(2) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. Page 2 of 8. Proposition ([]) Let q = pα (α > ) be a power of the prime p > and χ be a nonprincipal Dirichlet character mod q. Then, for any integers N, n, H with < H, we have na χ(a)e H / q/ log q. q a=N+ N+H . (.). At almost the same time, Liu [] showed independently the following. Proposition Let q be a prime power, χ , ψ be a multiplicative and additive character mod q, respectively, with χ non-principal. Then, for any integers N, H with < H, we have N+H . χ(a)ψ(a) H / q/+ .. a=N+. Now let q ≥ be a ﬁxed integer, A = A(q) ≤ q, B = B(q) ≤ q, and H = H(q) ≤ q. Deﬁne (A, B, H) = a ∈ Z | (a, q) = , ab ≡ . (mod q), ≤ a ≤ A, ≤ b ≤ B, |a – b| ≤ H .. It is a direct generalization of the set of so-called H-ﬂat numbers mod q, which was studied extensively by Xi (see [] and references therein). This paper deals with general partial Gaussian sums of the following type: Gk (χ, A, B, H; q) =. a∈(A,B,H). na ak χ(a)e , q. ) we can also where k ≥ is an arbitrary ﬁxed integer. For the sake of periodicity of e( na q restrict n to be ≤ n ≤ q. Then with the aid of the estimates for the general Kloosterman sums and the properties of trigonometric sums, we shall obtain upper bound estimates as follows. Theorem Let q ≥ be an integer and χ a non-principal Dirichlet character mod q. Then . nABd(q) Bd(q)(log q)(log H) + log q , + Gk (χ, A, B, H; q) A q d(q) q q k /. which is uniformly nontrivial for any positive integer n such that n < q/ . Taking n a constant, we can immediately obtain the following. Corollary Let q ≥ be an integer and χ a non-principal Dirichlet character mod q. Then we have . ABd(q) Bd(q)(log q)(log H) + log q . + Gk (χ, A, B, H; q) A q d(q) q q k /. Taking k = , B = q in Corollary , we obtain the following..

(3) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. Page 3 of 8. Corollary Let q ≥ be an integer and χ a non-principal Dirichlet character mod q. Then, for any positive integers A, H such that A, H ≤ q, we have G (χ, A, q, H; q) q/ d (q) log q.. (.). Remark It is easy to see that (.) is stronger than (.) for any integer H such that q/+ < H ≤ q. It is also stronger than (.) for any integer H such that q/ d/ (q) < H ≤ q. These results reveal that more cancelations occurred in the number set (A, B, H). Taking A = H = q in Corollary , we obtain the following. Corollary Let q ≥ be an integer and χ a non-principal Dirichlet character mod q. Then we have G (χ, q, q, q; q) =. q a=. na q/ d (q) log q. χ(a)e q. This is a little stronger than the classical result of the complete Gauss sums in the case when (n, q) > .. 2 Some lemmas To prove the theorem, we need the following lemmas. Lemma Let q, n, , r be integers with q > , q > n, and ≥ . Deﬁne h(r, ; n) = n ra a= a e( q ). Then we have. h(r, ; n). ⎧ ⎨=. n+ +. ⎩. + O(n ), q | r,. n , | sin(π s/q)|. q r,. where s = min(r, q – r) with ≤ r ≤ q – . . Proof See Lemma of [] or Lemma . of []. Lemma Let q be a positive integer. Then we have Kχ (m, n; q) ≤ q/ (m, n, q)/ d(q),. ) is the general Kloosterman sum, with aā ≡ where Kχ (m, n; q) = a (mod q) χ(a)e( ma+nā q (mod q) and (m, n, q) the greatest common divisor of m, n, q. . Proof See Lemma of [].. 3 Proof of the Theorem Now we come to prove the theorem. Note that Gk (χ, A, B, H; q) =. . na . a χ(a)e q. . t≤H a–b≡t. a≤A,b≤B (mod q),ab≡. k. (mod q).

(4) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. Page 4 of 8. Applying the trigonometric sums identity ⎧ ⎨ q q, q | m, ma = e ⎩, q m, q a= we obtain Gk (χ, A, B, H; q) na ak χ(a)e = q a∈(A,B,H). =. t≤H a–b≡t. a≤A,b≤B (mod q),ab≡. mt e – = q m≤q t≤H q =. na ak χ(a)e q. (mod q). a≤A,b≤B ab≡ (mod q). mt e – q m≤q t≤H q. (m + n)a – mb ak χ(a)e q. a,b≤q ab≡ (mod q). (m + n)a – mb χ(a)e q. r(a – c) s(b – d) e e q q r≤q c≤A d≤B s≤q mt (m + r + n)a – (m – s)b e – χ(a)e = q r,s≤q m≤q t≤H q q ×. . ck. a,b≤q ab≡ (mod q). × =. q. =. q + + +. rc sd ck e – e – q q c≤A d≤B mt Kχ (m + r + n, s – m; q)h(–r, k; A)h(–s, ; B) e – q r,s≤q m≤q t≤H mt e – Kχ (m + n, –m; q)h(–q, k; A)h(–q, ; B) q m≤q t≤H mt Kχ (m + r + n, –m; q)h(–r, k; A)h(–q, ; B) e – q r≤q– m≤q t≤H q mt Kχ (m + n, s – m; q)h(–q, k; A)h(–s, ; B) e – q s≤q– m≤q t≤H q mt Kχ (m + r + n, s – m; q)h(–r, k; A)h(–s, ; B). e – q r,s≤q– m≤q t≤H q . Then from Lemma and Lemma , we have mt Kχ (m + n, –m; q)h(–q, k; A)h(–q, ; B) e – q m≤q t≤H q . m min H, q m≤q q. – . Kχ (m + n, –m; q) · h(–q, k; A) · h(–q, ; B).

(5) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. . HAk+ Bq–/ d(q). Page 5 of 8. (m + n, q)/. m≤q/H. . (m + n, q)/ , m q/H<m≤q–. + Ak+ Bq–/ d(q). where x = mina∈Z |x – a|. Combining the estimates . (m + n, q)/ =. m≤q/H. . . . m≤q/H d|(m+n). d|q. =. . d/. . d/. d|q. . m≤q/(dH)+n/d. H – qd(q) + nd(q) and (m + n, q)/ / = d m m q/H<m≤q– q/H<m≤q– . d|q. d|(m+n). =. . q q–+n n Hd + d <m≤ d. d|q. . . d/. . d. –/. dm – n. q q–+n n Hd + d <m≤ d. d|q. n + m dm. d(q) log H + nd(q), we have mt Kχ (m + n, –m; q)h(–q, k; A)h(–q, ; B) e – q m≤q t≤H q nAk+ Bq–/ d (q) + Ak+ Bq–/ d (q) log H.. (.). Applying Lemma and Lemma again, we obtain mt e – Kχ (m + r + n, –m; q)h(–r, k; A)h(–q, ; B) q r≤q– m≤q t≤H q m min H, q r≤q– m≤q q HBq–/ d(q). . Kχ (m + r + n, –m; q) · h(–r, k; A) · h(–q, ; B). (m + r + n, –m, q)/ ·. r≤q– m≤q/H. + Bq–/ d(q). – . . Ak | sin( πqr )|. (m + r + n, –m, q)/ Ak · m | sin( πqr )| r≤q– q/H<m≤q–.

(6) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. HAk Bq–/ d(q) + Ak Bq–/ d(q). Page 6 of 8. (m + r + n, –m, q)/ r r≤q– m≤q/H (m + r + n, –m, q)/ . r m r≤q– q/H<m≤q–. Combining (m + r + n, –m, q)/ r r≤q– m≤q/H =. . r m≤q/H r≤q. d/. d|q. d|(r+n). =. . d/. d|q. d|m. (n+)/d≤r≤(q+n–)/d. q/H. . . d–/. d|q. dr – n. (n+)/d≤r≤(q+n–)/d. . . m≤q/(Hd). n + r dr. q+n– H – qd(q) log n+ and. (m + r + n, –m, q)/ r m r≤q– q/H<m≤q– =. d|q. d/. r q/H<m≤q– m r≤q– d|(r+n). =. d|q. d|m. . d–/. (n+)/d≤r≤(q+n–)/d. . dr – n. q/(Hd)<m≤q/d. m. . d–/ (log H) n r( – dr ) d|q r≤q/d q+n– d(q)(log H) log , n+ we have mt Kχ (m + r + , –m; q)h(–r, k; A)h(–q, ; B) e – q r≤q– m≤q t≤H q q+n– k –/ A Bq d (q)(log H) log . n+. (.). Similarly, we get the estimate mt e – Kχ (m + , s – m; q)h(–q, k; A)h(–s, ; B) q s≤q m≤q t≤H q q– Ak+ q–/ d (q) log . n+. (.).

(7) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. Page 7 of 8. Noting that mt Kχ (m + r + n, s – m; q)h(–r, k; A)h(–s, ; B) e – q r,s≤q– m≤q t≤H q q–/ τ (q). m min H, q r,s≤q– m≤q . HAk q–/ d(q) + Ak q/ d(q). – . · (m + r + n, s – m, q)/. Ak · | sin( πqr )| | sin( πqs )|. (m + r + n, s – m, q)/ rs m≤q/H r,s≤q–. (m + r + n, s – m, q)/ . rs q/H<m≤q– m r,s≤q–. Using the estimates (m + r + n, s – m, q)/ rs r,s≤q– m≤q/H =. =. . . . d|q. m≤q/H. r≤q– d|(m+r+n). . . d/. d/. r s≤q– s d|(s–m). . m≤q/H (m+n+)/d≤r≤(q+m+n–)/d. d|q. dr – m – n. (–m)/d≤s≤(q–m–)/d. ds + m. H – qd(q)(log q) log(q + n – ) and (m + r + n, s – m, q)/ rs q/H<m≤q– m r,s≤q– =. . d/. d|q. =. d|q. d/. . m q/H<m≤q–. r≤q– d|(m+r+n). . m q/H<m≤q– . r s≤q– s d|(s–m). (m+n+)/d≤r≤(q+m+n–)/d. dr – m – n. (–m)/d≤s≤(q–m–)/d. ds + m. d(q) log q log(q + n – ), we have mt Kχ (m + r + n, s – m; q)h(–r, k; A)h(–s, ; B) e – q r,s≤q– m≤q t≤H q Ak q/ d(q) log q log(q + n – ).. Now combining (.)-(.), we have Gk (χ, A, B, H; q) Ak q/ d(q). nABd(q) Bd(q)(log q)(log H) + log + q . q q. This completes the proof of the theorem.. (.).

(8) Ren et al. Journal of Inequalities and Applications (2016) 2016:295. Competing interests The authors declare that they have no competing interests. Authors’ contributions DH drafted the manuscript. GR and TZ participated in its design and coordination and helped to draft the manuscript. All authors read and approved the ﬁnal manuscript. Author details 1 College of Mathematics and Information Science, Xianyang Normal University, Xianyang, Shaanxi 712000, P.R. China. 2 School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, P.R. China. 3 School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi 710119, P.R. China. Acknowledgements The article was supported by the National Natural Science Foundation of China (Grant Nos. 11201275, 11471258), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20106101120001), the Natural Science Foundation of Shaanxi Province of China (Grant Nos. 2011JQ1010, 2016JM1017), the Scientiﬁc Research Program Funded by Shaanxi Provincial Education Department (Grant No. 15JK1794) and the Fundamental Research Funds for the Central Universities (Grant No. GK201503014). The authors want to show their great thanks to the anonymous referee for his/her helpful comments and suggestions. Received: 2 April 2016 Accepted: 9 November 2016 References 1. Burgess, DA: Partial Gaussian sums. Bull. Lond. Math. Soc. 20(6), 589-592 (1988) 2. Burgess, DA: Partial Gaussian sums. II. Bull. Lond. Math. Soc. 21(2), 153-158 (1989) 3. Burgess, DA: Partial Gaussian sums. III. Glasg. Math. J. 34(2), 253-261 (1992) 4. Liu, CL: On incomplete Gaussian sums. Acta Math. Sin. New Ser. 12(2), 141-150 (1996) 5. Xi, P, Yi, Y: On character sums over ﬂat numbers. J. Number Theory 130(5), 1234-1240 (2010) 6. Xu, ZF: Distribution of the diﬀerence of an integer and its m-th power mod n over incomplete intervals. J. Number Theory 133(12), 4200-4223 (2013) 7. Xu, ZF, Zhang, TP: High-dimensional D.H. Lehmer problem over short intervals. Acta Math. Sin. Engl. Ser. 30(2), 213-228 (2014). Page 8 of 8.

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