On the sign of the real part of the Riemann zeta function

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Transcript of On the sign of the real part of the Riemann zeta function

  • On the sign of the real part of theRiemann zeta function

    Richard P. BrentANU

    Joint work with

    Juan Arias de Reyna (University of Seville)Jan van de Lune (Hallum, The Netherlands)

    19 March 2012

    In memory of Alf van der Poorten

    Arias de Reyna, Brent and van de Lune Sign of

  • Abstract

    The real part of the Riemann zeta function (s) is usuallypositive in the half-plane to the right of the critical line

  • Alf van der Poorten

    Like almost all number theorists, Alf van der Poorten wasinterested in the Riemann zeta function. For example, Alf gavea very clear exposition of Aprys proof of the irrationality of(3), in his 1979 paper A proof that Euler missed . . . Aprysproof of the irrationality of (3).Thus, although my only joint work with Alf was on continuedfractions of algebraic numbers, I decided to talk today on atopic related to the Riemann zeta function.

    Arias de Reyna, Brent and van de Lune Sign of

  • Summary

    I Notation and definitions.I Motivation

  • Some notation and definitions

    P is the set of primes and p P is a prime.

    We always have s = + it C.log (s) denotes the main branch defined in the usual way onthe open set G equal to the complex plane C with cuts alongthe half-lines (+ i, + i] for each zero or pole + i of(s) with 1/2. Thus log (s) is real and positive in (1,+).For s G, we can define arg (s) by

    log (s) = log |(s)|+ i arg (s).

    |B| denotes the Lebesgue measure of a set B.Usually > 1/2 is fixed, t R is regarded as the independentvariable, and b = p.

    Arias de Reyna, Brent and van de Lune Sign of

  • Motivation

    Several authors (Gram, Titchmarsh, Edwards, . . .) haveobserved that

  • The Euler product

    Recall the Euler product formula

    (s) =

    p

    (1 ps

    )1,

    valid for > 1.Taking logarithms, we get

    log (s) =

    p

    log(1 ps

    )=

    p

    ps + O(1).

    There is a sense in which log (s) can be approximated by ashort Dirichlet series

    px p

    s in the region 1/2 1, butwe wont discuss this today.

    Arias de Reyna, Brent and van de Lune Sign of

  • RemarksLet 0 = 1.1923473371 be the real root in (1,+) of

    p

    arcsin(p) =

    2.

    It was shown by Jan van de Lune (1983) that( 0) 0 .

    Also, for any (1, 0), there exist arbitrarily large t such that

  • Figure from Van de Lune (1983)

    From the Figure, we see that the prime p contributes at mostarcsin(p) to arg ( + it). If ( + it) < 0 then we must have|arg ( + it)| > /2.

    Arias de Reyna, Brent and van de Lune Sign of

  • Some numerical results

    If we compute (s) for randomly chosen s with =

  • The seventh interval where
  • Asymptotic densities

    Fix > 1/2, and define

    d+() := limT+

    12T|{t [T ,+T ] : 0}| ,

    d() := limT+

    12T|{t [T ,+T ] :

  • Approximation of d() via arg (s)

    It is easier to work with

    d() = limT+

    12T|{t [T ,+T ] : |arg ( + it)| > /2}| .

    Observe that

  • A mean-value result

    Using Chebyshevs inequality and a mean-value result3 for|

  • Intuition

    Informally, the idea is that, for a prime p and large t ,

    pit = exp(it log(p))

    behaves like a random variable distributed uniformlyon the unit circle.Moreover, for different primes, the random variables areindependent (because the log p are independent over Q).A theorem of Bohr and Jessen (to be stated soon) justifies thisintuition.

    Arias de Reyna, Brent and van de Lune Sign of

  • The probability space

    Let T = {z C : |z| = 1} denote the unit circle with the usualprobability measure (that is d2 if we identify T with theinterval [0,2) via z = exp(i)).Define := TP with the product measure P = P .Each point of is a sequence = (xp)pP , with each xp T.This formalises the idea that the xp may be considered as arandom variables, uid4 on the unit circle.

    4Uniformly and independently distributed.Arias de Reyna, Brent and van de Lune Sign of

  • The measure P of Bohr and Jessen

    Before stating the theorem of Bohr and Jessen we need todefine a measure P.Fix > 1/2. The sum of random variables

    S = pP

    log(1 pxp) :=pP

    k=1

    1k

    pkxkp

    converges almost everywhere, so S is a well-defined randomvariable.The measure P of Bohr and Jessen is defined to be thedistribution of S, i.e. for each Borel set B C we have

    P(B) := P{ = (xp) : S() B}.

    Arias de Reyna, Brent and van de Lune Sign of

  • A theorem of Bohr and Jessen

    In modern language, Bohr and Jessen (1930/31) showed that,for each rectangle B with sides parallel to the axes,

    P(B) = limT

    12T|{t [T ,+T ], log ( + it) B}|

    (and the limit exists).It is easy to deduce that the same result holds forJordan-measurable sets B C.Bohr and Jessen also showed that P is absolutely continuouswith respect to the Lebesgue measure on C.

    Arias de Reyna, Brent and van de Lune Sign of

  • The measure on RWe can specialise5 to sets B of the form R B. For Jordansubsets B R, define

    (B) := P(R B).

    Thend() = (R\[/2, /2]) ,

    d+() =

    (kZ

    (2k /2, 2k + /2)

    ).

    The measure is the distribution function of the randomvariable =S:

    (B) = P(R B) = P{ : S() R B}= P{ : =S() B}.

    5Since R is not bounded, we need a limiting argument to justify this.Arias de Reyna, Brent and van de Lune Sign of

  • The characteristic function

    Recall that the characteristic function (y) of a random variableX is defined by

    (y) := E[exp(iXy)] .

    This is just a Fourier transform; we omit a factor 2 in theexponent to agree with the statistical literature.The characteristic function associated with is thecharacteristic function of the random variable =S:

    (x) :=

    eix=S() d .

    Arias de Reyna, Brent and van de Lune Sign of

  • as an infinite product over the primes

    We have(x) =

    p

    I(p, x) ,

    where (as usual) the product is over all primes p, and

    I(b, x) :=1

    2

    20

    eix arg(1b1eit ) dt .

    Arias de Reyna, Brent and van de Lune Sign of

  • Sketch of the proof

    By definition

    (x) =

    eix=S() d =

    p

    eix arg(1pxp) d.

    By independence the integral of the product is the product ofthe integrals:

    (x) =

    p

    eix arg(1pxp) d.

    Each random variable xp is distributed as ei on the unitcircle.

    Arias de Reyna, Brent and van de Lune Sign of

  • I(b, x) as an integral

    Assume b > 1. Recall the definition

    I(b, x) :=1

    2

    20

    eix arg(1b1eit ) dt .

    Then we have two equivalent expressions for I(b, x) as adefinite integral:

    I(b, x) =1

    0

    cos(

    x arctan(

    sin b cos

    ))d

    =2

    10

    cos(

    x arcsinub

    ) du1 u2

    .

    Arias de Reyna, Brent and van de Lune Sign of

  • Connection with Bessel functions

    Suppose b is large in the first representation

    I(b, x) =1

    0

    cos(

    x arctan(

    sin b cos

    ))d .

    Approximating arctan(sin /(b cos )) by sin()/b, we get

    I(b, x) 1

    0

    cos(x

    bsin

    )d = J0(x/b)

    from an integral representation of the Bessel function J0.A more detailed asymptotic analysis shows that I(b, x) hasinfinitely many real zeros near the points

    {(3/4 + k)/arcsin(1/b) : k Z0}.

    Arias de Reyna, Brent and van de Lune Sign of

  • What does look like?

    (x) is a product:

    (x) =

    p

    I(p, x) .

    Each factor in the product has infinitely many real zeros, andthe same is true for (x).We have the bound

    |(x)| C exp

    (c x

    1/

    log x

    )

    for some positive constants C = C(), c = c() and x 2.The best possible constants are not known, but empirically wecan take C = c = 1 for all x 7 and > 1/2.

    Arias de Reyna, Brent and van de Lune Sign of

  • Pictures of 1.02(x)Due to the exponential decay of |(x)| it is difficult to give asingle plot that shows its behaviour. Following are plots for = 1.02 and x in the intervals [0,5], [5,10], . . ., [25,30].

    Arias de Reyna, Brent and van de Lune Sign of

  • x [5,10]

    Arias de Reyna, Brent and van de Lune Sign of

  • x [10,15]

    Arias de Reyna, Brent and van de Lune Sign of

  • x [15,20]

    Arias de Reyna, Brent and van de Lune Sign of

  • x [20,25]

    Arias de Reyna, Brent and van de Lune Sign of

  • x [25,30]

    Arias de Reyna, Brent and van de Lune Sign of

  • Digression the hypergeometric differential equation

    The hypergeometric differential equation is the second-orderlinear equation

    z(1 z)w + (c (a + b + 1)z)w abw = 0,

    where primes denote differentiation with respect to z. Herea,b, c are constants. In a neighbourhood of the origin, thehypergeometric function

    2F1(a,b; c; z) =

    n=0

    (a)n(b)n(c)n

    zn

    n!

    is a solution; a second solution is

    z1c2F1(1 + a c,1 + b c; 2 c; z).

    Arias de Reyna, Brent and van de Lune Sign of

  • cos(2x arcsin u) as a hypergeometric function

    For |u| < 1 and all x C,

    cos(2x arcsin u) = 1 +

    n=1

    (2u)2n

    (2n)!

    n1j=0

    (j2 x2).

    To prove this, let f (u) := LHS, g(u) := RHS above. Then

    (1 u2)f (u) uf (u) + 4x2f (u) = 0.

    Also, g(u) satisfies the same differential equation. Sinceg(0) = f (0) = 1 and g(0) = f (0) = 0, the two solutionscoincide.Notes. When x Z the series reduces to a polynomial.The differential equation can