Denote by χ a Dirichlet character modulo q≥3, and ¯a means a⋅¯a≡1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form
q∑a=1′χ(ma+¯a),
where q∑a=1′ denotes the summation over all 1≤a≤q with (a,q)=1. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [
Citation: Wenjia Guo, Xiaoge Liu, Tianping Zhang. Dirichlet characters of the rational polynomials[J]. AIMS Mathematics, 2022, 7(3): 3494-3508. doi: 10.3934/math.2022194
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Denote by χ a Dirichlet character modulo q≥3, and ¯a means a⋅¯a≡1modq. In this paper, we study Dirichlet characters of the rational polynomials in the form
q∑a=1′χ(ma+¯a),
where q∑a=1′ denotes the summation over all 1≤a≤q with (a,q)=1. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [
Let q≥3 be an integer, and χ be a Dirichlet character modulo q. The characters of the rational polynomial are defined as follows:
N+M∑x=N+1χ(f(x)), |
where M and N are any given positive integers, and f(x) is a rational polynomial. For example, when f(x)=x, for any non-principal Dirichlet character χ mod q, Pólya [1] and Vinogradov [2] independently proved that
|N+M∑x=N+1χ(x)|<√qlnq, |
and we call it Pólya-Vinogradov inequality.
When q=p is an odd prime, χ is a p-th order character modulo p, Weil [3] proved
N+M∑x=N+1χ(f(x))≪p12lnp, |
where f(x) is not a perfect p-th power modulo p, A≪B denotes |A|<kB for some constant k, which in this case depends on the degree of f.
Many authors have obtained numerous results for various forms of f(x). For example, W. P. Zhang and Y. Yi [4] constructed a special polynomial as f(x)=(x−r)m(x−s)n and deduced
|q∑a=1χ((a−r)m(a−s)n)|=√q, |
where (r−s,q)=1, and χ is a primitive character modulo q. This shows the power of q in Weil's result is the best possible!
Also, when χ is a primitive character mod q, W. P. Zhang and W. L. Yao [5] obtained
q∑a=1χ(am(1−a)m)=√q¯χ(4m), |
where q is an odd perfect square and m is any positive integer with (m,q)=1.
When q=pα11pα22⋯pαss is a square full number with pi≡3mod4, χ=χ1χ2…χs with χi being any primitive even character mod pαii(i=1,2,…,s), W. P. Zhang and T. T. Wang [6] obtained the identity
|q∑a=1′χ(ma2k−1+n¯a)|=√q∏p|q(1+(mn(2k−1)p)), | (1.1) |
where a⋅¯a≡1modq, and (∗p) denotes the Legendre symbol. Besides, k, m and n also satisfying some special conditions. Other related work about Dirichlet characters of the rational polynomials can be found in references [7,8,9,10,11,12,13,14]. Inspired by these, we will study the sum
q∑a=1′χ(ma+¯a). |
Following the way in [6], we obtain W. P. Zhang and T. T. Wang's identity (1.1) under a more relaxed situation. Then by adding some new ingredients, we derive some new identities for the fourth power mean of it.
Noting that if χ is an odd character modulo q, m is a positive integer with (m,q)=1, we can get
q∑a=1′χ(ma+¯a)=q∑a=1′χ(−ma+¯(−a))=−q∑a=1′χ(ma+¯a). |
That is to say, under this condition,
q∑a=1′χ(ma+¯a)=0. |
So, we will only discuss the case of χ an even character. To the best of our knowledge, the following identities dealing with arbitrary odd square-full number cases are new and have not appeared before.
Theorem 1.1. Let q=pα11pα22⋯pαss be an odd square-full number, χi be any primitive even character mod pαii (i=1,2,…,s) and χ=χ1χ2⋯χs. Then for any integer m with (m,q)=1, we have the identity
|q∑a=1′χ(ma+¯a)|=√q∏p∣q(1+(mp)), |
where ∏p∣q denotes the product over all distinct prime divisors p of q.
Remark 1.1. It is obvious that Theorem 1.1 is W. P. Zhang and T. T. Wang's identity (1.1) with k=n=1 by removing the condition pi≡3mod4 (i=1,2,…,s). Besides, using our results, we can directly obtain the absolute values of the sums of Dirichlet characters satisfying some conditions, which avoids complex calculations. What's more, the result of Theorem 1.1 also shows that the order of q in Weil's result can not be improved.
To understand the result better, we give the following examples:
Example 1.1. Let q=32, χ be a Dirichlet character modulo 9 defined as follows:
χ(n)={e2πi⋅ind2n3,if (n,9)=1;0,if (n,9)>1. |
Obviously, χ is a primitive even character modulo 9. Taking m=1,2, then we have
|9∑a=1′χ(ma+¯a)|=|9∑a=1′χ(a+¯a)|=|3χ(2)+3χ(7)|=|3e2πi3+3e2πi⋅43|=6,|9∑a=1′χ(ma+¯a)|=|9∑a=1′χ(2a+¯a)|=|2χ(3)+2χ(6)+2χ(9)|=0. |
Example 1.2. Let q=52, χ be a primitive even character modulo 25 defined as follows:
χ(n)={e2πi⋅ind2n5,if (n,25)=1;0,if (n,25)>1. |
Taking m=1,2, then we have
|25∑a=1′χ(ma+¯a)|=|25∑a=1′χ(a+¯a)|=|5χ(2)+5χ(23)|=|5e2πi5+5e2πi⋅115|=10,|25∑a=1′χ(ma+¯a)|=|25∑a=1′χ(2a+¯a)|=|4χ(2)+4χ(3)+4χ(7)+4χ(8)+4χ(12)|=|4e2πi5+4e2πi⋅75+4e2πi⋅55+4e2πi⋅35+4e2πi⋅95|=0. |
Example 1.3. Let q=132, χ be a primitive even character modulo 169 defined as follows:
χ(n)={e2πi⋅ind2n13,if (n,169)=1;0,if (n,169)>1. |
Taking m=1,2, then we have
|169∑a=1′χ(ma+¯a)|=|169∑a=1′χ(a+¯a)|=|4χ(1)+26χ(2)+4χ(4)+4χ(9)+4χ(12)+4χ(14)+4χ(17)+4χ(22)+4χ(25)+4χ(27)+4χ(30)+4χ(35)+4χ(38)+4χ(40)+4χ(43)+4χ(48)+4χ(51)+4χ(53)+4χ(56)+4χ(61)+4χ(64)+4χ(66)+4χ(69)+4χ(74)+4χ(77)+4χ(79)+4χ(82)|=|8+8eπi13+34e2πi13+8e3πi13+8e4πi13+8e5πi13+8e6πi13+8e7πi13+8e8πi13+8e9πi13+8e10πi13+8e11πi13+8e12πi13|=26, |
|169∑a=1′χ(ma+¯a)|=|169∑a=1′χ(2a+¯a)|=|4χ(2)+4χ(3)+4χ(5)+4χ(8)+4χ(10)+4χ(11)+4χ(15)+4χ(16)+4χ(18)+4χ(21)+4χ(23)+4χ(24)+4χ(28)+4χ(29)+4χ(31)+4χ(34)+4χ(36)+4χ(37)+4χ(41)+4χ(42)+4χ(44)+4χ(47)+4χ(49)+4χ(50)+4χ(54)+4χ(55)+4χ(57)+4χ(60)+4χ(62)+4χ(63)+4χ(67)+4χ(68)+4χ(70)+4χ(73)+4χ(75)+4χ(76)+4χ(80)+4χ(81)+4χ(83)|=|12+12eπi13+12e2πi13+12e3πi13+12e4πi13+12e5πi13+12e6πi13+12e7πi13+12e8πi13+12e9πi13+12e10πi13+12e11πi13+12e12πi13|=0. |
The above examples can be easily achieved by our Theorem 1.1. From Theorem 1.1, we may immediately obtain the following two corollaries:
Corollary 1.1. Let q=pα11pα22⋯pαss be an odd square-full number, χi be any primitive even character mod pαi (i=1,2,…,s) and χ=χ1χ2⋯χs. Then for any integer m with (m,q)=1, we have the identity
|q∑a=1′χ(ma+¯a)|={2ω(q)√q, if m is a quadratic residue modulo q;0, otherwise, |
where ω(q) denotes the number of all distinct prime divisors of q.
Corollary 1.2. Let q=pα11pα22⋯pαss be an odd number with αi≥1 (i=1,2,…,s), χi be any primitive even character mod pαii and χ=χ1χ2⋯χs. Then for any integer m with (m,q)=1, we have the inequality
|q∑a=1′χ(ma+¯a)|≤2ω(q)√q. |
Theorem 1.2. Let q=pα11pα22⋯pαss be an odd square-full number, χi be any primitive even character mod pαii (i=1,2,…,s) and χ=χ1χ2⋯χs. Then for any integers k and m with k≥1 and (m,q)=1, we have the identity
∑∗χmodqχ(−1)=1|q∑a=1′χ(ma+¯a)|2k=qk2ω(q)J(q)∏p∣q(1+(mp))2k, |
where J(q) denotes the number of primitive characters modulo q, and ∑χmodq∗ denotes the summation over all primitive characters modulo q.
Example 1.4. Taking q=52, m=1,2, then we have
∑∗χmod25χ(−1)=1|25∑a=1′χ(ma+¯a)|2k=∑∗χmod25χ(−1)=1|25∑a=1′χ(a+¯a)|2k=8⋅102k,∑∗χmod25χ(−1)=1|25∑a=1′χ(ma+¯a)|2k=∑∗χmod25χ(−1)=1|25∑a=1′χ(2a+¯a)|2k=0, |
which can be easily achieved by our Theorem 1.2.
Taking k=2 in Theorem 1.2, we may immediately obtain the followings:
Corollary 1.3. Let q=pα11pα22⋯pαss be an odd square-full number, χi be any primitive even character mod pαii (i=1,2,…,s) and χ=χ1χ2⋯χs. Then for any integer m with (m,q)=1, we have the identity
∑∗χmodqχ(−1)=1|q∑a=1′χ(ma+¯a)|4=q22ω(q)J(q)∏p∣q(1+(mp))4. |
Corollary 1.4. Let q=pα11pα22⋯pαss be an odd square-full number, χi be any primitive even character mod pαii (i=1,2,…,s) and χ=χ1χ2⋯χs. Then we have the identity
∑∗χmodqχ(−1)=1|q∑a=1′χ(ma+¯a)|4={8ω(q)q2J(q), if m is a quadratic residue modulo q;0, otherwise. |
Theorem 1.3. Let p be an odd prime, χ be any non-principal character mod p. Then for any integer m with (m,p)=1, we have the identity
∑χmodpχ(−1)=1|p−1∑a=1χ(ma+¯a)|4={2p3−6p2+4−4(p2−3p+2)(mp)+(p−1)E,if p≡3mod4;2p3−6p2+4−4(p2+p−2)(mp)+(p−1)E,if p≡1mod4, |
where
E=p−1∑a=1p−1∑b=1((a2b−1)(b−1)bp)p−1∑d=1((¯a2d−1)(d−1)dp). |
Remark 1.2. From [8], we know that when f(x) is a polynomial of odd degree n≥3, Weil's estimate ([15,16])
|p−1∑x=0(f(x)p)|≤(n−1)√p, |
implies that E<4p2−8p. Noting that q∑a=1′χ(ma+¯a) can be regarded as a dual form of Kloosterman sums, which defined as q∑a=1′e2πima+ˉaq, we can obtain some distributive properties of q∑a=1′χ(ma+¯a) from Theorem 1.2 and 1.3.
From Theorem 1.3, we also have the following corollaries:
Corollary 1.5. Let p be an odd prime, χ be any non-principal character mod p. Then for any quadratic residue m mod p, we have the identity
∑χmodpχ(−1)=1|p−1∑a=1χ(ma+¯a)|4={2p3−10p2+12p−4+(p−1)E,if p≡3mod4;2p3−10p2−4p+12+(p−1)E,if p≡1mod4. |
Corollary 1.6. Let p be an odd prime, χ be any non-principal character mod p. Then for any quadratic non-residue m mod p, we have the identity
∑χmodpχ(−1)=1|p−1∑a=1χ(ma+¯a)|4={2p3−2p2−12p+4+(p−1)E,if p≡3mod4;2p3−2p2+4p−4+(p−1)E,if p≡1mod4. |
To prove our Theorems, we need some Lemmas as the following:
Lemma 2.1. Let q, q1, q2 be integers with q=q1q2 and (q1,q2)=1, χi be any non-principal character mod qi (i=1,2). Then for any integer m with (m,q)=1 and χ=χ1χ2, we have the identity
q∑a=1′χ(ma+¯a)=q1∑b=1′χ1(mb+¯b)q2∑c=1′χ2(mc+¯c). |
Proof. From the properties of Dirichlet characters, we have
q∑a=1′χ(ma+¯a)=q1q2∑a=1′χ1χ2(ma+¯a)=q1∑b=1′q2∑c=1′χ1χ2(m(bq2+cq1)+¯bq2+cq1)=q1∑b=1′q2∑c=1′χ1(m(bq2+cq1)+¯bq2+cq1)χ2(m(bq2+cq1)+¯bq2+cq1)=q1∑b=1′χ1(mbq2+¯bq2)q2∑c=1′χ2(mcq1+¯cq1)=q1∑b=1′χ1(mb+¯b)q2∑c=1′χ2(mc+¯c). |
This completes the proof of Lemma 2.1.
Lemma 2.2. Let p be an odd prime, α and m be integers with α≥1 and (m,p)=1. Then for any primitive even character χ mod pα, we have the identity
pα∑a=1′χ(ma+¯a)=χ1(m)τ2(¯χ1)τ(¯χ)(1+χ02(m)τ2(χ02¯χ1)τ2(¯χ1)), |
where χ02=(∗p), τ(χ)=pα∑a=1χ(a)e(apα), χ1 is a primitive character mod pα and χ=χ21.
Proof. For any primitive even character χ mod pα, there exists one primitive character χ1 mod pα such that χ=χ21. From the properties of Gauss sum, we can obtain
pα∑a=1′χ(ma+¯a)=1τ(¯χ)pα∑a=1′pα∑b=1¯χ(b)e(b(ma+¯a)pα)=1τ(¯χ)pα∑a=1¯χ(a)pα∑b=1¯χ(b)e(b(ma2+1)pα)=1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1¯χ(a)e(bma2pα)=1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1¯χ1(a2)e(bma2pα)=1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1(1+χ02(a))¯χ1(a)e(bmapα)=1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1¯χ1(a)e(bmapα)+1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1χ02(a)¯χ1(a)e(bmapα):=B1+B2. |
Now we compute B1 and B2 respectively.
B1=1τ(¯χ)pα∑b=1¯χ(b)e(bpα)pα∑a=1¯χ1(a)e(bmapα)=1τ(¯χ)pα∑b=1¯χ(b)χ1(bm)e(bpα)pα∑a=1¯χ1(bma)e(bmapα)=χ1(m)τ(¯χ1)τ(¯χ)pα∑b=1¯χ(b)χ1(b)e(bpα)=χ1(m)τ(¯χ1)τ(¯χ)pα∑b=1¯χ1(b)e(bpα)=χ1(m)τ2(¯χ1)τ(¯χ). |
Similarly, we have
B2=χ1(m)χ02(m)τ2(χ02¯χ1)τ(¯χ). |
Therefore, we can obtain
pα∑a=1′χ(ma+¯a)=χ1(m)τ2(¯χ1)τ(¯χ)(1+χ02(m)τ2(χ02¯χ1)τ2(¯χ1)). |
Lemma 2.3. Let p be an odd prime. Then for any integer n, we have the identity
p∑a=1(a2+np)={−1,if (n,p)=1;p−1,if (n,p)=p. |
Proof. See Theorem 8.2 of [17].
Lemma 2.4. Let p be an odd prime. Then we have the identity
p−2∑a=2p−1∑b=1((a2b−1)(b−1)bp)=2×(−1)p−12+2. |
Proof. From the properties of character sum, we have
p−2∑a=2p−1∑b=1((a2b−1)(b−1)bp)=p−1∑b=1(b−1p)p−2∑a=2((a2b−1)bp)=p−1∑b=1(b−1p)p−2∑a=2(b2(a2−¯b)p)=p−1∑b=1(b−1p)p−2∑a=2(a2−¯bp)=p−1∑b=1(b−1p)(p∑a=1(a2−¯bp)−(1−¯bp)−((p−1)2−¯bp)−(p2−¯bp))=p−1∑b=1(b−1p)(−1−2(1−¯bp)−(−¯bp))=−p−1∑b=1(b−1p)−2p−1∑b=1(b−1p)(1−¯bp)−p−1∑b=1(b−1p)(−¯bp)=−p−2∑b=0(bp)−2p−1∑b=1(b−1p)((1−¯b)b2p)−p−1∑b=1(¯b−1p)=−2p−2∑b=0(bp)−2p−1∑b=1((b−1)2bp)=−2(p−1∑b=0(bp)−(p−1p))−2×(−1)=2×(−1)p−12+2. |
This completes the proof of Lemma 2.4.
Now we come to prove our Theorems.
Firstly, we prove Theorem 1.1. With the help of Lemma 2 in [6], when α≥2, we have
τ2(χ02¯χ1)τ2(¯χ1)=(1p)2=1, |
which implies from Lemma 2.2, we can obtain
|pα∑a=1′χ(ma+¯a)|=|χ1(m)τ2(¯χ1)τ(¯χ)(1+(mp))|=√pα(1+(mp)). |
Then, applying Lemma 2.1, we can obtain
|q∑a=1′χ(ma+¯a)|=|pα11∑a1=1′χ1(ma1+¯a1)|⋯|pαss∑as=1′χs(mas+¯as)|=√q∏p∣q(1+(mp)). |
This completes the proof of Theorem 1.1.
Then, from Lemma 2.1 and Lemma 2.2, we can prove Theorem 1.2 as following:
∑∗χmodqχ(−1)=1|q∑a=1′χ(ma+¯a)|2k=∑∗χ1modpα11χ1(−1)=1|pα11∑a1=1′χ1(ma1+¯a1)|2k⋯∑∗χsmodpαssχs(−1)=1|pαss∑as=1′χs(mas+¯as)|2k=s∏i=1[12J(pαii)pkαii|1+(mpi)|2k]=qk2ω(q)J(q)∏p∣q(1+(mp))2k. |
Finally, we prove Theorem 1.3. For any integer m with (m,p)=1, we have
p−1∑a=1χ(ma+¯a)=p−1∑u=1χ(u)p−1∑a=1am+¯a≡umodp1=p−1∑u=1χ(u)p−1∑a=1a2m2−amu+m≡0modp1=p−1∑u=1χ(u)p−1∑a=0(2am−u)2≡u2−4mmodp1=p−1∑u=1χ(u)p−1∑a=0a2≡u2−4mmodp1=p−1∑u=1χ(u)(1+(u2−4mp))=p−1∑u=1χ(u)(u2−4mp)=χ(2)p−1∑u=1χ(u)(u2−mp). |
So from the orthogonality of Dirichlet characters and the properties of reduced residue system modulo p, we have
∑χmodpχ(−1)=1|p−1∑a=1χ(ma+¯a)|4=∑χmodpχ(−1)=1|χ(2)p−1∑u=1χ(u)(u2−mp)|2|χ(2)p−1∑u=1χ(u)(u2−mp)|2=∑χmodpχ(−1)=1p−1∑a=1p−1∑b=1p−1∑c=1p−1∑d=1χ(ac¯bd)(a2−mp)(b2−mp)(c2−mp)(d2−mp)=∑χmodpχ(−1)=1p−1∑a=1p−1∑b=1p−1∑c=1p−1∑d=1χ(ac)(a2b2−mp)(b2−mp)(c2d2−mp)(d2−mp)=p−1∑a=1p−1∑b=1p−1∑c=1p−1∑d=1(a2b2−mp)(b2−mp)(c2d2−mp)(d2−mp)∑χmodpχ(−1)=1χ(ac)=p−12p−1∑a=1p−1∑b=1p−1∑c=1p−1∑d=1a≡¯cmodp(a2b2−mp)(b2−mp)(c2d2−mp)(d2−mp)+p−12p−1∑a=1p−1∑b=1p−1∑c=1p−1∑d=1a≡−¯cmodp(a2b2−mp)(b2−mp)(c2d2−mp)(d2−mp)=(p−1)p−1∑a=1p−1∑b=1p−1∑d=1(a2b2−mp)(b2−mp)(¯a2d2−mp)(d2−mp)=(p−1)p−1∑a=1p−1∑b=1(1+(bp))(a2b−mp)(b−mp)p−1∑d=1(1+(dp))(¯a2d−mp)(d−mp)=(p−1)p−1∑a=1p−1∑b=1(a2b−1p)(b−1p)p−1∑d=1(¯a2d−1p)(d−1p)+(p−1)p−1∑a=1p−1∑b=1(a2b−1p)(b−1p)p−1∑d=1(mp)((¯a2d−1)(d−1)dp)+(p−1)p−1∑a=1p−1∑b=1(mp)((a2b−1)(b−1)bp)p−1∑d=1(¯a2d−1p)(d−1p)+(p−1)p−1∑a=1p−1∑b=1(mp)((a2b−1)(b−1)bp)p−1∑d=1(mp)((¯a2d−1)(d−1)dp):=A1+A2+A3+A4. |
Now we compute A1, A2, A3, A4 respectively. Noticing that χ(−1)=1, from the properties of the complete residue system modulo p, we have
p−1∑b=1(a2b−1p)(b−1p)=p−1∑b=0(a2b−1p)(b−1p)−1=p−1∑b=0(4a2p)((a2b−1)(b−1)p)−1=p−1∑b=0((2a2b−a2−1)2−(a2−1)2p)−1=p−1∑b=0(b2−(a2−1)2p)−1. |
Applying Lemma 2.3, we can get
A1=(p−1)p−1∑a=1p−1∑b=1(a2b−1p)(b−1p)p−1∑d=1(¯a2d−1p)(d−1p)=(p−1)p−1∑a=1(p−1∑b=0(b2−(a2−1)2p)−1)(p−1∑d=0(d2−(¯a2−1)2p)−1)=(p−1)[2p−1∑b=0(b2p)p−1∑d=0(d2p)+p−2∑a=2p−1∑b=0(b2−(a2−1)2p)p−1∑d=0(d2−(¯a2−1)2p)]−2(p−1)p−1∑a=1p−1∑b=0(b2−(a2−1)2p)+(p−1)2=2p3−6p2+4. |
Then, we compute A2. With the aid of Lemma 2.4, we have
A2=(p−1)p−1∑a=1p−1∑b=1(a2b−1p)(b−1p)p−1∑d=1(mp)((¯a2d−1)(d−1)dp)=(p−1)p−1∑a=1[p−1∑b=0(b2−(a2−1)2p)−1]p−1∑d=1(mp)((¯a2d−1)(d−1)dp)=(p−1)2p−1∑d=1(mp)((d−1)2dp)−(p−1)p−2∑a=2p−1∑d=1(mp)((¯a2d−1)(d−1)dp)+(p−1)2p−1∑d=1(mp)(((p−1)2d−1)(d−1)dp)−(p−1)p−1∑a=1p−1∑d=1(mp)((¯a2d−1)(d−1)dp)=(p2−3p+2)[p−1∑d=1(mp)((d−1)2dp)+p−1∑d=1(mp)(((p−1)2d−1)(d−1)dp)]−2(p−1)p−2∑a=2p−1∑d=1(mp)((a2d−1)(d−1)dp)=2(p2−3p+2)(mp)p−1∑d=1((d−1)2dp)−4(p−1)[(−1)p−12+1](mp)=2(p2−3p+2)(mp)p−1∑b=2(bp)−4(p−1)[(−1)p−12+1](mp)=−2(p2−3p+2)(mp)−4(p−1)[(−1)p−12+1](mp). |
Similarly, we have
A3=−2(p2−3p+2)(mp)−4(p−1)[(−1)p−12+1](mp). |
Note that
A4=(p−1)p−1∑a=1p−1∑b=1((a2b−1)(b−1)bp)p−1∑d=1((¯a2d−1)(d−1)dp), |
which completes the proof of Theorem 1.3.
Three Theorems are stated in the main results. The Theorem 1.1 obtains an exact computational formula for q∑a=1′χ(ma+¯a), which broadens the scope of q by removing the condition p≡3mod4 in the previous article, where p is the prime divisor of q. The Theorem 1.2 derives a new identity for the mean value of it by adding some different ingredients. What's more, the Theorem 1.3 bridges the fourth power of Dirichlet characters with Legendre symbols of certain polynomials, which may be useful in the related future research. However, due to some technical reasons, we can only deal with the odd square-full number q case.
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China (No. 11871317), and the Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China (No. 2021JC-29).
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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