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An identity involving certain Hardy sums and Ramanujan’s sum
Advances in Difference Equations volume 2013, Article number: 261 (2013)
The main purpose of this paper is using the properties of Gauss sums and the mean value theorem of the Dirichlet L-functions to study one kind of a hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and to give an exact computational formula for it.
Let c be a natural number and d be an integer prime to c. The classical Dedekind sums
describe the behavior of the logarithm of the eta-function (see  and ) under modular transformations. Funakura  gave an analogous transformation formula for the logarithm of the classical theta-function
That is, put with , , and . Then we have
where is defined as
The sums (and certain related ones) are sometimes called Hardy sums. They are closely connected with Dedekind sums. Regarding the properties of and related sums, some authors studied them and obtained many interesting results; see [4–7] and . For example, Wenpeng Zhang  proved the following conclusion: Let p be an odd prime. Then, for any fixed positive integer m, we have the asymptotic formula
where is the Riemann zeta-function and .
On the other hand, we define Ramanujan’s sum as follows (see Theorem 8.6 of ):
The main purpose of this paper is using the properties of Gauss sums and the mean square value theorem of the Dirichlet L-functions to study a hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and to give an exact computational formula for it. That is, we shall prove the following.
Theorem Let be an odd square-full number. Then we have the identity
where denotes the Euler function, and denotes the product over all distinct prime divisors of q.
It is very interesting that the value is equal to zero in our theorem if we use instead of .
For a general odd number , whether there exits a computational formula for
is an open problem. Interested readers are welcome to study it with us.
2 Several lemmas
In this section, we shall give several lemmas, which are necessary in the proof of our theorem. Hereinafter, we shall use many properties of Gauss sums and character sums, all of which can be found in , so they will not be repeated here. First we have the following lemma.
Lemma 1 Let be an odd number, and let χ be any non-principal character . Then, for any integer m with , we have the identity
where denotes the classical Gauss sums, and .
Proof For any non-principal character , from the definition of and the properties of Gauss sums, we have
This proves Lemma 1. □
Lemma 2 Let be an integer. Then, for any integer a with , we have the identity
where denotes the Dirichlet L-function corresponding to character .
Proof See Lemma 2 of . □
Lemma 3 Let and . Then we have the identity
Proof This formula is an immediate consequence of (5.9) and (5.10) in . □
Lemma 4 Let be an odd number and with . Then we have the identity
Proof From Lemma 2 and Lemma 3, we have
where λ is the principal character .
From the Euler infinite product formula, we have
where denotes the product over all primes .
Now, combining (2.1), (2.2) and Lemma 2, we have the identity
This proves Lemma 4. □
Lemma 5 Let be a square-full number (i.e., and a prime implies ). Then, for any non-primitive character , we have the identity
Proof It is clear that from the multiplicative properties of Gauss sums we know that we only need to prove , a power of prime, where . Suppose that χ is a non-primitive character , then χ must be a character . So, from the definition of Gauss sums and the properties of a complete residue system and trigonometric sums, we have
This proves Lemma 5. □
Lemma 6 Let be an odd square-full number. Then we have the identities
where denotes the summation over all odd primitive characters .
Proof From the definition of , we have the computational formula
From the reciprocity formula of , we know that for any positive integer a with , we have the identity (see )
Applying this formula, we have
From (2.3), Lemma 2 with and the Möbius inversion formula, we have
Then, using formula (2.5) and the Möbius inversion formula, we have
From (2.4), Lemma 2 with and the Möbius inversion formula, we also have
Then, using (2.7) and the Möbius inversion formula, we have
Now Lemma 6 follows from (2.6) and (2.8). □
3 Proof of the theorems
In this section, we shall complete the proof of our theorem. Note that if χ is a primitive character , then , and . From Lemma 1, Lemma 2, Lemma 4, Lemma 5 and Lemma 6, we have
where we have used the identity
This completes the proof of our theorem.
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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 N.S.F. (11071194, 61202437) of P.R. China, and partly by the Fundamental Research Funds for the Central Universities of P.R. China (CHD2010JC101)
The authors declare that they have no competing interests.
WW carried out the hybrid mean value problem involving certain Hardy sums and Ramanujan’s sum and gave an exact computational formula. DH participated in the research and summary of the study. All authors read and approved the final manuscript.