:0) π p n ( x + h ) − π √ nx (cid:1)(cid:0) π √ n √ x + h − π √ n √ x (cid:1) dx ≪ U n − + U h , N THE DIVISOR PROBLEM WITH CONGRUENCE CONDITIONS 13 which yields Z UU F dx = − π X n ≤ y n X n = hl sin (cid:0) θ ( h, l )+ π (cid:1) X n = h ′ l ′ cos (cid:0) θ ( h ′ , l ′ )+ π (cid:1) Z UU x (5.12) × sin (cid:0) π p n ( x + h ) − π √ nx (cid:1) sin (cid:0) π p n ( x + h )+4 π √ nx (cid:1) dx ≪ X n ≤ y d ( n ) n ( U n − + U h ) ≪ U. From (5.10)-(5.12), we get(5.13) Z UU F ( x ) dx ≪ U h log √ Uh . Combining (5.7), (5.9) and (5.13), we obtain Z UU F ( x ) dx ≪ U h log √ Uh + U log y, which together with (5.5), (5.6) yields(5.14) Z UU (cid:0) R ( x + h ; y ) − R ( x ; y ) (cid:1) ( x ) dx ≪ U h log √ Uh + U log y. From (5.3), (5.4), and (5.14), it follows that J ( U, h ) ≪ U h log √ Uh + U log y, which implies(5.15) Z ≪ T h log √ Th + T L , via a splitting argument. Then Lemma 5.2 follows from (5.1), (5.2), and (5.15). (cid:3) Proof of Theorem 2.3 In this section, we will give a proof of Theorem 2.3 by following the approachof [16]. We still write ∆ ∗ ( q q x ) = ∆( q q x ; r , q , r , q ). Define∆ ∗ + ( t ) = 12 (cid:0) | ∆ ∗ ( t ) | + ∆ ∗ ( t ) (cid:1) , ∆ ∗− ( t ) = 12 (cid:0) | ∆ ∗ ( t ) | − ∆ ∗ ( t ) (cid:1) . We need the following two lemmas. Lemma 6.1. Z TT ∆ ∗ ± ( q q t ) dt ≫ T . Proof. From Corollary 2.1 with k = 2 , 4, by H¨older’s inequality, we get T ≪ Z TT ∆ ∗ ( q q t ) dt ≪ (cid:16) Z TT | ∆ ∗ ( q q t ) | dt (cid:17) (cid:16) Z TT ∆ ∗ ( q q t ) dt (cid:17) ≪ (cid:16) Z TT | ∆ ∗ ( q q t ) | dt (cid:17) T , which yields(6.1) Z TT | ∆ ∗ ( q q t ) | dt ≫ T . From (1.7), we see Z TT ∆ ∗ ( q q t ) dt ≪ T . Thus, from the definition of ∆ ∗± ( q q t ), we have Z TT ∆ ∗± ( q q t ) dt ≫ T . Then by Cauchy-Schwarz’s inequality, we get T ≪ (cid:16) Z TT dt (cid:17) (cid:16) Z TT ∆ ∗ ± ( q q t ) dt (cid:17) ≪ T (cid:16) Z TT ∆ ∗ ± ( q q t ) dt (cid:17) , which immediately implies Lemma 6.1. (cid:3) Lemma 6.2. Suppose ≤ H ≤ √ T . Then Z TT max h ≤ H (cid:0) ∆ ∗± ( q q ( t + h )) − ∆ ∗± ( q q t ) (cid:1) dt ≪ H T L . Proof. Since | ∆ ∗± ( q q ( t + h )) − ∆ ∗± ( q q t ) | ≤ | ∆ ∗ ( q q ( t + h )) − ∆ ∗ ( q q t ) | , it is sufficient to prove that I = Z TT max h ≤ H (cid:0) ∆ ∗ ( q q ( t + h )) − ∆ ∗ ( q q t ) (cid:1) dt ≪ H T L . For 0 < u < u ≪ T , it easy to see that∆ ∗ ( q q u ) − ∆ ∗ ( q q u ) ≥ − O (cid:0) ( u − u ) log T (cid:1) . Write H = 2 λ b , such that λ ∈ N and 1 ≤ b < 2. Then for each t ∈ [ T, T ], we havemax h ≤ H (cid:12)(cid:12) ∆ ∗ ( q q ( t + h )) − ∆ ∗ ( q q t ) (cid:12)(cid:12) ≪ max ≤ j ≤ λ (cid:12)(cid:12) ∆ ∗ ( q q ( t + jb )) − ∆ ∗ ( q q t ) (cid:12)(cid:12) + L . Similar to the argument of the proof of Lemma 2 of [4], by using Lemma 5.2, wewe can deduce that I ≪ λ X µ ≤ λ X ≤ ν ≤ µ Z T + ν λ − µ bT + ν λ − µ b (cid:0) ∆ ∗ ( q q ( t + 2 λ − µ b )) − ∆ ∗ ( q q t ) (cid:1) dt + T L ≪ λ X µ ≤ λ X ≤ ν ≤ µ (cid:0) λ − µ bT L + T L (cid:1) ≪ λ X µ ≤ λ (cid:0) λ bT L + 2 µ T L (cid:1) ≪ λ H T L + λH T L ≪ H T L . Thus we get Lemmma 6.2. (cid:3) N THE DIVISOR PROBLEM WITH CONGRUENCE CONDITIONS 15 Now we finish the proof of Theorem 2.3. Let P ( t ) = ∆ ∗± ( q q t ) and Q ( t ) = δt for a sufficiently small δ > 0, and ω ( t ) = P ( t ) − h ≤ H (cid:0) P ( t + h ) − P ( t ) (cid:1) − Q ( t ) . Then Z TT ω ( t ) dt ≫ T − O (cid:0) H T L (cid:1) − O (cid:0) δ T (cid:1) ≫ T , (6.2)from Lemma 6.1 and Lemma 6.2, by taking H = δT L − . For any point t , where ω ( t ) > h ∈ [0 , H ], we see that P ( t + h ) has the same sign as P ( t ),and | P ( t + h ) | > | Q ( t ) | .Let S = { t ∈ [ T, T ] : ω ( t ) > } . From Corollary 2.1 and (6.2), using Cauchy-Schwarz’s inequality, we have T ≪ Z TT ω ( t ) dt ≤ Z S ω ( t ) dt ≤ Z S ∆ ∗ ± ( q q t ) dt ≤| S | (cid:16) Z TT ∆ ∗ ( q q t ) dt (cid:17) ≪ | S | T, which implies | S | ≫ T. Thus the proof of Theorem 2.3 is completed. (cid:3) Proof of Theorem 2.4 Suppose k ≥ T ≫ ( q q ) ε is a large parameter. Set δ = (cid:26) − , if C k ≥ , , if C k < , where C k is defined in (2.1).By Theorem 2.3, there exists t ∈ [ T, T ] such that δ ∆( q q u ; r , q , r , q ) > c t for any u ∈ [ t, t + H ], with H = c √ T L − . Thus c k H t k < Z t + H t δ k ∆ k ( q q u ; r , q , r , q ) du = δ k C k (cid:0) ( t + H ) k − t k (cid:1) + δ k (cid:0) F k (cid:0) q q ( t + H ); r , q , r , q (cid:1) −F k ( q q t ; r , q , r , q ) (cid:1) , which yields δ k (cid:16) F k (cid:0) q q ( t + H ); r , q , r , q (cid:1) −F k (cid:0) q q t ; r , q , r , q (cid:1)(cid:17) >c k H t k − δ k C k (cid:0) k (cid:1) t k H + O ( H t k − ) = C ∗ k H t k (cid:0) O ( H T − ) (cid:1) , with C ∗ k = c k − δ k C k (cid:0) k (cid:1) > . Thus we get (cid:12)(cid:12) F k (cid:0) q q ( t + H ); r , q , r , q (cid:1) −F k (cid:0) q q t ; r , q , r , q (cid:1)(cid:12)(cid:12) ≫ H T k , which immediatly implies Theorem 2.4. (cid:3) References [1] H. Cram´er. ¨Uber zwei S¨atze des Herrn G. H. Hardy. Mathematische Zeitschrift , 15(1):201–210, 1922.[2] A. Good. Ein ω -Result at f¨ur das quadratische Mittel der Riemannschen Zetafunktion aufder kritischen Linie. Inventiones mathematicae , 41(3):233–251, 1977.[3] D. R. Heath-Brown. The distribution and moments of the error term in the Dirichlet divisorproblem. Acta Arith , 60(4):389–415, 1992.[4] D. R. Heathbrown and K. Tsang. Sign changes of E (T), δ (x), and P (x). Journal of NumberTheory , 49(1):73–83, 1994.[5] M. N. Huxley. Exponential sums and lattice points III. Proceedings of the London Mathe-matical Society , 87(03):591–609, 2003.[6] M. N. Huxley. Exponential sums and the Riemann zeta function V. Proceedings of the LondonMathematical Society , 90(01):1–41, 2005.[7] A. Ivi´c. Large values of the error term in the divisor problem. Inventiones mathematicae ,71(3):513–520, 1983.[8] A. Ivi´c and P. Sargos. On the higher moments of the error term in the divisor problem. IllinoisJournal of Mathematics , 51(2):353–377, 2007.[9] L. R. Jia and W. G. Zhai. A weighted divisor problem. Journal of Number Theory , 178:60 –93, 2017.[10] M. Jutila. On the divisor problem for short intervals. Ann. Univer. Turkuensis Ser. AI ,186:23–30, 1984.[11] K. Liu. On higher-power moments of the error term for the divisor problem with congruenceconditions. Monatshefte f¨ur Mathematik , 163(2):175–195, 2011.[12] W. M¨uller and W. G. Nowak. Third power moments of the error terms corresponding tocertain arithmetic functions. manuscripta mathematica , 87(1):459–480, 1995.[13] H. E. Richert. Ein Gitterpunktproblem. Mathematische Annalen , 125(1):467–471, 1952.[14] Z. Shan. Hilbert inequality (in Chinese). Chinese Science Bulletin , 29(01):021, 1984.[15] K. M. Tsang. Higher-power moments of δ ( x ), E ( t ) and P ( x ). Proceedings of the LondonMathematical Society , 65:65–84, 1992.[16] Kai-Man Tsang and Wenguang Zhai. Sign changes of the error term in Weyl’s law for Heisen-berg manifolds. Transactions of the American Mathematical Society , 364(5):2647–2666, 2012.[17] G. Vorono¨ı. Sur une fonction transcendante et ses applications `a la sommation de quelquess´eries. In Annales scientifiques de l’ ´Ecole Normale Sup´erieure , volume 21, pages 207–267,1904.[18] W. G. Zhai. On higher-power moments of ∆( x )(II). Acta Arith , 114:35–54, 2004. Department Of Mathematical Sciences, Tsinghua University, Beijing 100084, People’sRepublic of China E-mail address : [email protected] Department of Mathematics, China University of Mining and Thechnology, Beijing100083, People’s Republic of China E-mail address : [email protected] School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’sRepublic of China E-mail address ::