On a class of Lyapunov's inequality involving $\lambda$-Hilfer Hadamard fractional derivative

1.   Introduction and main results

Let $q\geq 3$ be an integer, and $\chi$ be a Dirichlet character modulo $q$. The characters of the rational polynomial are defined as follows:

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 $\chi$ mod $q$, Pólya ^[1] and Vinogradov ^[2] independently proved that

and we call it Pólya-Vinogradov inequality.

When $q = p$ is an odd prime, $\chi$ is a $p$-th order character modulo $p$, Weil ^[3] proved

where $f(x)$ is not a perfect $p$-th power modulo $p$, $A\ll B$ 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

where $(r-s, q) = 1$, and $\chi$ is a primitive character modulo $q$. This shows the power of $q$ in Weil's result is the best possible!

Also, when $\chi$ is a primitive character mod $q$, W. P. Zhang and W. L. Yao ^[5] obtained

where $q$ is an odd perfect square and $m$ is any positive integer with $(m, q) = 1$.

When $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ is a square full number with $p_{i}\equiv 3 \bmod 4$, $\chi = \chi_{1}\chi_{2}\ldots\chi_{s}$ with $\chi_{i}$ being any primitive even character mod $p^{\alpha_{i}}_{i}$$(i = 1, 2, \ldots, s)$, W. P. Zhang and T. T. Wang ^[6] obtained the identity

where $a\cdot\overline{a} \equiv 1 \bmod q$, and $\left(\frac{\ast}{p}\right)$ 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

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 $\chi$ is an odd character modulo $q$, $m$ is a positive integer with $(m, q) = 1$, we can get

That is to say, under this condition,

So, we will only discuss the case of $\chi$ 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 $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd square-full number, $\chi_{i}$ be any primitive even character mod $p^{\alpha_{i}}_{i}$ $(i = 1, 2, \ldots, s)$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then for any integer $m$ with $(m, q) = 1$, we have the identity

where $\prod_{p\mid q}$ 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 $p_{i} \equiv 3 \bmod 4$ $(i = 1, 2, \ldots, s)$. 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 $q = 3^{2}$, $\chi$ be a Dirichlet character modulo $9$ defined as follows:

Obviously, $\chi$ is a primitive even character modulo $9$. Taking $m = 1, 2$, then we have

Example 1.2. Let $q = 5^{2}$, $\chi$ be a primitive even character modulo $25$ defined as follows:

Taking $m = 1, 2$, then we have

Example 1.3. Let $q = 13^{2}$, $\chi$ be a primitive even character modulo $169$ defined as follows:

Taking $m = 1, 2$, then we have

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 $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd square-full number, $\chi_{i}$ be any primitive even character mod $p^{\alpha}_{i}$ $(i = 1, 2, \ldots, s)$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then for any integer $m$ with $(m, q) = 1$, we have the identity

where $\omega(q)$ denotes the number of all distinct prime divisors of $q$.

Corollary 1.2. Let $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd number with $\alpha_{i}\geq 1$ $(i = 1, 2, \ldots, s)$, $\chi_{i}$ be any primitive even character mod $p^{\alpha_{i}}_{i}$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then for any integer $m$ with $(m, q) = 1$, we have the inequality

Theorem 1.2. Let $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd square-full number, $\chi_{i}$ be any primitive even character mod $p^{\alpha_{i}}_{i}$ $(i = 1, 2, \ldots, s)$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then for any integers $k$ and $m$ with $k \geq 1$ and $(m, q) = 1$, we have the identity

where $J(q)$ denotes the number of primitive characters modulo $q$, and ${{\sum\limits_{\chi \,\bmod \,q}}^{*}}$ denotes the summation over all primitive characters modulo $q$.

Example 1.4. Taking $q = 5^{2}$, $m = 1, 2$, then we have

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 $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd square-full number, $\chi_{i}$ be any primitive even character mod $p^{\alpha_{i}}_{i}$ $(i = 1, 2, \ldots, s)$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then for any integer $m$ with $(m, q) = 1$, we have the identity

Corollary 1.4. Let $q = p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}\cdots p^{\alpha_{s}}_{s}$ be an odd square-full number, $\chi_{i}$ be any primitive even character mod $p^{\alpha_{i}}_{i}$ $(i = 1, 2, \ldots, s)$ and $\chi = \chi_{1}\chi_{2}\cdots\chi_{s}$. Then we have the identity

Theorem 1.3. Let $p$ be an odd prime, $\chi$ be any non-principal character mod $p$. Then for any integer $m$ with $(m, p) = 1$, we have the identity

where

Remark 1.2. From ^[8], we know that when $f(x)$ is a polynomial of odd degree $n\geq 3$, Weil's estimate (^[15,16])

implies that $E < 4p^{2}-8p$. Noting that $\sum\limits^{q}_{a = 1}'\chi(ma+\overline{a})$ can be regarded as a dual form of Kloosterman sums, which defined as $\sum\limits^{q}_{a = 1}'e^{ 2\pi i\frac{ma+\bar{a}}{q} }$, we can obtain some distributive properties of $\sum\limits^{q}_{a = 1}'\chi(ma+\overline{a})$ 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, $\chi$ be any non-principal character mod $p$. Then for any quadratic residue $m$ mod $p$, we have the identity

Corollary 1.6. Let $p$ be an odd prime, $\chi$ be any non-principal character mod $p$. Then for any quadratic non-residue $m$ mod $p$, we have the identity

2.   Some Lemmas

To prove our Theorems, we need some Lemmas as the following:

Lemma 2.1. Let $q$, $q_{1}$, $q_{2}$ be integers with $q = q_{1}q_{2}$ and $(q_{1}, q_{2}) = 1$, $\chi_{i}$ be any non-principal character mod $q_{i}$ $(i = 1, 2)$. Then for any integer $m$ with $(m, q) = 1$ and $\chi = \chi_{1}\chi_{2}$, we have the identity

Proof. From the properties of Dirichlet characters, we have

This completes the proof of Lemma $2.1$.

Lemma 2.2. Let $p$ be an odd prime, $\alpha$ and $m$ be integers with $\alpha\geq 1$ and $(m, p) = 1$. Then for any primitive even character $\chi$ mod $p^{\alpha}$, we have the identity

where $\chi^{0}_{2} = \left(\frac{\ast}{p}\right)$, $\tau(\chi) = \sum\limits_{a = 1}^{p^{\alpha}}\chi(a)e\left(\frac{a}{p^{\alpha}}\right)$, $\chi_{1}$ is a primitive character mod $p^{\alpha}$ and $\chi = \chi^{2}_{1}$.

Proof. For any primitive even character $\chi$ mod $p^{\alpha}$, there exists one primitive character $\chi_{1}$ mod $p^{\alpha}$ such that $\chi = \chi^{2}_{1}$. From the properties of Gauss sum, we can obtain

Now we compute $B_{1}$ and $B_{2}$ respectively.

Similarly, we have

Therefore, we can obtain

Lemma 2.3. Let $p$ be an odd prime. Then for any integer $n$, we have the identity

Proof. See Theorem 8.2 of ^[17].

Lemma 2.4. Let $p$ be an odd prime. Then we have the identity

Proof. From the properties of character sum, we have

This completes the proof of Lemma $2.4$.

3.   Proof of Theorems

Now we come to prove our Theorems.

Firstly, we prove Theorem $1.1$. With the help of Lemma $2$ in ^[6], when $\alpha\geq 2$, we have

which implies from Lemma $2.2$, we can obtain

Then, applying Lemma $2.1$, we can obtain

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:

Finally, we prove Theorem $1.3$. For any integer $m$ with $(m, p) = 1$, we have

So from the orthogonality of Dirichlet characters and the properties of reduced residue system modulo $p$, we have

Now we compute $A_{1}$, $A_{2}$, $A_{3}$, $A_{4}$ respectively. Noticing that $\chi(-1) = 1$, from the properties of the complete residue system modulo $p$, we have

Applying Lemma $2.3$, we can get

Then, we compute $A_{2}$. With the aid of Lemma $2.4$, we have

Similarly, we have

Note that

which completes the proof of Theorem $1.3$.

4.   Conclusions

Three Theorems are stated in the main results. The Theorem 1.1 obtains an exact computational formula for $\sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}),$ which broadens the scope of $q$ by removing the condition $p \equiv 3 \bmod 4$ 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.

Acknowledgments

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).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.