Bulletin T.CXXIX de l’Académie serbe des sciences et des arts - 2004 Classe des sciences mathématiques et naturelles sciences mathématiques, N\underline{o} 29

THE CIRCLE AND DIVISOR PROBLEM

A. IVIĆ

(Presented at the 2nd Meeting, held on March 26, 2004)

A b s t r a c t. New proofs for the classical bounds

P (x) ≪ x^{1/3}, Δ(x) ≪ x^{1/3} log x

are given. Here P (x) denotes the error term in the classical circle, and Δ(x) in the classical divisor problem.

AMS Mathematics Subject Classification (2000): 11N37

Key Words: circle problem, divisor problem, Voronoi summation formula, Selberg class

Recently S.D. Miller and W. Schmid [7] gave a proof of the bound

P (x) ≪_{ε} x^{1/3+ε}  (1)

in the classical circle problem. Here as usual P (x) = ∑_{n≤x} r(n) − πx, r(n) is the number of representations of the natural number n as a sum of two integer squares, and ε > 0 denotes arbitrarily small numbers. The bound (1) was obtained by the use of the Voronoi summation formula

\begin{matrix}∑ & ′ \\ a≤n≤b \end{matrix} r(n)f(n) = π ∫_{a}^{b} f(x) dx + ∑_{n=1}^{∞} r(n) ∫_{a}^{b} f(x)J_{0}(2π\sqrt{xn}) dx, (2)

where f(x) is a suitable smooth function, J is the Bessel function, and ∑′ denotes that at n = a and n = b the summand is to be halved if a or b is an integer. In [7], (2) was proved by a two-dimensional Poisson summation formula, but it may be proved analogously like the classical Voronoi formula (see e.g., [2, Chapter 3]). Then, on p. 20, the authors say: “With more effort, one can remove ε from these bounds (i.e. (1), and the analogous bound in the divisor problem), and get Voronoi’s result

P (x) ≪ x^{1/3}.”  (3)

The aim of this note is to give a new, simple proof of (3) by using (2). We start by noting that, in view of the non-negativity of r(n), we have

∑_{n=1}^{∞} f_{−}(n)r(n) ≤ ∑_{X<n≤2X} r(n) ≤ ∑_{n=1}^{∞} f_{+}(n)r(n),  (4)

where f_{−}(x) is a smooth, non-negative function supported in [X, 2X] such that f(x) = 1 for x ∈ [X + G, 2X − G] (X^{ε} ≤ G ≤ \sqrt{X}), while similarly f_{+}(x) is supported in [X −G, 2X +G] and satisfies f(x) = 1 for x ∈ [X, 2X]. If henceforth we denote by f(x) either f_{−}(x) or f_{+}(x), then f^{(r)}(x) ≪_{r} G^{−r} (r = 0, 1, 2, …), and by (2) we have

∑_{n=1}^{∞} f(n)r(n) = πX + O(G) + ∑_{n=1}^{∞} r(n) ∫_{X−G}^{2X+G} f(x)J_{0}(2π\sqrt{xn}) dx. (5)

From the theory of Bessel functions we need only the relation (see e.g., N.N. Lebedev [6])

\frac{d}{dz} [z^{ν}J_{ν}(z)] = z^{ν}J_{ν−1}(z)  (6)

and the asymptotic expansion (k ∈ ℕ is arbitrary, but fixed, and | arg z| ≤ π − ε)

\begin{aligned}J_{ν}(z) & = (\frac{2}{πz} )^{1/2} cos(z − \frac{1}{2}νπ − \frac{1}{4}π) (∑_{j=0}^{k} c_{j}(ν)z^{−2j} + O(|z|^{−2k−2})) \\ & + (\frac{2}{πz} )^{1/2} sin(z − \frac{1}{2}νπ − \frac{1}{4}π) (∑_{j=1}^{k} d_{j}(ν)z^{1−2j} + O(|z|^{−2k−1})) , \end{aligned} (7)

with suitable constants c_{j}(ν), d_{j}(ν). By using (7) and the first derivative test (see e.g., [2, Lemma 2.1]) it is seen that the integral on the right-hand side of (5) is ≪ X^{1/4}n^{−3/4}, hence for Y ≥ 2 we have

∑_{n≤Y} r(n) ∫_{X−G}^{2X+G} f(x)J_{0}(2π\sqrt{xn}) dx ≪ ∑_{n≤Y} r(n)X^{1/4}n^{−3/4} ≪ (XY)^{1/4}

on using partial summation and ∑_{n≤x} r(n) ≪ x. Furthermore, by using (6) (with ν = 1, 2), performing two integrations by parts and noting that the support of f′′ has measure ≪ G, we obtain that

∑_{n>Y} r(n) ∫_{X−G}^{2X+G} f(x)J_{0}(2π\sqrt{xn}) dx

= ∑_{n>Y} \frac{r(n)}{π^{2}n} ∫_{X−G}^{2X+G} f′′(x)xJ_{2}(2π\sqrt{xn}) dx

≪ ∑_{n>Y} r(n)n^{−5/4}G^{−1}X^{3/4} ≪ X^{3/4}G^{−1}Y^{−1/4}.

Therefore (5) yields

\begin{aligned}∑_{n=1}^{∞} f(n)r(n) & = πX + O(G) + O((XY)^{1/4}) + O(X^{3/4}G^{−1}Y^{−1/4}) \\ & = πX + O(X^{1/3}) \end{aligned}

on choosing G = Y = X^{1/3}. From (4) and the above estimates we have then

∑_{X<n≤2X} r(n) = πX + O(X^{1/3}),  (8)

and (3) follows from (8) on replacing X by 2^{−j}X and summing over j = 1, 2, . . ..

An analogous reasoning gives also the classical bound

Δ(X) ≪ X^{1/3} log X, Δ(X) = ∑_{n≤X} d(n) − X(log X + 2γ − 1),

in the Dirichlet divisor problem, where d(n) is the number of divisors of n and γ = −Γ′(1) is Euler’s constant. This follows with the use of the corresponding Voronoi formula (see [2, Chapter 3]) for the divisor function, instead of (2). Namely we have

\begin{matrix}∑ & ′ \\ a≤n≤b \end{matrix} d(n)f(n) = ∫_{a}^{b}(log x + 2γ)f(x) dx + ∑_{n=1}^{∞} d(n) ∫_{a}^{b} f(x)α(xn) dx, (9)

where 0 < a < b < ∞, f(x) ∈ C^{2}[a, b], and in standard notation of Bessel functions

α(x)= 4K_{0}(4π\sqrt{x}) − 2πY_{0}(4π\sqrt{x})  (10)

= −\sqrt{2}x^{−1/4} (sin(4π\sqrt{x} − π/4) − (32π)^{−1} cos(4π\sqrt{x} − π/4)) + O(x^{−5/4}).

In case we are dealing with a Voronoi formula for a multiplicative function which is not necessarily non-negative, then often one can use the theorem of P. Shiu [8] on sums of multiplicative functions in short intervals. For example, such is the function a(n), the n-th Fourier coefficient of a holomorphic cusp form with respect to the full modular group, which is a normalized eigenfunction for the Hecke operators (see e.g., M. Jutila [4] for Voronoitype formulas for these functions). More generally, Voronoi-type formulas for smooth functions f can be obtained for a wide class of arithmetic functions (e.g., the so-called S-class of A. Selberg; see Kaczorowski-Perelli [5] for an comprehensive account). The author [3] obtained such a formula for the divisor function d_{r}(n), generated by ζ^{r}(s), r ∈ ℕ. The key to obtaining Voronoi-type formulas in such situations is the relation

\frac{1}{2πi} ∫_{σ−i∞}^{σ+i∞} F (s)G(s) ds = ∫_{0}^{∞} f(x)g(x) dx

which holds (see e.g., E.C. Titchmarsh [9]) under suitable conditions if F (s) and G(s) are Mellin transforms of f(x) and g(x), respectively. The ideas used in this note in proving (1) work well also in the general case, enabling one to get bounds for corresponding error terms without the “ε”-factor. The simple (explicit) form of the right-hand side of (2) (and (9)-(10)) is due to the fact that

\frac{2^{s−p−1}Γ(\frac{1}{2}s)}{Γ(p − \frac{1}{2}s + 1)} (0 < ℜe s = σ < p + \frac{3}{2})

is the Mellin transform of x^{−p}J_{p}(x), and a corresponding Mellin pair exists also in the case of the Voronoi formula (9). In the general case the function appearing in the summation formula is a ‘generalized’ Bessel function (see e.g., J.L. Hafner [1]), and its asymptotics may be found by the method developed in Hafner’s paper or by the author in [3].

REFERENCES

[1] J. L. H a f n e r, On the representation of summatory function of a class of arithmetic functions, in “Analytic Number Theory” (ed. M.I. Knopp), LNM 899, Springer Verlag, Berlin etc., 1981, 145-165.

[2] A. I v i ć, The Riemann zeta-function, Dover, Mineola (New York), 2003.

[3] A. I v i ć, On the ternary additive divisor problem and the sixth moment of the zetafunction, in “Sieve Methods, Exponential Sums, and their Applications in Number Theory” (eds. G.R.H. Greaves et al.), Cambridge University Press, Cambridge, 1996, 205-243.

[4] M. J u t i l a, A method in the theory of exponential sums, TATA Institute of Fundamental Research Lecture Notes 80, Springer Verlag, Berlin etc., 1987.

[5] J. K a c z o r o w s k i and A. P e r e l l i, The Selberg class; a survey, in “Number Theory in Progress” (eds. K. Györy et al.), Walter de Gruyter, Berlin–New York, 1999, 953-992.

[6] N. N. L e b e d e v, Special functions & their applications, Dover, New York, 1972.

[7] S. D. M i l l e r and W. S c h m i d, Summation formulas, from Poisson and Voronoi

   to the present, arXiv:math.NT/0304187, 2003.

[8] P. S h i u, A Brun–Titchmarsh theorem for multiplicative functions, J. reine angew. Math. 31(1980), 161-170.

[9] E. C. T i t c h m a r s h, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1948.

Aleksandar Ivić Katedra Matematike RGF-a Universiteta u Beogradu Djušina 7, 11000 Beograd Serbia (Yugoslavia) e-mail: aivic@mat.bg.ac.yu, ivic@rgf.bg.ac.yu