On the mean square of the divisor function in short intervals

The meaning of “short” comes from the condition h = o(x) as x → ∞, so that h is indeed much smaller in comparison with x. As in analytic number theory one is usually interested in the averages of error terms, where the averaging usually smoothens the irregularities of distribution of the function in question, we shall be interested in mean square estimates of (1.2), both discrete and continuous. To this end we therefore define the following means (to stress the analogy between the discrete and the continuous, x is being kept as both the continuous and the integer variable): (1.3)
1. Introduction Let, for a fixed integer k ≥ 2, dk (n) denote the (generalized) divisor function which denotes the number of ways n can be written as a product of k factors. This is a well-known multiplicative function (dk (mn) = dk (m)dk (n) for coprime m, n = 1 ∈ N, and d2 (n) ≡ d(n) is the classical number of divisors function). Besides this definition one has the property that dk (n) is generated by ζ k (s), where ζ (s) is the Riemann zeta-function, defined as
where Pk−1 (t) is a polynomial of degree k − 1 in t, all of whose coefficients can be calculated explicitly (e.g., P1 (t) = log t + 2γ − 1, where γ = −Γ′ (1) = 0.577215 . . . is Euler’s constant). Thus ∆k (x) represents the error term in the asymptotic formula for the summatory function of dk (n), and a vast literature on this subject exists (see e.g., [2] or [6]). Here we shall be concerned with the “short difference” (1.2) ∆k (x + h) − ∆k (x) (1 ≪ h ≪ x, h = o(x) as x → ∞).
1
2
A. Ivi´ c
and this connects the problems related to dk (n) to zeta-function theory. Let as usual (1.1)
n≤x
dk (n) = Res
s=1
xs ζ k ( s ) + ∆k (x) = xPk−1 (log x) + ∆k (x), s
k
(X, h) : =
X ≤x≤2X
(∆k (x + h) − ∆k (x)) ,
2
2X
(1.4)
Ik (X, h) : =
X
(∆k (x + h) − ∆k (x)) dx.
2
The problem is then to find non-trivial upper bounds for (1.3)–(1.4), and to show for which ranges of h = h(k, X ) (= o(X ) ) they are valid. Namely the “trivial” bound in all cases is the function X 1+ε h2 , where here and later ε > 0 denotes arbitrarily small positive constants, not necessarily the same ones at each occurrence. This comes from the elementary bound dk (n) ≪ε nε and (1.5) ∆k (x + h) − ∆k (x) = dk (n) + xPk−1 (log x) − (x + h)Pk−1 (log(x + h))
x<n≤x+h
≪ ε xε
x<n≤x+h
1 + h logk−1 x ≪ε xε h (1 ≪ h ≪ x),
On the mean square of the divisor function in short intervals
3
where we used the mean value theorem. Note that, from the work of P. Shiu [5] on multiplicative functions, one has the bound dk (n) ≪ h logk−1 x ( xε ≤ h ≤ x) ,
3 /2
√
x h
(xε ≤ h ≤ x1/2−ε ).
4 Here, as usual,
T
A. Ivi´ c
E (T ) :=
0
1 + it)|2 dt − T |ζ ( 2
log
T + 2γ − 1 2π
1 + it)| (see e.g., represents the error term in the mean square formula for |ζ ( 2 Chapter 15 of [2] for a comprehensive account), while f (x) = Ω(g (x)) means that limx→∞ f (x)/g (x) = 0. Namely Jutila (op. cit.) has shown that the integral in (1.10) equals the expression on the right-hand side of (1.8), hence the conclusion follows from Theorem 2 and the above discussion. The omega-result (1.11) shows that the difference E (x + h) − E (x) cannot be too small in a fairly wide range for h. An omega-result analogous to (1.11) holds for ∆(x + h) − ∆(x) as well, namely
2X
(∆k (x + h) − ∆k (x))2 dx

(h = h(X ) ≫ 1, h = o(x) as X → ∞)
X
where h lies in a suitable range. For k ≥ 2 a fixed integer, ∆k (x) is the error term in the asymptotic formula for the summatory function of the divisor function dk (n), generated by ζ k (s).
arXiv:0708.1601v1 [math.NT] 12 Aug 2007
ON THE MEAN SQUARE OF THE DIVISOR FUNCTION IN SHORT INTERVALS
´ Aleksandar Ivic
Abstract. We provide upper bounds for the mean square integral Z
x<n≤x+h
hence in this range (1.5) can be improved a bit, and thus we can also consider (1.6) ∆k (x + h) − ∆k (x) ≪ h logk−1 x ( xε ≤ h ≤ x)
as the “trivial” bound. It should be mentioned that the cases k = 2 of (1.3) and (1.4) have been treated by Coppola–Salerno [1] and M. Jutila [4], respectively, so that we shall concentrate here on √the case when k > 2. In these papers it had 1 ε been shown that, for X ≤ h ≤ 2 X, L = log X , (1.7) 8 (X, h) = 2 Xh log3 2 π
∞
ζ (s) =
n=1
n−s
(σ = ℜe s > 1),
and otherwise by analytic continuation. Namely
∞ k ∞ k −s
ζ (s) =
n=1
n
=
n=1
dk (n)n−s
(σ = ℜe s > 1),
1991 Mathematics Subject Classification. 11 M 06, 11 N 37. Key words and phrases. The divisor functions in short intervals, the Riemann zeta-function, mean square bounds. Typeset by AMS-TEX
2X
√
X h
√ + O(XhL5/2 L),
(1.8)
1 I2 (X, h) = 4π 2
X n≤ 2 h
d2 (n) n3 /2
x
X
1 /2
exp 2πih
n x
2
−1
dx
+ Oε (X 1+ε h1/2 ). From (1.8) Jutila deduces (a ≍ b means that a ≪ b ≪ a) √ X 3 (1.9) I2 (X, h) ≍ Xh log (X ε ≤ h ≤ X 1/2−ε ), h but it is not obvious that I2 (X, h) is asymptotic to the main term on the right-hand side of (1.7). This, however, is certainly true, and will follow from our Theorem 2 and from (1.7). Theorem 2 says that, essentially, the sums k (X, h) and Ik (X, h) √ are of the same order of magnitude. It is also true that, for X ε ≤ h ≤ 1 X, L = 2 log X , √ 2X √ 8 X 2 (1.10) (E (x + h) − E (x)) dx = 2 Xh log3 + O(XhL5/2 L), π h X implying in particular that (1.11) E (x + h) − E (x) = Ω √ h log