Error term of the mean value theorem for binary Egyptian fractions

Abstract In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.


Introduction
Let ∈ + a n , and ( ) = a n , 1. Egyptian fractions concern the representation of rational numbers as the finite sum of distinct unit fractions: is always soluble. This conjecture is still open, although much work has been carried out. See [2][3][4], for example, for more details about the ternary conjecture. Refer to [5] for more information on Egyptian fractions. Considering the binary Egyptian fractions when = k 2, define ( ) = ( ) ∈ = + R n a x y a n x y ; card , : 1 1 .
The Diophantine equation is not always necessary to have a solution. For example, for n with all its prime factor p of the form ≡ ( ) p a 1 mod , it has no such representation. Thus, it is natural to consider the mean value of ( ) R n a ; . For > X 0, define ; . n X n a , 1 Huang and Vaughan [6] proved that Here ( ) ζ s denotes the Riemann zeta function. In [7], Jia got a more explicit expression with better error term for ( ) C a 0 . See [8][9][10] for more results about ( ) R n a ; . In [6], Huang and Vaughan employed for the first time in this area, of complex analytic technique from multiplicative number theory, and gave an innovative counting function with a different criterion with Croot et al. in [11]. Their estimate (3) holds uniformly for X and ∈ a and the error term is almost optimal. The bound for the error here is as strong as can be established on generalized Riemann hypothesis (GRH) with their method. The aim of this paper is to improve the estimate (3) concerning X.
Theorem 1. Let > X 0 and ( ) X a Δ ; be defined in (1). We have Remark 1. Theorem 1 is better than (3) when a is small. Obviously, Theorem 1 is a PNT (prime number theorem) type result, which depends on the zero-free region of the Riemann zeta-function. So in some sense, it is the best possible result under the present methods in the analytic number theory. It is impossible to improve the exponent 1/2 in Theorem 1 without better zero-free region of the Riemann zeta-function.
Under Riemann hypothesis (RH), we can prove the following power saving result.
Here the implied constant depends only on ε.
Remark 2. One main tool in [6] is the well-known Perron formula. Our proof relies on the convolution method, the results concerning divisor problems with Dirichlet characters (see [12,13]) and moment results of the Dirichlet L-functions.
In the authors' another work [14], the mean square of the error term under RH was studied. An asymptotic formula can be obtained, which suggests that the average size of the error term is ( ) Both Theorems 1 and 2 depend on zeros of ( ) ζ s . To avoid this point, it is interesting to consider the average value in short intervals. For > X Y , 0, we define ; .
X n X Y n a , 1 Then we can get the following estimate.
. We have with same α and β as above, where (⋅ ) D a ; is defined by (2).
Remark 3. For the proof of Theorem 3, we use the technique of Zhai [15] who considered the short interval distribution of a class of integers, and his lemma (see Lemma 8) can help us to get some saving in short interval. The constraint for > X a 90 is needed for the character sum (see Lemma 2). We can also use Lemma 1 to get a slightly worse result if it is removed from the theorem.
In what follows, … c c , ,

Preliminary
In this section, we list some lemmas which will be needed in the proof. We cite a theorem of Friedlander and Iwaniec [12].   .
The following lemma is due to Nowak [13], which has more precise estimation than the theorem given by Friedlander and Iwaniec for the special case given below. where (⋅) P 2 is a polynomial of degree 2.    Proof. Well-known. □ The estimation for the mean value of the Möbius function depends on the zero-free region of Riemann zeta function (see for example Theorem 12.7 [16]).
Error term of the mean value theorem for binary Egyptian fractions  1253 The following results about moments of zeta function and Dirichlet L-functions are well-known (see [17] and Theorem 10.1 of [18]).
where ∑ ⁎ indicates that the sum is over the primitive characters modulo a.
Proof. Let χ ⁎ modulo q be a primitive character. Then we have Then according to Lemma 6, we get The following lemma is due to [15], which gives an upper bound for some summation in short intervals.
First of all, we assume that where χ 0 is the principal character modulo a and ( ) F s χ can be analytically continued to . If χ is a non-principal character modulo q ( χ ⁎ is the primitive character induced from χ), we have Hence, we can write with (⋅ ) d χ ; is a parameter to be determined. For the first sum, by using (6) and Lemma 3, we get Proof. We can apply similar argument to the proof above and get , exp log log log , where C 1 and C 2 are absolute constants. By using (6) again, we have Thanks to (7) and (9), the first term on the right is log log log α ε 2 1 2 9 3 5 1 5 and the second term on the right is α ε ε 10 2 0 Therefore, we have Error term of the mean value theorem for binary Egyptian fractions  1257 By an appeal to (8), (10) and (14) exp log log log . Therefore, we obtain