A NOTE ON THE ZEROS OF ZETA AND L-FUNCTIONS EMANUEL CARNEIRO, VORRAPAN CHANDEE AND MICAH B. MILINOVICH 1 Abstract. Let πS(t) denote the argument of the Riemann zeta-function at the point s = 2 + it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound (and prove some related results) for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. 1. Introduction Let ζ(s) denote the Riemann zeta-function and, if t does not correspond to an ordinate of a zero of ζ(s), let 1 S(t) = arg ζ( 1 +it); π 2 where the argument is obtained by continuous variation along straight line segments joining the points 1 2; 2 + it, and 2 + it, starting with the value zero. If t does correspond to an ordinate of a zero of ζ(s), set 1 S(t) = lim S(t+") + S(t−") : 2 "!0 The function S(t) arises naturally when studying the distribution of the nontrivial zeros of the Riemann zeta-function. For t > 0, let N(t) denote the number of zeros ρ = β + iγ of ζ(s) with ordinates satisfying 1 0 < γ ≤ t where any zeros with γ = t are counted with weight 2 . Then, for t ≥ 1, it is known that t t t 7 1 N(t) = log − + + S(t) + O : 2π 2π 2π 8 t In 1924, assuming the Riemann hypothesis, Littlewood [12] proved that log t S(t) = O log log t as t ! 1. The order of magnitude of this estimate has never been improved. Using the Guinand-Weil explicit formula for the zeros of ζ(s) and the classical Beurling-Selberg majorants and minorants for characteristic functions of intervals, Goldston and Gonek [9] established Littlewood's estimate in the form 1 log t jS(t)j ≤ +o(1) (1.1) 2 log log t as t ! 1. The authors [4] recently sharpened this inequality and proved the following theorem. Theorem 1. Assume the Riemann hypothesis. Then 1 log t jS(t)j ≤ +o(1) 4 log log t for sufficiently large t, where the term of o(1) is O(log log log t= log log t). 2000 Mathematics Subject Classification. 11M06, 11M26, 11M36, 11M41, 41A30. Key words and phrases. Riemann zeta-function, automorphic L-functions, Beurling-Selberg extremal problem, extremal functions, exponential type. 1 In this note, we give a new and much simpler proof of Theorem 1. As in the work of Goldston and Gonek mentioned above, we use the explicit formula in conjunction with the classical majorants and minorants of exponential type for characteristic functions of intervals that were constructed by Beurling and Selberg. In [4], two proofs of Theorem 1 are given. The more direct of these proofs proceeds as follows. Assuming the Riemann hypothesis, for t not corresponding to an ordinate of a zero of ζ(s), it is shown in [4] that 1 X S(t) = F (t−γ) + O(1); π γ where 1 x F (x) = arctan − ; (1.2) x 1 + x2 1 and the sum runs over the nontrivial zeros ρ = 2 + iγ of ζ(s) counted with multiplicity. Drawing upon the work of Carneiro and Littmann [5], upper and lower bounds for S(t) are established by replacing F (x) in the sum over zeros by (optimally chosen) real entire majorants and minorants of F (x) and then applying the explicit formula. Using similar ideas, Chandee and Soundararajan [6] proved that log 2 log t log jζ( 1 +it)j ≤ +o(1) 2 2 log log t as t ! 1, also under the assumption of the Riemann hypothesis. The theory of extremal functions of exponential type has also been recently used in [3] to improve the upper and lower bounds for the pair correlation of zeros of ζ(s) under the Riemann hypothesis. In Section 4, we discuss how to extend Theorem 1 to a large class of L-functions and prove a number of other related results. If an L-function is self-dual, our new proof of Theorem 1 generalizes in a natural way. However, if an L-function is not self-dual, then our new proof does not generalize but we show that an analogue of Theorem 1 is still possible using a modification of our previous approach in [4]. 2. Beurling-Selberg majorants and minorants For functions R 2 L1(R), let Z 1 Rb(ξ) = R(x) e−2πixξ dx −∞ denote the Fourier transform of R. In the 1930s, Beurling observed that the real entire functions 2 ( 1 ) 2 sin πz X sgn(m) 2 sin πz H±(z) = + ± π (z−m)2 z πz m=−∞ satisfy the inequalities H−(x) ≤ sgn(x) ≤ H+(x) for all real x, have Fourier transforms Hb ±(ξ) supported in [−1; 1], and satisfy Z 1 ± H (x)−sgn(x) dx = 1: −∞ Moreover, Beurling showed that among all real entire majorants and minorants for sgn(x) with Fourier transforms supported in [−1; 1], the functions H±(x) minimize the L1(R)-distance to sgn(x). The details of this construction can be found in [20]. For t > 0, let χ[−t;t](x) denote the characteristic function of the interval [−t; t] (normalized at the end- points). From the above properties of Beurling's functions H±(z), Selberg observed that the real entire functions 1 n o R±(z) = H±∆(t+z) + H±∆(t−z) (2.1) 2 2 − + satisfy the inequalities R (x) ≤ χ[−t;t](x) ≤ R (x) for all real x and have Fourier transforms supported in [−∆; ∆]. We summarize these (and some other relevant) properties of the functions R±(z) in the following lemma. Lemma 2. Let t and ∆ be positive real numbers. The even real entire functions R±(z) defined in (2.1) satisfy the following properties: − + (i) R (x) ≤ χ[−t;t](x) ≤ R (x) for all real x ; Z 1 ± 1 (ii) R (x) − χ[−t;t](x) dx = ; −∞ ∆ (iii) R±(z) e2π∆jIm zj ; (iv) R±(x) min(1; ∆−2(jxj − t)−2) for jxj > t ; (v) Rb±(ξ) = 0 for jξj ≥ ∆ ; sin 2πtξ 1 (vi) Rb±(ξ) = + O for jξj ≤ ∆. πξ ∆ The implied constants in (iii), (iv) and (vi) are absolute. Proof. This is a specialization of [9, Lemma 2]. For a further discussion on these Beurling-Selberg extremal functions, as well as proofs of the above assertions, see the survey articles of Montgomery [14], Selberg [19] and Vaaler [20]. 3. Proof of Theorem 1 The functional equation for the Riemann zeta-function, in symmetric form, can be written as ΓR(s)ζ(s) = ΓR(1−s)ζ(1−s) s − 2 s where ΓR(s) = π Γ( 2 ). Lemma 3. For all t > 0, we have 1 Z t Γ0 R 1 N(t) = ( 2 +iu) du + S(t) + 1: 2π −t ΓR Proof. Using the functional equation for ζ(s), the argument principle, and the Schwarz reflection principle, it can be shown that 1 N(t) = arg Γ ( 1 +it) + S(t) + 1 π R 2 for any t > 0. See, for instance, [15, Theorem 14.1]. Since Z t Γ0 R 1 1 ( 2 +iu) du = 2 arg ΓR( 2 +it); −t ΓR the lemma follows. 1 Lemma 4. Let h(s) be analytic in the strip jIm sj ≤ 2 + " for some " > 0, and assume that jh(s)j (1 + jsj)−(1+δ) for some δ > 0 when jRe sj ! 1. Then X ρ− 1 1 1 1 Z 1 Γ0 h 2 = h + h − + h(u) Re R ( 1 +iu) du i 2i 2i π Γ 2 ρ −∞ R (3.1) 1 1 X Λ(n) log n log n − p bh + bh − ; 2π n 2π 2π n=2 3 where the sum on the left-hand side runs over the nontrivial zeros ρ of ζ(s) and Λ(n) is the von Mangoldt function defined to be log p if n = pk, p a prime and k ≥ 1, and zero otherwise. Proof. This is [9, Lemma 1]. For a similar formula, see [10, Equation (25.10)]. We now deduce Theorem 1 from Lemmas 2, 3 and 4. Proof of Theorem 1. Assume the Riemann hypothesis and that ∆ ≥ 1 and t ≥ 10. By Lemma 2 (i), we have X n + o X + 0 ≤ R (γ) − χ[−t;t](γ) = R (γ) − 2 N(t) (3.2) γ γ 1 + where the sum runs over the ordinates of the nontrivial zeros ρ = 2 +iγ of ζ(s). Note that R (u) is even and satisfies the conditions of Lemma 4. In fact, the growth condition required in Lemma 4 follows by Lemma 2 (iii)-(iv) via the Phragm´en-Lindel¨ofprinciple. Using inequality (3.2) in conjunction with Lemma 3 and (3.1), we derive that 1 Z 1 n o Γ0 1 1 Λ(n) log n 1 + R 1 X + + S(t) ≤ R (u)−χ[−t;t](u) ( +iu) du − p Rb + R − 1: (3.3) 2π Γ 2 2π n 2π 2i −∞ R n=2 + 1 π∆ Lemma 2 (iii) implies that R 2i e , while Lemma 2 (v)-(vi) imply that the sum over prime powers in (3.3) is X Λ(n) 1 1 1 p + eπ∆ n log n ∆ ∆ n≤e2π∆ by the Prime Number Theorem and partial summation. Therefore 1 Z 1 n o Γ0 + R 1 π∆ S(t) ≤ R (u)−χ[−t;t](u) ( 2 +iu) du + O e : (3.4) 2π −∞ ΓR It remains to estimate the integral in (3.4).