MEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION. S.M. Gonek University of Rochester

Transcription

1 MEAN VALUE THEOREMS AND THE ZEROS OF THE ZETA FUNCTION S.M. Gonek University of Rochester June 1, 29/Graduate Workshop on Zeta functions, L-functions and their Applications 1

2 2 OUTLINE I. What is a mean value theorem? II. Mean values and zeros. III. A sample of important estimates. IV. Application: A simple zero density estimate. V. Application: Levinson s method. VI. Application: The number of simple zeros.

3 I. What is a mean value theorem? 3 In general it is an estimate for an average of a function. For example 1 2π 2π f(z + Re iθ ) 2 dθ. For a function with a Dirichlet series, F (s) = n=1 a nn s, the average is typically along a vertical segment: F (σ + it) 2 dt, or F (σ + it) dt. Note: 1) The path of integration might not be in the half plane of convergence. 2) It is customary not to divide by T.

4 4 There are many variants, for example, an average over a discrete set of points: R F (σ r + it r ) 2 (σ r + it r C). r=1 Example 1. Take F (s) = ζ(s) k, σ 1/2, and k a positive integer. We are interested in the means I k (σ, T ) = ζ(σ + it)k 2 dt = ζ(σ + it) 2k dt. Example 2. Take F (s) = (ζ (s)) k, S a set of zeros ρ = β + iγ of ζ(s). One can consider the means ζ (ρ) 2k. ρ S

5 Example 3. Let 5 F (s) = F N (s) = N a n n s n=1 be a Dirichlet polynomial of length N. One can show that N a n n σ it 2 dt = (T + O(N log N)) n=1 N n=1 a n 2 n 2σ. This is the classical mean value theorem for Dirichlet polynomials. A stronger version, due to H. L. Montgomery and R. C. Vaughan, is N a n n σ it 2 dt = n=1 N n=1 a n 2 (T + O(n)). n2σ

6 6 II. Mean values and zeros. Mean value estimates are used to study the zeros in a variety of ways. One direct link between them and the zeros of an analytic function is given by Theorem. (Jensen s Formula) Let f(z) be analytic for z R and suppose that f(). Let r 1, r 2,..., r n be the moduli of all the zeros of f(z) inside z R. Then log( f() Rn r 1 r 2 r n ) = 1 2π 2π log f(re iθ ) dθ. That is, the distribution of zeros of f(z) inside the circle is related to the mean of log f(z) on the circle. There is an analogous result for rectangles that is more useful when working with Dirichlet series.

7 Theorem. (Littlewood s Lemma) Let f(s) be analytic and 7 nonzero on the rectangle C with vertices σ, σ 1, σ 1 + it, σ + it. Then 2π ρ C dist(ρ) = log f(σ + it) dt log f(σ 1 + it) dt + σ1 σ arg f(σ + it ) dσ σ1 σ arg f(σ) dσ, where the sum runs over the zeros ρ of f(s) in C and dist(ρ) is the distance from ρ to the left edge of the rectangle.

8 8 Proof of Littlewood s Lemma. Let C denote the rectangle C together with the loops L ρ around each zero ρ (see the figure). Then log f(s) ds = log f(s) ds + C C ρ C L ρ log f(s) ds. Now log f(s) is analytic and single valued in C, so C log f(s) ds =. Therefore, C log f(s) ds = ρ C L ρ log f(s) ds. If ρ = β + iγ, and the circle in L ρ has radius r, then β r 2π log f(s) ds = log f(σ + iγ ) dσ + log f(re iθ ) ire iθ dθ L ρ σ β r σ log f(σ + iγ + ) dσ.

9 9 The second integral as r +. The third integral is β r ( log f(σ + iγ ) + 2πi ) dσ. σ Hence, as r +, log f(s) ds 2πi L ρ β σ dσ = 2πi(β σ ). Therefore C log f(s) ds = 2πi ρ C (β σ ). We may write this as 2πi ρ C (β σ ) = log f(σ 1 + it) idt log f(σ + it) idt + σ1 σ log f(σ) dσ σ1 The result follows on equating imaginary parts. σ log f(σ + it ) dσ.

10 1 Only the first term on the right in Littlewood s Lemma will be significant for us, so we will write 2π dist(ρ) = ρ C log f(σ + it) dt log f(σ 1 + it) dt as + σ1 σ arg f(σ + it ) dσ σ1 σ arg f(σ) dσ, 2π ρ C dist(ρ) = log f(σ + it) dt + E and ignore E. Usually we cannot estimate the integral directly, so we use the following trick: 1 T log f(σ + it) dt = 1 2T log( f(σ + it) 2 ) dt 1 2 log( 1 T f(σ + it) 2 dt), ( The average of the log is the log of the average. ) Note: Our mean values appear!

11 III. A sample of important estimates. 11 Recall that I k (σ, T ) = ζ(σ + it) 2k dt. The case k = 1. If σ > 1/2, as T. I 1 (σ, T ) = ζ(σ + it) 2 dt c 1 (σ) T, Hardy and Littlewood (1918) proved that if σ = 1/2, then I 1 (1/2, T ) T log T. In particular, ζ(σ + it) is smaller on average for σ > 1/2 then for σ = 1/2. Since ζ( it) has many zeros, we should expect ζ(s) to be very erratic on σ = 1 2.

12 12 The case k = 2. If σ > 1/2, as T. I 2 (σ, T ) = ζ(σ + it) 4 dt c 2 (σ) T, Ingham (1926) proved that I 2 (1/2, T ) T 2π 2 log4 T. The case k > 2. No asymptotic has yet been proven. Balasubramanian and Ramachandra have shown that I k (1/2, T ) T log k2 T, and we expect that I k (1/2, T ) c k T log k2 T.

13 13 Conrey and Ghosh conjectured that That is, that c k = a kg k Γ(k 2 + 1). Here a k = p I k (1/2, T ) a kg k Γ(k 2 + 1) T logk2 T. ( ( 1 1 ) (k 1) 2 k 1 p r= and g k is some (then unknown) constant. ( ) ) 2 k 1 p r J. Keating and N. Snaith used random matrix theory to conjecture the value of g k for all k. When k is an integer their conjecture is r Conjeture. (Keating Snaith) g k = (k 2!) k 1 j= j! (j + k)!.

14 14 Another important mean is (1) ζ (j) (σ + it)m N (σ + it) 2 dt, where M N (s) = 1 n N µ(n) n s log n (1 log N ). M N (s) approximates 1/ζ(s) when σ > 1. This continues for σ 1 in some sense. Hence, ζ(s)m N (s) should be tamer than ζ(s) on σ = 1 2. General estimates for means like (1) were proved by Conrey, Ghosh, and Gonek with N = T θ and θ < 1/2. Later, Conrey used Kloosterman sum techniques to show these formulas also hold for θ < 4/7.

15 15 Assuming the Riemann Hypothesis and the Generalized Lindelöf Hypothesis are true, Conrey, Ghosh, and Gonek also proved discrete versions of this, including estimates for sums like <γ<t ζ (ρ)m N (ρ) 2. Here γ runs over the ordinates of the zeros ρ = iγ of ζ(s). Again N = T θ and θ < 1/2.

16 16 IV. Application: A simple zero density estimate. Let N(σ, T ) = ρ=β+ıγ <γ T σ<β 1 We want an upper bound for N(σ, T ) when 1 2 < σ 1 is fixed. 1.

17 17 We apply Littlewood s Lemma on the rectangle C with vertices σ, 2, 2 + it, σ + it, where 1 2 < σ 1 is fixed. ρ C dist(ρ) = 1 2π log( ζ(σ + it) ) dt + E. Let σ be a real number with σ < σ < 1. On the one hand, ρ C On the other hand, 1 2π by our trick. dist(ρ) ρ C σ β log( ζ(σ + it) ) dt = 1 4π dist(ρ) (σ σ )N(σ, T ). T 4π log( 1 T log( ζ(σ + it) 2 ) dt ζ(σ + it) 2 ) dt

18 18 Hence (σ σ )N(σ, T ) T 4π log( 1 T The integral is ζ(σ + it) 2 ) dt + E. I 1 (σ, T ) c 1 (σ ) T. Therefore, (σ σ )N(σ, T ) T 4π log c 1(σ ), and N(σ, T ) T. Since N(T ) T 2π log T, we see that 1 N(σ, T )/N(T ) = O( log T ) for any fixed σ > 1/2.

19 19 We may interpret this as saying that only an infinitesimal proportion of the zeros are to the right of any line Res = σ > 1/2. This was the first zero density estimate. It was proved by H. Bohr and E. Landau (1914). Since then, much stronger results have been proven, typically of the form N(σ, T ) << T λ(σ), where λ(σ) < 1 and λ(σ) is decreasing for σ > 1/2.

20 2 V. Application: Levinson s method. Recall that N(T ) = #{ρ = β + iγ ζ(ρ) =, < γ < T } T 2π log T and let N (T ) = # {ρ = 12 } + iγ ζ(ρ) =, < γ < T denote the number of zeros on the critical line up to height T. G. H. Hardy (1914) : N (T ) (as T ) G. H. Hardy-J. E. Littlewood (1921) : N (T ) > ct A. Selberg (1942) : N (T ) > c N(T ) N. Levinson (1974) : N (T ) > 1 3 N(T ) J. B. Conrey (1989) : N (T ) > 2 5 N(T )

21 Levinson s method begins with the following fact first proved by Speiser. 21 Theorem. (Speiser) RH ζ (s) in < σ < 1 2. In the early seventies, N. Levinson and H. L. Montgomery proved a quantitative version of this: Theorem. (Levinson-Montgomery) ζ(s) and ζ (s) have the same number of zeros inside C up to O(log T ). Proof. arg ζ ζ (s) = O(log T ), C and arg ζ ζ (s) = 2π(# zeros of ζ (s) in C # zeros of ζ(s) in C). C

22 22 Sketch of Levinson s method ζ (s) has N (T ) zeros here So does ζ(s) (up to O(log T )) and here Therefore N(T ) = N (T ) + 2N (T ) + O(log T ), or N (T ) = N(T ) 2N (T ) + O(log T ). We know N(T ), so we need an upper bound for N (T ).

23 23 N (T ) = the no. of zeros of ζ (s) here = the no. of zeros of ζ (1 s) here By the functional equation for ζ(s), ζ (s) has the same zeros in 1/2 < σ < 2, < t < T as where L(s) 1 2π log s. G(s) = ζ(s) + ζ (s) L(s), So we need an upper bound for the number of zeros of G(s) in the rectangle on the right.

24 24 We apply Littlewood s Lemma to C a with a = 1 2 c log T and c >. 1 2π log G(a + it) = dt + E dist(ρ ) zeros of G C a zeros of G C a β >1/2 dist(ρ ) (1/2 a)n (T ).

25 25 Thus, (1/2 a)n (T ) 1 2π = 1 4π T 4π log ( 1 T log GM(a + it) dt + E log GM(a + it) 2 dt + E ) GM(a + it) 2 dt + E, where M(s) = n T θ a n n s, a n = µ(n)n a 1/2 ( 1 log n ) log T θ, approximates 1/ζ(s). Thus, we require an estimate for GM(a + it) 2 dt. This is similar to mean values mentioned before.

26 26 Levinson (1974) : One can take θ = 1/2 ɛ. This gives N (T ) > 1 N(T ) (ast ). 3 J. B. Conrey (1989) : One can take θ = 4/7 ɛ. This gives N (T ) > 2 N(T ) (ast ). 5 As a function of θ, the asymptotic estimate for GM(a + it) 2 dt. is the same in both cases. D. Farmer has argued that this remains true for θ arbitrarily large. Farmer s conjecture imples that N (T ) N(T ).

27 VI. Application: The number of simple zeros. 27 Let N s (T ) = #{ρ = β + iγ ζ(ρ) =, ζ (ρ), < γ < T } We believe that for all T >, N(T ) = N (T ) = N s (T ). H. Montgomery (1974) : RH = N s (T ) > 2 3 N(T ) via the pair correlation method. Conrey, Ghosh, Gonek (1999): RH + GLH = N s (T ) > 19 N(T ) = (.73...)N(T ). 27

28 28 Sketch of the method This time we use discrete mean values. By the Cauchy Schwarz inequality <γ<t ζ (1/2+iγ) 2 ( <γ T 1/2+iγ is simple )( 1 <γ<t ζ (1/2 + iγ) 2), An asymptotic estimate for the means provides a lower bound for N s (T ). But this only leads to N s (T ) > ct. We lose in applying the Cauchy Schwarz inequality. To minimize the loss we mollify ζ (1/2 + iγ) by a Dirichlet polynomial M N (s). <γ<t ζ (ρ)m N (ρ) 2 ( <γ T ρ=1/2+iγ is simple )( 1 <γ<t ζ (ρ)m N (ρ) 2), An elaboration of the method shows that on the same hypotheses at least 95.5% of the zeros of ζ(s) are either simple or double.