Dirichlet polynomial approximations to zeta functions

In this paper we consider L-functions satisfying certain standard conditions, their approximation by Dirichlet polynomials and, especially, lower bounds for the lengths of the polynomials that provide good approximations. 1. Introduction For ~(s) the Riemann zeta-function one has the Dirichlet series representation valid for where s = u + it. By the absolute convergence of this series one sees that, even for x not very large, the Dirichlet polynomial L n-s gives a rather good n2 approximation to ~(s), with a remainder which is o( 1 ) as x -~ oo. This is a nice property, since one would expect the finite sum to be easier to work with for purposes of estimation. However, one is of course more interested in estimating ~(s) in the critical strip 0 u 1. Here the above polynomial still provides [T, § 4.11 ] (at least away from the pole) a useful approximation to ~(s), moreover the smoothed polynomials * Supported in part by NSERC Grant A5123. Pervenuto alla Redazione il 2 Luglio 1994. 518 do an even better job, but all of these only for x > (1 + 0 ( 1 )) l! l, and this limits their usefulness for application. 27r Thus the question arises whether shorter approximations of the same quality to ~(s), or for that matter to L-series and general zeta functions, exist. In this paper we investigate approximations by Dirichlet polynomials to L-functions of a fairly general type, and show in many cases that it is not possible to achieve a very good level of approximation by means of polynomials essentially shorter than the known approximations. Thus we may view such a result as a first step toward understanding the analytic complexity of a zeta function. We shall consider L-functions L(s) having the following properties (compare, for example, [S]): (HI) L(s) is given by an absolutely convergent Dirichlet series in the half-plane u > 1, with coefficients an satisfying a, = 1 and an « n°(1). (H2) L(s) is meromorphic of finite order in the whole complex plane, has only finitely many poles and satisfies a functional equation 1 where with constants satisfying From the fact that L(s) is of finite order with finitely many poles and satisfies a functional equation of the above type and from the Phragmen-Lindelof principle, it follows that L(s) has, away from the poles, polynomial growth in any fixed vertical strip. Moreover L(s), for a 1 has order not less than 2 1 and for a 0 has order precisely where It now follows by a well-known argument (cf. [T, § 9.4]) that the number N(T; L) of non-trivial zeros (that is, those not located at the poles of the r factors) of L(s) satisfying 0 t T, is given asymptotically by 1 For a function f (s) we define 7(s)=f(-~). 519 where cL is a constant depending on L. Since we assume a 1 = 1 we may compute the constants explicitly and write this in the form where One should remark that the choice of the parameter Q and the Gamma factors in the above decomposition of are not uniquely determined due to the multiplication formula for the Gamma function. However the key quantities A and CL used in this paper are uniquely determined by L(s). The next assumption that we make about our L-function is that it satisfy a weak zero-density estimate. Let N(u, T; L) denote the number of non-trivial zeros p = {3 + iq of L with 0 q T and {3 > ~ . Then we assume: (H3) For any fixed 6 > 0, we have Our two main results place a limitation on the length of the Dirichlet polynomial (actually, may be replaced by any fixed positive constant) if it is to be a 2 useful approximation to L(s). Specifically, we prove THEOREM 1. Let L(s) satisfy assumptions (Hl)-(H3), and let e,,-’ > 0. Suppose that we have on the segment Our basic strategy is to use a well known lemma of Littlewood to compare, in a suitable rectangle, the number of zeros of the function L(s) with that of the approximating polynomial These should be nearly equal if the approximation is sufficiently good. On the other hand we shall be able to estimate the former using (1.1). This will give a contradiction provided that we can give a smaller upper bound to the number of zeros of D2(s) in case x is not too large. Such a result is provided by the following: 520 PROPOSITION 1. Let Dx(s) given by (1.2) satisfy Then, uniformly for a a oo we have Let also N(a, T, T + H; Dx) denote the number of zeros of D2(s) satisfying u > a, T t T + H, where H T. Then, uniformly for -H a l, we have where the implied constant is absolute. The exponent 2A given in Theorem 1 is sharp, as will be seen in the next section. Nevertheless, a slightly different argument using Rouche’s theorem shows that the bound can be made still more precise if one is willing to strengthen the assumptions to some extent. Specifically, we have: THEOREM 2. Let L(s) satisfy assumptions (HI), (H2), and also, for every 6 > 0, the strengthened zero density bound Suppose that we have on the segment (u = -c’, T t (I + Then x > (1 + o(I»CLT2A, with CL as in ( 1.1’). In Theorem 2, not only the exponent, but even the constant CL is the best possible. We remark that the assumption of (1.3) on the segment with u = -g’ is stronger than the assumption on a corresponding segment with a = 1 -s’; see Proposition 3. In the event that one assumes a stronger version as in Theorem 2, but is willing to settle for the weaker conclusion of Theorem 1, then it is possible to give a somewhat simpler proof which combines the principle of the argument with the result of Proposition 1. Throughout the paper, implied constants may depend on L(s) which is considered to be fixed. It would be of interest to have analogous results that are uniform in the parameter Q. The paper is organized as follows. In Section 2 we give a number of examples illustrating the sharpness of our results. In Section 3 we give an alternative argument that is considerably shorter than the proof of the main 521 theorems, but which gives only weaker bounds except in the case A 1. The remaining sections are devoted to the proof of the Theorems. In Section 4, we give the proof of Proposition 1, bounding the number of zeros of Dirichlet polynomials. In Section 5 we prove Proposition 3 which shows that, given an approximation of the type hypothesized by the theorems, that approximation continues to hold for all larger values of u. In Section 6 we prove a number of consequences of our hypothesis of a zero-density bound. We find, with good localization, thin horizontal strips on which there holds a Lindelof strength bound for L, and within each of these, a horizontal line on which holds a similar bound for L-1. These bounds, which are needed for our application of the Littlewood lemma and the Rouche theorem, improve earlier results which would not have sufficed. Finally, in Section 7, we combine the above preparations to complete the proofs of our results. 2. Some Examples and Remarks EXAMPLE 1. Zeros of Dirichlet polynomials The finite Euler product has length by the prime number theorem and has, for T t 2T, zeros on the imaginary axis at t = 2n7r/ log p. These number again by the prime number theorem. Thus the bound given in Proposition 1 is asymptotically sharp. EXAMPLE 2. Approximate functional equation As is well known, it is possible to approximate the Riemann zeta function, using two Dirichlet polynomials rather than one, in a way which allows shorter polynomials, namely: 522 for x y : , and s in any fixed vertical strip away from the pole at Here appears in the functional equation It is likely that there should be an analogue to our Theorem for approximate functional equations of this type stating that such an approximate functional equation can only hold for L(s) in the range T t 2T provided that xy > At first we hoped that our method, based on counting zeros, would lead to this result, but were stopped by the following example which shows that the analogue for Proposition 1 (at least in its obvious form) does not hold. Take L(s) to be ~(s); take x = y = 1. Then the "approximation" is 1 + x(s) and, despite the fact that x and y are bounded, this has asymptotically (in fact, with an error term only O(log T)) as many zeros as ~(s) itself inside the rectangle 0 Q 1, T t 2T. EXAMPLE 3. Existence of approximations It is well-known that smoothed truncations of a Dirichlet series can provide very good approximations. Let u(x) be a C°° function with compact support in (0, 1], such that -