MEAN NUMBER OF REAL ZEROS OF A RANDOM TRIGONOMETRIC POLYNOMIAL . III

If al, a2,...,an are independent, normally distributed random variables with mean 0 and variance 1, and if n is the mean number of zeros on the interval (0, 2r) of the trigonometric polynomial alcos x + 21/2a2cos 2x +...n1/2ancos nx, then n 2 /2{(2n + I) + D + (2n + i)102 + (2n + i)203} + O{(2n + i)-3}, in which D-0.378124, D2 = -1/2, D30.5523. After tabulation of 5D values of un when ni(i)40, we find that the approximate formula for obtained from the above result when the error term is neglected, produces 5D values that are in error by at most i0 -5 when n >_ 8, and by only about 0.1% when n2.


Introduction
Suppose that n is an integer, greater than 1, that aj (j-1,2,...,n) are independent, normally distributed random variables with mean 0 and variance 1, that p is a real number greater than 1 2' and that unp is the mean value of the number of zeros on the interval (0, 2r) of the random trigonometric polynomial E jPajcos jx.
(1.3) r--0 It follows that the error term O(n1/2) in the Das result is actually O(1) when p is a nonnegative 1 integer. In this paper we will prove that a relation of the form (1.3) is also valid when p-.
This proof emulates the analysis in [7], although that analysis actually fails when p is not a positive integer.
After a statement of the basic formulas on which our analysis rests, we devote Section 2 to the derivation of a series representatio n of n 1/2 that converges when n is sufficiently large.
(Henceforth we will omit the subscript 1/2.) Asyltotic representations of the first four coefficients in that series are derived in Section 3, and are used to deduce (1.3) when p-1 / 2 . We tabulate in Section 4 5D values of n when n-1(1)40. We find that the approximation to n, obtained from (1.3) when the O{(2n / 1) -3} term is neglected, produces 50 values that differ from the tabulated values by at most 10-5 when n _> 8 and by only about 0.1% when n-2. In Section 5 we show that the series representation of n, derived in Section 2, actually converges when n >_ 2.
If we use (2.2) and Lemmas 2 and 4, we obtain the following lemma. Lemma5: It is true when O <_ x <_ r/2 and n >_ n o that 2Fn(x (2n + 1)G(z) (2n + 1)-rut, Moreover, the series (2.23) converges absolutely and uniformly in x and n. The final lemma in this section is a consequence of (2.1) and Lemma 5. Lemma6: It is true when n >_ n o that u n (2n + 1)E (2n + 1)-rvr, In the next four lemmas we will exhibit constants Srm(0 _< m < 3-r, r-0,1,2,3) and S r (r 0, 1, 2, 3) such that rn--0 when r-0,1,2,3. In the proofs of these lemmas, it will be convenient to use Tq(z) as a generic symbol for a trigonometric sine polynomial of degree q not necessarily the same at each occurrence.
Hence equations (3.44) through (3.47) are valid for all positive z. Each of the integrals with respect to x on the interval (0, zr/2) of each of the six terms on the right hand side of (3.43) is, therefore, O{(2n-4-1)-1}. In view of (2.26), this remark suffices to prove Lemma 10.
Our principal result, stated in the following theorem, is an immediate consequence of Lemmas 6 through 10 and the observation that S O + S 1 + S 2 -4-5' 3 0.

The Integer n o
Although it is not logically necessary to know a specific value for the integer n o in the theorem and Lemmas 2, 4, 5 and 6, it is interesting to observe that n o can actually be chosen as small as 2. We begin the proof of this assertion with the following lemma.
The lemma is an immediate consequence of Lemmas 11 and 12, and the fact that (2.17) is implied by (5.6). (Although adequate, this result is rather crude. It is possible to show, with the help of numerical calculations similar to those we will use below in the proofs of Lemmas 14 and 15, that the left hand side of (5.6) does not exceed 0.299394.) The analysis to show that (2.22) is true when n > 2 and 0 < x < 7r/2, so that Lemma 4 is true when n o 2, is somewhat more recondite. We begin with the following assertion.

Therefore, Lemmas
The first sentence in Lemma 16 is an immediate consequence of Lemmas 14 and 15. Therefore, Lemma 4 is true when n o 2. This remark and Lemma 13 then show that Lemmas 5 and 6 are true when n o -2.