Open Access

An identity involving certain Hardy sums and Ramanujan’s sum

Advances in Difference Equations20132013:261

https://doi.org/10.1186/1687-1847-2013-261

Received: 24 July 2013

Accepted: 9 August 2013

Published: 26 August 2013

Abstract

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.

MSC:11L05, 11L10.

Keywords

Gauss sumsRamanujan’s sumcertain Hardy sumshybrid mean valuecomputational formula

1 Introduction

Let c be a natural number and d be an integer prime to c. The classical Dedekind sums
S ( d , c ) = j = 1 c ( ( j c ) ) ( ( d j c ) ) ,
where
( ( x ) ) = { x [ x ] 1 2 , if  x  is not an integer; 0 , if  x  is an integer,
describe the behavior of the logarithm of the eta-function (see [1] and [2]) under modular transformations. Funakura [3] gave an analogous transformation formula for the logarithm of the classical theta-function
θ ( z ) = n = + exp ( π i n 2 z ) , Im ( z ) > 0 .
That is, put V z = ( a z + b ) ( c z + d ) with a , b , c , d Z , c > 0 , and a d b c = 1 . Then we have
log θ ( V z ) = log θ ( z ) + 1 2 log ( c z + d ) 1 4 π i + 1 4 π i S 1 ( d , c ) ,
(1.1)
where S 1 ( d , c ) is defined as
S 1 ( d , c ) = j = 1 c 1 ( 1 ) j + 1 + [ d j c ] .
The sums S 1 ( d , c ) (and certain related ones) are sometimes called Hardy sums. They are closely connected with Dedekind sums. Regarding the properties of S 1 ( d , c ) and related sums, some authors studied them and obtained many interesting results; see [47] and [8]. For example, Wenpeng Zhang [7] proved the following conclusion: Let p be an odd prime. Then, for any fixed positive integer m, we have the asymptotic formula
h = 1 p 1 | S 1 ( h , p ) | 2 m = p 2 m ζ 2 ( 2 m ) ( 1 1 4 m ) ζ ( 4 m ) ( 1 + 1 4 m ) + O ( p 2 m 1 exp ( 6 ln p ln ln p ) ) ,

where ζ ( s ) is the Riemann zeta-function and exp ( y ) = e y .

On the other hand, we define Ramanujan’s sum R q ( c ) as follows (see Theorem 8.6 of [9]):
R q ( c ) = k = 1 q ( k , q ) = 1 e 2 π i k c q = d | ( c , q ) d μ ( q / d ) ,

where μ ( n ) is the famous Möbius function. Some related properties of R q ( c ) can also be found in [9, 10] and [11].

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 q > 1 be an odd square-full number. Then we have the identity
h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) S 1 ( 2 h , q ) = ϕ 2 ( q ) p | q ( 1 + 1 p ) ,

where ϕ ( q ) denotes the Euler function, and p | q 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 R q ( h + 1 ) instead of R q ( 2 h + 1 ) .

For a general odd number q 3 , whether there exits a computational formula for
h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) S 1 ( 2 h , q )

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 [9], so they will not be repeated here. First we have the following lemma.

Lemma 1 Let q > 1 be an odd number, and let χ be any non-principal character mod q . Then, for any integer m with ( m , q ) = 1 , we have the identity
c = 1 q χ ( c ) R q ( m c + 1 ) = χ ¯ ( m ) τ ( χ ) τ ( χ ¯ ) ,

where τ ( χ ) = a = 1 q 1 χ ( a ) e ( a q ) denotes the classical Gauss sums, and e ( y ) = e 2 π i y .

Proof For any non-principal character χ mod q , from the definition of R q ( c ) and the properties of Gauss sums, we have
c = 1 q χ ( c ) R q ( m c + 1 ) = c = 1 q b = 1 q ( b , q ) = 1 χ ( c ) e ( b ( m c + 1 ) q ) = b = 1 q ( b , q ) = 1 e ( b q ) c = 1 q χ ( c ) e ( m b c q ) = χ ¯ ( m ) b = 1 q ( b , q ) = 1 χ ¯ ( b ) e ( b q ) c = 1 q χ ( m b c ) e ( m b c q ) = χ ¯ ( m ) τ ( χ ) b = 1 q χ ¯ ( b ) e ( b q ) = χ ¯ ( m ) τ ( χ ) τ ( χ ¯ ) .

This proves Lemma 1. □

Lemma 2 Let q > 2 be an integer. Then, for any integer a with ( a , q ) = 1 , we have the identity
S ( a , q ) = 1 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( a ) | L ( 1 , χ ) | 2 ,

where L ( 1 , χ ) denotes the Dirichlet L-function corresponding to character χ mod d .

Proof See Lemma 2 of [12]. □

Lemma 3 Let q > 0 and ( h , q ) = 1 . Then we have the identity
S 1 ( h , q ) = 8 S ( h + q , 2 q ) + 4 S ( h , q ) .

Proof This formula is an immediate consequence of (5.9) and (5.10) in [6]. □

Lemma 4 Let q > 1 be an odd number and 0 < h < q with ( h , q ) = 1 . Then we have the identity
S 1 ( 2 h , q ) = 20 S ( 2 h , q ) + 8 S ( 4 h , q ) + 8 S ( h , q ) .
Proof From Lemma 2 and Lemma 3, we have
S 1 ( 2 h , q ) = 8 S ( 2 h + q , 2 q ) + 4 S ( 2 h , q ) = 4 π 2 q d | 2 q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h + q ) | L ( 1 , χ ) | 2 + 4 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h ) | L ( 1 , χ ) | 2 = 4 π 2 q d | q ( 2 d ) 2 ϕ ( 2 d ) χ mod 2 d χ ( 1 ) = 1 χ ( 2 h + d ) | L ( 1 , χ ) | 2 = 16 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h + q ) λ ( 2 h + q ) | L ( 1 , χ λ ) | 2 = 16 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h ) | L ( 1 , χ λ ) | 2 ,
(2.1)

where λ is the principal character mod 2 .

From the Euler infinite product formula, we have
| L ( 1 , χ λ ) | 2 = p 1 | 1 χ ( p 1 ) λ ( p 1 ) p 1 | 2 = p 1 > 2 | 1 χ ( p 1 ) p 1 | 2 = | 1 χ ( 2 ) 2 | 2 p 1 | 1 χ ( p 1 ) p 1 | 2 = ( 5 4 χ ( 2 ) 2 χ ¯ ( 2 ) 2 ) | L ( 1 , χ ) | 2 ,
(2.2)

where p 1 denotes the product over all primes p 1 .

Now, combining (2.1), (2.2) and Lemma 2, we have the identity
S 1 ( 2 h , q ) = 16 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h ) | L ( 1 , χ λ ) | 2 = 16 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 χ ( 2 h ) ( 5 4 χ ( 2 ) 2 χ ¯ ( 2 ) 2 ) | L ( 1 , χ ) | 2 = 20 S ( 2 h , q ) + 8 S ( 4 h , q ) + 8 S ( h , q ) .

This proves Lemma 4. □

Lemma 5 Let q > 1 be a square-full number (i.e., q 4 and a prime p | q implies p 2 | q ). Then, for any non-primitive character χ mod q , we have the identity
a = 1 q χ ( a ) e ( a q ) = 0 .
Proof It is clear that from the multiplicative properties of Gauss sums we know that we only need to prove q = p α , a power of prime, where α 2 . Suppose that χ is a non-primitive character mod p α , then χ must be a character mod p α 1 . So, from the definition of Gauss sums and the properties of a complete residue system mod p α and trigonometric sums, we have
a = 1 p α χ ( a ) e ( a p α ) = r = 0 p 1 a = 1 p α 1 χ ( r p α 1 + a ) e ( r p α 1 + a p α ) = a = 1 p α 1 χ ( a ) e ( a p α ) r = 0 p 1 e ( r p ) = 0 .

This proves Lemma 5. □

Lemma 6 Let q > 3 be an odd square-full number. Then we have the identities
χ mod q χ ( 1 ) = 1 | L ( 1 , χ ) | 2 = π 2 12 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) ; χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 = π 2 24 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) ,

where χ ( 1 ) = 1 χ mod q denotes the summation over all odd primitive characters χ mod q .

Proof From the definition of S ( a , q ) , we have the computational formula
S ( 1 , q ) = a = 1 q 1 ( a q 1 2 ) 2 = ( q 1 ) ( q 2 ) 12 q .
(2.3)
From the reciprocity formula of S ( a , q ) , we know that for any positive integer a with ( a , q ) = 1 , we have the identity (see [4])
S ( a , q ) + S ( q , a ) = q 2 + a 2 + 1 12 a q 1 4 .
Applying this formula, we have
S ( 2 , q ) = q 2 + 2 2 + 1 24 q 1 4 S ( q , 2 ) = ( q 1 ) ( q 5 ) 24 q .
(2.4)
From (2.3), Lemma 2 with a = 1 and the Möbius inversion formula, we have
q 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 | L ( 1 , χ ) | 2 = π 2 d | q μ ( d ) q d S ( 1 , q d ) = π 2 d | q μ ( d ) ( q d 1 ) ( q d 2 ) 12 = π 2 12 ϕ ( q ) [ q p | q ( 1 + 1 p ) 3 ]
or
χ mod q χ ( 1 ) = 1 | L ( 1 , χ ) | 2 = π 2 12 ϕ 2 ( q ) q 2 [ q p | q ( 1 + 1 p ) 3 ] .
(2.5)
Then, using formula (2.5) and the Möbius inversion formula, we have
χ mod q χ ( 1 ) = 1 | L ( 1 , χ ) | 2 = d | q μ ( d ) ( χ mod q / d χ ( 1 ) = 1 | L ( 1 , χ ) | 2 ) = π 2 12 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) .
(2.6)
From (2.4), Lemma 2 with a = 2 and the Möbius inversion formula, we also have
q 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 = π 2 d | q μ ( d ) q d S ( 2 , q d ) = π 2 d | q μ ( d ) ( q d 1 ) ( q d 5 ) 24 = π 2 24 ϕ ( q ) [ q p | q ( 1 + 1 p ) 6 ]
or
χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 = π 2 24 ϕ 2 ( q ) q 2 [ q p | q ( 1 + 1 p ) 6 ] .
(2.7)
Then, using (2.7) and the Möbius inversion formula, we have
χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 = d | q μ ( d ) ( χ mod q / d χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 ) = π 2 24 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) .
(2.8)

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 mod q , then | τ ( χ ) | = q , and τ ( χ ) τ ( χ ¯ ) = χ ¯ ( 1 ) τ ( χ ) τ ( χ ) ¯ = χ ¯ ( 1 ) q . From Lemma 1, Lemma 2, Lemma 4, Lemma 5 and Lemma 6, we have
h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) S 1 ( 2 h , q ) = h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) ( 20 S ( 2 h , q ) + 8 S ( 4 h , q ) + 8 S ( h , q ) ) = 20 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) χ ( h ) χ ( 2 ) | L ( 1 , χ ) | 2 + 8 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) χ ( h ) χ ( 4 ) | L ( 1 , χ ) | 2 + 8 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 h = 1 q ( h , q ) = 1 R q ( 2 h + 1 ) χ ( h ) | L ( 1 , χ ) | 2 = 20 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 τ ( χ ) τ ( χ ¯ ) | L ( 1 , χ ) | 2 + 8 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 τ ( χ ) τ ( χ ¯ ) χ ( 2 ) | L ( 1 , χ ) | 2 + 8 π 2 q d | q d 2 ϕ ( d ) χ mod d χ ( 1 ) = 1 τ ( χ ) τ ( χ ¯ ) χ ¯ ( 2 ) | L ( 1 , χ ) | 2 = 20 π 2 q 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 | L ( 1 , χ ) | 2 8 π 2 q 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 8 π 2 q 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 χ ¯ ( 2 ) | L ( 1 , χ ) | 2 = 20 π 2 q 2 ϕ ( q ) π 2 12 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) 8 π 2 q 2 ϕ ( q ) π 2 24 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) 8 π 2 q 2 ϕ ( q ) π 2 24 ϕ 3 ( q ) q 2 p | q ( 1 + 1 p ) = ϕ 2 ( q ) p | q ( 1 + 1 p ) ,
where we have used the identity
χ mod q χ ( 1 ) = 1 χ ( 2 ) | L ( 1 , χ ) | 2 = χ mod q χ ( 1 ) = 1 χ ¯ ( 2 ) | L ( 1 , χ ) | 2 .

This completes the proof of our theorem.

Declarations

Acknowledgements

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)

Authors’ Affiliations

(1)
School of Science, Chang’an University, Xi’an, Shaanxi, China
(2)
Department of Mathematics, Northwest University, Xi’an, Shaanxi, China

References

  1. Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.MATHGoogle Scholar
  2. Chowla S: On Kloosterman’s sums. Norske Vid. Selsk. Forh. 1967, 40: 70–72.MathSciNetMATHGoogle Scholar
  3. Funakura T: On Kronecker’s limit formula for Dirichlet series with periodic coefficients. Acta Arith. 1990, 55: 59–73.MathSciNetMATHGoogle Scholar
  4. Guy RK: Unsolved Problems in Number Theory. 2nd edition. Springer, New York; 1994.View ArticleMATHGoogle Scholar
  5. Malyshev AV: A generalization of Kloosterman sums and their estimates. Vestn. Leningr. Univ. 1960, 15: 59–75. (in Russian)MathSciNetGoogle Scholar
  6. Xi P, Yi Y: On character sums over flat numbers. J. Number Theory 2010, 130: 1234–1240. 10.1016/j.jnt.2009.10.011MathSciNetView ArticleMATHGoogle Scholar
  7. Zhang W: On a problem of D. H. Lehmer and its generalization. Compos. Math. 1993, 86: 307–316.MathSciNetMATHGoogle Scholar
  8. Zhang W: A problem of D. H. Lehmer and its mean square value formula. Jpn. J. Math. 2003, 29: 109–116.MathSciNetMATHGoogle Scholar
  9. Zhang W: A problem of D. H. Lehmer and its generalization (II). Compos. Math. 1994, 91: 47–56.MathSciNetMATHGoogle Scholar
  10. Zhang W: A mean value related to D. H. Lehmer’s problem and the Ramanujan’s sum. Glasg. Math. J. 2012, 54: 155–162. 10.1017/S0017089511000498MathSciNetView ArticleMATHGoogle Scholar
  11. Zhang W: On the mean values of Dedekind sums. J. Théor. Nr. Bordx. 1996, 8: 429–442. 10.5802/jtnb.179View ArticleMathSciNetMATHGoogle Scholar
  12. Zhang W: On the difference between an integer and its inverse modulo n (II). Sci. China Ser. A 2003, 46: 229–238. 10.1360/03ys9024MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Wang and Han; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.