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Lectures on Mean Values of the Riemann Zeta Function

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Lectures on Mean Values of the Riemann Zeta Function Lectures on Mean Values of The Riemann Zeta Function By A.Ivic Tata Institute of Fundamental Research Bombay 1991 Lectures on Mean Values of The Riemann Zeta Function By A. Ivic Published for the Tata Institute of Fundamental Research SPRINGER-VERLAG Berlin Heidelberg New York Tokyo Author A. Ivic S.Kovacevica, 40 Stan 41 Yu-11000, Beograd YUGOSLAVIA © Tata Institute of Fundamental Research, 1991 ISBN 3-350-54748-7-Springer-Verlag, Berlin. Heidelberg. New York. Tokyo ISBN 0-387-54748-7-Springer-Verlag, New York, Heidelberg. Berlin. Tokyo No part of this book may e reproduced in any form by print, microfilm or any other mans with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by Anamika Trading Company Navneet Bhavan, Bhavani Shankar Road Dadar, Bombay 400 028 and Published by H.Goetze, Springer-Verlag, Heidelberg, Germany PRINTED IN INDIA Preface These lectures were given at the Tata Institute of Fundamental Research in the summer of 1990. The specialized topic of mean values of the Riemann zeta-function permitted me to go into considerable depth. The central theme were the second and the fourth moment on the critical line, and recent results concerning these topic are extensively treated. In a sense this work is a continuation of my monograph [1], since except for the introductory Chapter 1, it starts where [1] ends. Most of the results in this text are unconditional, that is, they do not depend on unproved hypothesis like Riemann’s (all complex zeros of ζ(s) have real parts equal to 1 ) or Lindelof’s¨ (ζ( 1 + it) tǫ). On the other hand, in many 2 2 ≪ problems concerning mean values one does not obtain the conjectured results even if one assumes some unproved hypothesis. For example, it 1 +ǫ does not seem possible to prove E(T) T 4 even if one assumes the ≪ Riemann hypothesis. Incidentally, at the moment of writing of this text, it is not yet known whether the Riemann or Lindelof hypothesis is true or not. Each chapter is followed by Notes, where some results not treated in the body of the text are mentioned, appropriate references: are given etc. Whenever possible, standard notation (explained at the beginning of the text) is used. I’ve had the pleasure to have in my audience well-known special- ists from analytic number theory such as R. Balasubramanian, K. Ra- machandra, A. Sankaranarayanan, T.N. Shorey and S. Srinivasan. I am grateful to all of them for their interest, remarks and stimulating discus- sions. The pleasant surroundings of the Institute and the hospitality of v vi 0. Preface the Tata people made my stay a happy one, inspite of heavy monsoon rains. I wish to thank also M.N. Huxley, M. Jutila, K. Matsumoto and T. Meurman for valuable remarks concerning the present lecture notes. Special thanks are due to K. Ramachandra, Who kindly invited me to the Tata Institute, and with whom I’ve been in close contact for many years. Finally, most of all I wish to thank Y. Motohashi, whose great work on the fourth moment influenced me a lot, and who kindly permitted me to include his unpublished material in my text. Belgrade, March 1991. Notation Owing to the nature of this text, absolute consistency in notation could not be attained, although whenever possible standard notation is used. The following notation will occur repeatedly in the sequel. k, l, m, n Natural numbers (positive integers). A, B, C, C1,... Absolute positive constants (not necessarily the same at each occurrence). s, z, w Complex variables Re s and Im s denote the real imaginary part of s, respectively; com- mon notation is σ = Res and t = Ims. t, x, y Real variables. Res F(s) The residue of F(s) at the point s = s . s=s ◦ ζ(s◦) Riemann’s zeta-function defined by ζ(s) = ∞ s n− for Res > 1 and for other values of n=1 Xs by analytic continuation. γ(s) The gamma-function, defined for Res > 0 by ∞ s 1 t Γ(s) = t − e− dt, otherwise by analytic 0 continuation.Z γ Euler’s constant, defined by ∞ x r = e− log x dx = 0.5772156649 .... 0 χ(s) = ζ(Zs)/ζ(1 s) = (2π)s/(2Γ(s) cos πs ). − 2 vii viii 0. Notation log ζ(σ + it) µ(σ) = lim sup (σ real). t log t exp z = ez.→∞ e(z) = e2πiz. log x = Loge x(= lnx). [x] Greatest integer not exceeding x. dk(n) Number of ways n can be written as a product of k fixed natural numbers. d(n) = d2(n) = number of divisors of n. A sum over all positive divisors of n. Xd n a σ a(n) = d . Xd n f (n) A sum taken over all natural numbers not ex- n k ceeding x; the empty sum is defined to be X≤ zero. ′ f (n) Same as above, only ′ denotes that the last n x summand is halved if x is an integer. X≤ ∗ A product taken over all possible values of the Yj index j; the empty sum is defined to be unity. T 1 2k Ik(T) = ζ( + it) dt(k 0). 0 2 ≥ E (T) TheZ error term in the asymptotic formula k when k 1 is an integer. ≥ T 1 2 T E(T)(= E1(T)) ζ( + it) dt T log (2γ 1)T. 0 2 − 2π − − Z T 2 Eσ(T) ζ(σ + it) dt ζ(2σ)T 0 − Z ζ(2σ 1)Γ(2γ 1) 2 2σ 1 − − sin(πσ)T <σ< 1 . 1 σ − 2 − − 2 1 ar sinh z = log(z + (z + 1) 2 ). 1 (x) = ′d(n) x(log x + 2γ 1) . △ n x − − − 4 X≤ ix k(x) Error term in the asymptotic formula for △ dk(n); 2(x) = (x). n x △ △ X≥ Jp(z), Kp(z), Yp(z) Bessel functions of index p. T G(T) = (E(T) πdt). 2 − Z T 1 3 G (T) = (Eσ(t) b(σ))dt <σ< . ◦ 2 − 2 4 B(σ) TheZ constant defined by (3.3). ! S (m, n; c) Kloosterman sum, defined as ≡ 1 d c,(d,c)=1,dd ≤ ≤ X ′ md + nd (mod c)e ′ . c ! cr(n) Ramanujan sum, defined as cr(n) = h e n . r 1 h r,(h,r)=1 ! 1 ≤ ≤X α j, x j, H j 2 Quantities appearing in the spectral theory of the non-Euclidean Laplace operator, defined in Section 5.3. f (x) f (x) g(x) as Means lim = 1, with x0 possibly infi- ∼ x x0 g(x) x x → → 0 nite. f (x) = O(g(x)) Means f (x) Cg(x) for g(x) > 0, x x | | ≤ ≥ 0 and some absolute constant C > 0. f (x) g(x) Means the same as f (x) = 0(g(x)). ≪ f (x) g(x) Means the same as g(x) = 0( f (x)). ≫ f (x) g(x) Means that both f (x) (g(x)) and g(x) ≍ ≪ ≪ f (x) hold. (a, b) Means the interval a < x < b. [a, b] Means the integral a x b. ≤ ≤ δ, ǫ An arbitrarily small number, not necessarily the same at each occurrence in the proof of a theorem or lemma. Cr[a, b] The class of functions having a continuous r- th derivative in [a, b]. x 0. Notation f (x) f (x) = O(g(x)) as Means that lim = 0, with x0 possibly x x0 g(x) x x → → 0 infinite. f (x) =Ω+(g(x)) Means that there exists a suitable constant C > 0 and a sequence xn tending to such ∞ that f (x) > Cg(x) holds for x = xn. f (x) =Ω (g(x)) Means that there exists a suitable constant − C > 0 and a sequence xn tending to such ∞ that f (x) < Cg(x) holds for x = xn. − f (x) =Ω (g(x)) Means that both f (x) = Ω+(g(x)) and f (x) = ± Ω (g(x)) holds. − f (x) =Ω(g(x)) Means that f (x) =Ω+(g(x)). | | Contents Preface v Notation vii 1 Elementary Theory 1 1.1 Basic Properties of ζ(s).................. 1 1.2 Elementary Mean Value Results . 4 1.3 Bounds Over Short Intervals . 10 1.4 Lower Bounds For Mean Values on... 16 2 The Error Terms In Mean Square Formulas 29 2.1 The Mean Square Formulas . 29 2.2 The Beginning of Proof . 32 2.3 Transformation of the Expressions for the Error Terms . 39 2.4 Evaluation of Some Exponential Integrals . 51 2.5 Completion of the Proof of the Mean Square Formulas . 56 2.6 The Mean Square Formula For E(T)........... 66 2.7 The Upper Bounds for the Error Terms . 83 2.8 Asymptotic Mean Square of the Product.... 102 3 The Integrals of the Error Terms in Mean Square Formulas113 3.1 The Formulas for the Integrated Error Terms . 113 3.2 Omega-Results ......................135 3.3 Mean Square Formulas . 156 3.4 The Zeros of E(T) π ..................160 − xi xii Contents 3.5 Some Other Results . 166 4 A General Study of Even Moments 181 4.1 The Error Term for the 2kth Moment . .181 4.2 The Approximate Functional Equation . 188 4.3 Evaluation of Ik(T) ....................198 4.4 Evaluation of the First Main Term . 207 4.5 The Mean Square Formula . 215 4.6 The Fourth Power Moment . 222 4.7 TheSixthPowerMoment. .233 5 Motohashi’s formula for the fourth moment 245 5.1 Introduction and Statement fo results . 245 5.2 Analytic Continuation .
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