An identity involving certain Hardy sums and Ramanujan’s sum
© Wang and Han; licensee Springer. 2013
Received: 24 July 2013
Accepted: 9 August 2013
Published: 26 August 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.
where is the Riemann zeta-function and .
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.
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 .
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.
where denotes the classical Gauss sums, and .
This proves Lemma 1. □
where denotes the Dirichlet L-function corresponding to character .
Proof See Lemma 2 of . □
Proof This formula is an immediate consequence of (5.9) and (5.10) in . □
where λ is the principal character .
where denotes the product over all primes .
This proves Lemma 4. □
This proves Lemma 5. □
where denotes the summation over all odd primitive characters .
Now Lemma 6 follows from (2.6) and (2.8). □
3 Proof of the theorems
This completes the proof of our theorem.
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)
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