On two sums related to the Lehmer problem over short intervals

1.   Introduction

Let $p$ be an odd primes, $a$ be an integer coprime to $p$. So there exists a $0 < \overline{a}\leq p$ such that $a\overline{a}\equiv1\mod p$. Let $N(p)$ denoted the number of integers $a$ ($0 < a < p$) such that $a\overline{a}\equiv1\mod p$ and $2\nmid a+\overline{a}$. D. H. Lehmer (see ^[1] problem F12 p. 381) asked a problem about the asymptotic formula of $N(p)$. To give an answer to this problem, Professor W. P. Zhang proved the following two theorems (see ^[2,3,4]):

Theorem 1.1. Let $N(p)$ denoted the number of integers $a$ ($0 < a < p$) such that $a\overline{a}\equiv1\mod p$ and $2\nmid a+\overline{a}$. We have

where the $S(a, b; p)$ are Kloosterman sums, defined as follows:

Theorem 1.2. For any odd number $q$, let $N(q)$ denotes the number of integers $a$ ($0 < a < q$) which coprime to $q$ such that $a\overline{a}\equiv1\mod q$ and $2\nmid a+\overline{a}$. We have

For more about Lehmer problem, one can see ^{[5,6,7,8,9,10,11,12]}. In 2015, H. Zhang and W. P. Zhang ^[13] investigated two new sums:

which related to the Lehmer problem, and gave the asymptotic formulae for these new sums.

In this paper, we give asymptotic formulae for these new sums over short intervals.

Theorem 1.3. Let $p$ be an odd primes, and $k > 2$ be an integer, and $\varepsilon > 0$ is a sufficient small real number. $|A_i|\gg p^{1/2+1/k+\varepsilon}$, $A_i = \{1, 2, \cdots, t_i\}$, $[1, t_i]\in(0, p)$, $i = 1, 2, \cdots, k$. Define $N(k, p)$ as follows:

then we have

where the implied constant of the first and second big 'O' terms depends on $k$.

Theorem 1.4. Let $p$ be an odd primes, $k > 4$ be an integer, and $\varepsilon > 0$ is a sufficient small real number. $p\geq n > p^{\frac{k+2}{2k-2}+\varepsilon}$ be an integer. $A_i =$ $\{1, 2, \cdots, t_i\}$, $i = 1, 2, \cdots, k-1$, $p^{\frac{k+2}{2k-2}+\varepsilon} < t_i < n$, $A_k = \{1, 2, \cdots, n\}$. Define $M(k, p)$ as follows:

then we have

where the implied constant of the first and second big 'O' terms depends on $k$.

Remark 1.5. The short interval result is usually more difficult than the whole interval result. This is why this problem is interesting and meaningful, and use some technique which are often used to obtain a short interval result such as Lemma 2.4 and formula (2.1). We use $1-e\left(\frac{a+b}{2}\right)$ instead of $(-1)^{a+b}$ to detect $2\mid a+b$.

2.   Some Lemmas

Before we prove the theorem, we introduce some Lemmas, and through out the paper, $p$ always denotes an odd prime number, and $\chi$ and $\psi$ always denote the character $\mod p$, $\chi_0$ denotes the principal character $\mod p$.

We define the generalized Kloosterman sums, as follows:

where $m$, $n$, $q$ be integers and $\chi$ is a Dirichlet character $\mod q$. And we define Gauss sums, as follows:

We have following estimates.

Lemma 2.1. Let $p$ be an odd primes, then we have

Proof. See ^[14].

Lemma 2.2. Let $a$, $b$, $q$ be natural numbers, $\chi$ runs through all the non-principal characters $\mod q$, then we have

Proof. See Lemma 3 in ^[15].

Similarly, we have

Lemma 2.3. Let $p$ be an odd primes, and $a$, $b$be natural numbers, $\psi$ runs through all the non-principal characters $\mod p$, $\chi$ is a character $\mod p$, then we have

Lemma 2.4. Let $N$ be a nature number, $\alpha$ be a real number, then

where $\|\alpha\|$ denotes the distance from $\alpha$ to the nearest integer.

Proof. See ^[16].

Lemma 2.5. Let $U$ and $\gamma$ be two positive real numbers, $p$ be a nonzero natural number, then

Proof. Since the inequality

hold for at most two $k$. They contribute at most $2U$. The others contribute at most

This completes the proof of the lemma.

Lemma 2.6. Let $p$ be an odd prime number, $A = \{1, 2, \cdots, s\}$, $[1, s]\in(0, p)$, then we have

when $\chi\neq\chi_0$, and

when $\chi = \chi_0$.

Proof. When $\chi = \chi_0$, we have

Let

We note that for $\psi\neq\chi_0$, we have

By Lemma 2.2 and 2.4, we have

By Lemma 2.5, we have

So when $\chi = \chi_0$, we have

When $\chi\neq\chi_0$, we have

We only estimate $S_2$, the estimation of $S_1$ is similar.

For $S_{21}$, by (2.1), we have

So we have

By Lemma 2.3 and 2.4, we have

By Lemma 2.5, we have

For $S_{22}$, it is simple, we have

So when $\chi\neq\chi_0$, we have

So we complete the proof of Lemma 2.6.

For convenience, we record the following four special cases of Lemma 2.6.

Corollary 2.7. Let $p$ be an odd prime number, $A = \{1, 2, \cdots, t\}$, $[1, t]\in(0, p)$, then we have

Corollary 2.8. Let $p$ be an odd prime number, $A = \{1, 2, \cdots, t\}$, $[1, t]\in(0, p)$, $\chi\neq\chi_0$, then we have

Corollary 2.9. Let $p$ be an odd prime number, $A = \{1, 2, \cdots, h\}$, $[1, h]\in(0, p)$, then we have

Corollary 2.10. Let $p$ be an odd prime number, $A = \{1, 2, \cdots, t\}$, $[1, t]\in(0, p)$, $\chi\neq\chi_0$, then we have

3.   Proofs of the theorems

Now we prove Theorem 1.3, we give asymptotic formulae for two sums over short intervals.

By Corollary 2.7, we have

By Corollary 2.8, we have

So we have

Note that the implied constant of the first big 'O' term is dependent on $k$. This completes the proof of Theorem 1.3.

Now we prove Theorem 1.4, we have

By Corollary 2.9 and 2.10, we have

By the following formula

and the definition of interval $A_k$, we have

Note that the implied constant of the first big 'O' term is dependent on $k$. This completes the proof of Theorem 1.4.

4.   Conclusions

In this paper, we investigated two problems which related to the Lehmer problem, and gave the asymptotic formulae for these new sums over short intervals.

Acknowledgements

The author would like to express his heartfelt thanks to the reviewers for their valuable comments and suggestions.

Conflict of interest

The author declares that there are no conflicts of interest regarding the publication of this paper.