Science Record

7

The method used here is still the two-dimensional Van der Corput’s method. But, instead of approximating a double sum by a double integral, we twice used a modified one-dimensional inversion formula, as stated in the following lemma.

Lemma. Let f(x) be an algebraic function defined in the interval (a,b). Let :

A tt A. vr eA <—< x)|<—, oi.

Let (@, B) be the image of (a, 4) under the transformation y =f (x), then

Prifay)—vay) >» eritls) — eni4 > MiG) + O Clog (B =a 2) als

a<n<b a<v<p

OG = a RU! +R).

where 7, is the solution of f(7y) =x

The proof of this paper is based on the speciality of the problem. For example, corresponding to the non-vanishing property of the Hessian in [8], we get easily that the number of “zero lines” of this paper does not exceed 3 (in other words, let «= .2'/y’, the number of positive roots of « is less than or equal to 3).

_it should be noticed that, first, the method used in this paper is a simplification of the method of approximation by double integral, and may be used either to simplify Professor S. H. Min’s proof on the order of

iL : : ; . ‘ c(4+ it) or to improve on his result; secondly, the method used in this

paper is a refinement of approximation by double integral. If we use the original method, even neglecting the difficulty of the vanishing of the Hessian (which is the main difficulty), we can only obtain

1 13 i5

S€R®N* log 4, but the corresponding estimation of this paper is that

x es S <r" (RN)® log ®t,

where , } 7 R N’ - 3 Pe) = a 2niV x , , a © S= > DS #V9, RR <IR, N<N<IN, @ <u cys x=R y=N 4

Lvidently, it is impossible to derive 0< a by the original method.