User stopple - MathOverflow

Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$

2013-01-29T16:55:03Z

Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example the answers to this question

I'm looking for references for computation of $\zeta^\prime(s)/\zeta(s)$, as well as the its derivative, in the critical strip $0<\text{Re}(s)<1$. (I'm sure they're out there but google scholar/MathSciNet searches return way too many irrelevant hits.)

Of course, both $\zeta(s)$ and $\zeta^\prime(s)$ are implemented in packages like Sage, one can just take the quotient and then use this to numerically estimate the derivative via the difference quotient, but this seems computationally wasteful. We have that $$ \frac{\zeta^\prime(s)}{\zeta(s)}=\log(2\pi)-1-\gamma/2-\frac{1}{s-1}-\frac12\frac{\Gamma^\prime(s/2+1)}{\Gamma(s/2+1)}+\sum_\rho\left(\frac{1}{s-\rho}+\frac{1}{\rho}\right). $$

Similarly, one gets $(\zeta^\prime(s)/\zeta(s))^\prime$ upon differentiating term by term.

The digamma function $\Gamma^\prime/\Gamma$ as well as the Riemann zeros $\rho$ are implemented in Mathematica. So I think what I'm asking is a reference to answer the following:

Given $\epsilon$, how many zeros do I need to take as a as a function of $t=\text{Im}(s)$ so the error is bounded by $\epsilon$?

Added: In Theorem 9.6(A) in Titchmarsh's "Theory of the Riemann Zeta Function", one can compute the relevant constants to show that $$ \left|\frac{\zeta^\prime(s)}{\zeta(s)}-\sum_{|\rho-s|\le 6}\frac{1}{s-\rho}\right|\le 4\log t. $$ So in answer to Joro's question below, yes the sum is dominated by the zeros close to $s$.

A naive implementation, using all the zeros up to height $2t$, gives the following graphic for the $\arg(\zeta^\prime/\zeta)$, interpreted as a color. Here $0<\sigma<1$, $1000< t<1010$.

Answer by Stopple for What do theta functions have to do with quadratic reciprocity?

2013-01-28T17:15:53Z

Going in the direction of more generality:

With $\theta(\tau)=\sum_n\exp(\pi i n^2 \tau)$, theta reciprocity describes how the function behaves under the linear fractional transformation $[\begin{smallmatrix} 0&1 \\ -1&0\end{smallmatrix}]$ . From this one can show it's an automorphic form (of half integral weight, on a congruence subgroup). Automorphic forms and more generally automorphic representations are linked by the Langlands program to a very general approach to a non-abelian class field theory. Your "Why should we expect ..." question is dead-on. This is very deep and surprising stuff.

In the direction of more specificity, the connection to the heat kernel is fascinating. (In this context, Serge Lang was a great promoter of 'the ubiquitous heat kernel.') The theta function proof is also discussed in Dym and McKean's 1972 book "Fourier Series and Integrals" and in Richard Bellman's 1961 book "A Brief Introduction to Theta Functions." Bellman points out that theta reciprocity is a remarkable consequence of the fact that when the theta function is extended to two variables, both sides of the reciprocity law are solutions to the heat equation. One is, for $t\to 0$ what physicists call a 'similarity solution' while the other is, for $t\to \infty$ the separation of variables solution. By the uniqueness theorem for solutions to PDEs, the two sides must be equal!

A special case of quadratic reciprocity is that an odd prime $p$ is a sum of two squares if and only if $p\equiv 1\bmod 4$. This can be be done via the theta function and is in fact given in Jacobi's original 1829 book "Fundamenta nova theoriae functionum ellipticarum."

Answer by Stopple for The Riemann Hypothesis and the Langlands program

2013-01-22T03:10:51Z

Number theory can seem to the beginner like a very random collection of results, and it is only fairly recently in its 5000 year history that the larger picture has begin to emerge. A report from the NAS in the early 1990s opened my eyes to the fact that number theory now is centered around three questions, each having to do with $L$ -functions.

The first area is the Riemann hypothesis, and its generalizations to more general $L$-functions. Questions about the vertical as well as horizontal distribution of the zeros initiated by Montgomery and influenced by random matrix theory fall in this area.

The second area is the Langlands program. Unlike the Riemann hypothesis, the Langlands program is named for the most advanced part of the theory, not the original question. But the roots of the Langlands program go all the way back to Gauss' Law of Quadratic Reciprocity: Given an odd prime $q$, let $\epsilon=\pm 1$ so that $\epsilon q\equiv 1\bmod 4$. Then for an odd prime $p$, the Legendre symbols $(\frac{p}{q})$ and $(\frac{\epsilon q}{p})$ are equal. More abstractly, the Galois representation arising from the Kronecker symbol $(\frac{\epsilon q}{*})$ has the same $L$ function as the Dirichlet character $(\frac{*}{q})$. Langlands interprets the latter as an automorphic from on $GL(1)$.

The third area is the Bloch-Beilinson conjectures, which include the Birch and Swinnerton-Dyer conjecture as a special case. The simplest manifestation of these is the Dirichlet Class Number Formula.

As GH's answer, different areas relate to each other. The possibility of a Landau-Siegel zero prevents us from getting the lower bound on class numbers we expect. The fact CM elliptic curves were known to be automorphic allowed Birch and Swinnerton-Dyer to be able to compute sufficient examples to make a conjecture.

Answer by Stopple for Are potential complex zeros not on the critical line of Dedekind zeta function in quadruples?

2013-01-14T16:19:38Z

The functional equation tells you that if $\rho$ is a zero, so is $1-\rho$. On the other hand, since the Dirichlet series coefficients are real, we have that for $\text{Re}(s)>1$, $\zeta(\bar{s})=\overline{\zeta(s)}$. By analytic continuation this holds for all $s\ne 1$. So if $\rho$ is a complex zero off the critical line, so is $\bar{\rho}$. This works for any Dirichlet series with functional equation and real coefficients, i.e. real quadratic character, elliptic curve,...

The Riemann zeros and the heat equation

2012-12-04T22:08:16Z

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as $$ \Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du, $$ where $\Phi(u)$ is defined as $$ 2\sum_{n=1}^\infty\left(2\pi^2n^4\exp(9u/2)-3\pi n^2\exp(5u/2)\right)\exp(-n^2\pi\exp(2u)). $$ This arises from integration by parts after writing $\Xi$ as the Mellin transform of the theta function, and then a change of variables from multiplicative to additive notation. In 1950 de Bruijn (building on work of Polya) introduced a deformation parameter $t$: $$ \Xi_t(x)=\int_0^\infty \exp(t u^2)\Phi(u)\cos(ux)\, du, $$ so that for $t=0$, $\Xi_0(x)$ is just $\Xi(x)/2$.

de Bruijn proved the following theorem about the zeros in $x$:

(i) For $t\ge 1/2$, $\Xi_t(x)$ has only real zeros.
(ii) If for some real $t$, $\Xi_t(x)$ has only real zeros, then $\Xi_{t^\prime}(x)$ also has only real zeros for any $t^\prime>t.$

In 1976 Newman showed that there exists a real constant $\Lambda$, $-\infty<\Lambda\le 1/2$, such that
(i) $\Xi_t(x)$ has only real zeros if and only if $t\ge\Lambda$.
(ii) $\Xi_t(x)$ has some complex zeros if $t<\Lambda$.

The constant $\Lambda$ is known as the de Bruijn-Newman constant. The Riemann hypothesis is the conjecture that $\Lambda\le 0$. Newman made the complementary conjecture that $\Lambda\ge 0$, with the often quoted remark

"This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so."

Given the significance of the de Bruijn-Newman constant $\Lambda$, much work has gone into estimating lower bounds, and the current record (Saouter et. al.) is $ -1.14\times 10^{-11}<\Lambda. $

A breakthrough occurred in the work of Csordas, Smith and Varga, "Lehmer pairs of zeros, the de Bruijn-Newman constant, and the Riemann Hypothesis", Constructive Approximation, 10 (1994), pp. 107-129. They realized that unusually close pairs of zeros of the Riemann zeta function, the so-called Lehmer pairs, could be used to give lower bounds on $\Lambda$. The idea of the proof is that the function $\Xi_t(x)$ satisfies the backward heat equation $$ \frac{\partial \Xi}{\partial t}+\frac{\partial^2 \Xi}{\partial x^2}=0, $$ from which they are able to draw conclusions about the differential equation satisfied by the $k$-th gap between the zeros as the deformation parameter $t$ varies.

They mention this PDE in a rather offhand way, as a remark on an alternate proof to one of the lemmas. In fact, it does not seem to be well known that the de Bruijn-Newman constant can be interpreted as a time variable in a heat flow. Is this well known? Or put more concretely, does anyone have a citation prior to 1994 which mentions this fact?

Answer by Stopple for Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

2012-11-06T16:16:40Z

For books I recommend Audrey Terras' "Harmonic Analysis on Symmetric Spaces and Applications, I." It begins with Fourier analysis on $\mathbb R^m$, includes automorphic forms both Maass and classical, and finishes with the Selberg Trace formula. The orientation is towards number theory, but there are lots of applications and extensive references to the literature. Also lots of exercises.

For an expository paper, I recommend H.P. McKean "Selberg's trace formula applied to a compact Riemann surface" in Comm. Pure & Appl. Math., v. 25 (1975), 225-246; with errata in v. 27 (1974) p. 134.

Both of these are older than the more modern references in the other answers, but closer perhaps in style to Hejhal's approach.

Answer by Stopple for What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-29T15:35:21Z

Another application of the computation of large numbers of Riemann zeros (beyond verification of the Riemann Hypothesis) is towards bounding the deBruijn-Newman constant $\Lambda$:

deBruijn introduced a deformation parameter $t$ in the Riemann $\Xi$ function so that $\Xi_0(x)=\Xi(x)$ and the Riemann zeros $x(t)$ flow according to the "backward heat equation." Together their work shows the existence of a constant $\Lambda$ such that, for $\Lambda\le t$ the function $\Xi_t(x)$ has only real zeros, while for $t<\Lambda$ there exist complex zeros. The Riemann Hypothesis is the conjecture that $\Lambda\le 0$. Newman made the complementary conjecture that $\Lambda\ge 0$, writing "This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Csordas, Smith, and Varga were able to analyze the ODEs governing the motion of the zeros, and use the fact that an very close pair of zeros, so-called Lehmer pairs, would give a lower bound on $\Lambda.$

The current best bound via this approach, due to Saouter, Gourdon, and Demichel, is that $$ -1.14\times 10^{-11}<\Lambda $$ based on a Lehmer pair at height about $7.95\times 10^{12}$

Answer by Stopple for What does the numerically verified part of the Riemann Hypothesis tell about prime numbers?

2012-10-29T03:42:45Z

The disproof of Mertens' conjecture (cited above) was certainly a computations tour de force using explicit values of the zeros of $\zeta(s)$. Another good example is the paper of Rosser and Schoenfeld "Sharper Bounds for the Chebyshev Functions $\theta(x)$ and $\psi(x)$" Math. Comp., v. 29 1975, pp. 243-269.

We know by the Prime Number Theorem that $\Psi(x)\sim x$. Rosser and Schoenfeld use values of zeros of $\zeta(s)$ to show, for example, that for $\log(x)>105$, we have $|\Psi(x)-x|<x\epsilon(x)$ , where, for $X=(\log(x)/9.6459 08801)^{1/2}$ $$ \epsilon(x)= 0.257634 \left(1 + \frac{0.96642}{X} \right) X^{3/4}\exp(-X). $$ The paper contains a number of results of this flavor, about the Chebyshev function $\theta(x)$, and about asymptotics of the $n$th prime $p_n$.

The reason it is difficult to convert results about low lying zeros to results about small primes is that the Explicit Formula, (mentioned in comments above) has the primes and zeros lying on opposite sides of a Fourier Transform. The Heisenberg Uncertainty Principle applies

http://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle

Answer by Stopple for Origin of the notation s=\sigma+it in analytic number theory

2012-09-12T23:00:30Z

In skimming through Narkiewicz "The Development of Prime Number Theory", one sees a reference on p. 155 (footnote 38) to a certain R. Lipschitz, who in Crelle in 1857 "studied the series $\sum_{n=1}^\infty\exp(nui)n^{-\sigma}$ for real values of $\sigma$." I checked the reference; Lipschitz was indeed using $\sigma$.

Lipschitz is referred to several times in this section of Narkiewicz for later work on functional equations of various $L$-functions.

Answer by Stopple for At what times were people interested in prime numbers

2012-09-10T22:41:58Z

For 2), it depends a little on how you interpret the question. Primes in the abstract are covered in Chapter XVIII of Dickson's History of the Theory of Numbers, vol I. There's not much between Euclid and Euler.

On the other hand, primes of special forms related to perfect numbers or amicable pairs were written about extensively in the 15 centuries before Fermat. Admittedly, often incorrectly or with little content. In Chapter I of Dickson, Carolus Bovillus (1470-1553) claims that $2^n-1$ is prime if $n$ is odd, giving the example $511=2^9-1$. (In fact $7|511$). But it was not all nonsense. For example, Thabit ibn Qurra (836-901) showed that if $$ p=3\cdot 2^{k-1}-1, q=3\cdot 2^k-1, r=9\cdot 2^{2k-1}-1 $$ are all primes, then $$ m=p\cdot q\cdot 2^k, n=r\cdot 2^k $$ form an amicable pair: $s(m)=n$ and $s(n)=m$, where $s(k)$ is the sum of the proper divisors of $k$.

Answer by Stopple for How should an analytic number theorist look at Bessel functions?

2012-08-30T19:15:00Z

Expanding on Qiaochu's and David's comments, from the point of view of automorphic forms and automorphic representations, a good reference is "Special Functions and the Theory of Group Representations" by N. Ja. Vilenkin and published by the AMS.

"Special Functions" by Andrews, Askey, and Roy published by CUP is another good reference with an anayltic number theory slant.

EDIT: The point here is that for a finite group $G$ and a representation $\pi$ into a finite dimensional vector space $V$ with orthonormal basis $\{\vec{e_i}\}$, the function $g\to \langle \vec{e_i},\pi(g)\vec{e_j}\rangle$ gives the $i,j$ coefficient of the matrix associated to $\pi(g)$, or 'matrix coefficient function'. For a Lie group $G$ with an infinite dimensional representation $\pi$ in a Hilbert space $H$, there is a natural analog, and the functions which arise this way play an obviously important role in harmonic analysis in $G$. Bessel functions can be interpreted this way.

Answer by Stopple for Discrete Mathematics textbooks for undergraduates

2012-08-01T15:09:38Z

I like Concrete Mathematics by Graham, Knuth and Patashnik:

This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.

Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study.

Answer by Stopple for Number Fields Arising from Newforms

2012-07-30T21:05:42Z

It's not true for any old modular form. Since the forms live in a vector space over $\mathbb C$, you can achieve any complex number as a coefficient.

Here's a partial answer to what is true. You need to have a cusp form that is an eigenfunction of the Hecke operators, normalized so the leading coefficient is $1$. Since the Hecke operators are self adjoint in the Peterson (sp?) inner product, the eigenvalues are real, and one can show these are the coefficients in the $q$ expansion as follows: for $p$ prime, the $m$th coefficient of $T_p f$ is $a_{mp}$, for all $m$, more or less from the definition of $T_p$. This is also $\lambda_p a_m$, and from this and $a_1=1$ one deduces $a_p=\lambda_p$ (take $m=1$.) The general case follows from the recursion for powers of primes, and multiplicativity.

This answer is not quite right because it doesn't explain how CM extensions can arise, but it's a start.

Answer by Stopple for Axioms for Riemann $\zeta$ function

2012-07-18T15:46:15Z

Hamburger's Theorem (see Titchmarsh 'Theory of the Riemann Zeta Function' $\S$ 2.13) is in some sense an axiomatic characterization of $\zeta(s)$ among all Dirichlet series by its functional equation. It says:

Let $f(s)=\sum_n a_n n^{-s}$ a Dirichlet series absolutely convergent for $\sigma>1$ such that for some polynomial $P(s)$, $G(s)=P(s)f(s)$ is an integral function of finite order. Suppose $$ f(s)\Gamma(s/2)\pi^{-s/2}=g(1-s)\Gamma((1-s)/2)\pi^{-(1-s)/2} $$ where $g(1-s)=\sum_n b_n n^{1-s}$ is absolutely convergent for $\sigma<-\alpha<0$. Then $$ f(s)=C\zeta(s) $$ for some constant $C$.

Hamburger's theorem was a motivation for Hecke's study of Dirichlet series with functional equations generally, leading to his work on automorphic forms.

Fourier Transform: Smoothness and Decay

2011-08-04T21:03:37Z

For an application to analytic number theory, I'm considering the Fourier transform of test functions $h(x)$ on $\mathbb R$ with compact support, so the transforms $\hat h(y)$ are smooth. But I want to add a mild condition on $h$ that will force $\hat h(y) \ll 1/y^2$ as $y\to\infty$. Based on the examples I know, it looks like "continuous and piecewise differentiable" works, but I'd like the largest space of test functions possible. I'm familiar with the definition of $L^2$-based Sobolev Spaces $H^k(\mathbb R)$, but $k=1$ seems too weak, and $k=2$ too strong.

What is the right space of test functions $h(x)$ so that $\hat h(y)\ll 1/y^2$

Answer by Stopple for explicit large gap for consecutive zeros of the Riemann zeta function

2012-06-12T16:59:43Z

Theorem 9.12 in Titchmarsh says (in his shorthand style) there exists a constant $A$ such that for all sufficiently large T, (etc.)

The proof uses the Borel-Caratheodory theorem, and can be made effective if you really really want it. Titchmarsh has a series of seven successive constants $A_1, A_2,\ldots A_6, A$ with the final $A$ being the constant you reference above. This is not conditional on the Riemann Hypothesis.

It's not clear how your actual question relates to your title.

Answer by Stopple for Quadratic reciprocity and Weil reciprocity theorem

2012-06-07T16:47:29Z

John Milnor's Introduction to Algebraic K-Theory, has (on p. 101) some useful background on the connection between $K$-theory and quadratic reciprocity:

Theorem 11.6 (Tate) The group $K_2\mathbb Q$ is canonically isomorphic to the direct sum $$ A_2\oplus A_3\oplus A_5\oplus\ldots $$ where $A_2=\{\pm1\}$ and for $p$ odd, $A_p=(\mathbb Z/p\mathbb Z)^\times$.
Milnor explains:

In fact the isomorphism will be given by the correspondence $$ \{x,y\}\to(x,y)_2\oplus(x,y)_3\oplus(x,y)_5\oplus\ldots $$ Tate remarks that his proof of the theorem is lifted directly from the argument which was used by Gauss in his first proof of the quadratic reciprocity law.

(The whole section beginning on p.99 is titled Gauss and Quadratic Reciprocity)

Forgive me if this is already familiar to you.

Answer by Stopple for Is there any way to generalize the Laplacian to finite groups?

2012-05-31T22:33:30Z

It's not clear to me that the Laplacian of the Cayley graph of the group is the right object. You might want to start by looking at Audrey Terras's book Fourier Analysis on Finite Groups and Applications (if you haven't already). She does not define the Laplacian of a finite group (but does talk about Cayley graphs and their Laplacians.)

To elaborate, on symmetric spaces in general, there is typically a whole algebra of differential operators which commute with the group action, not just polynomials in the Laplacian, which are relevant for the harmonic analysis (see e.g. Harmonic Analysis on Symmetric Spaces and Applications, I, II.)

Answer by Stopple for Linear (in)dependence of $\Im(\rho_n)$ and fundamental theorem of arithmetic

2012-05-04T22:24:18Z

The duality to which you refer does not mean that linear dependence of the zeros is equivalent to linear dependence of primes (or even their logarithms.)

You might look at Odlyzko and te Riele's disproof of the Mertens' conjecture, and the related literature - the truth of Mertens would have implied linear dependence of the zeros. This goes back to Ingham.

EDIT: Since you thought this was helpful, here's an expanded version. The duality between primes and zeros in the Riemann's Explicit Formula does not match up a single zero with a single prime - if it did you could imagine taking linear combinations on both sides to get a result of the kind you asked about.

Instead, the Explicit Formula does almost exactly the opposite. Because a test function on the primes side corresponds to the (more or less) Fourier Transform on the zeros side, the
Heisenberg Uncertainty Principle comes into play. The more you localize on one side, say the zeros, the more spread out the test function becomes on the primes side (and v. versa.)

Kronecker theorems on linear forms.

2012-02-29T23:50:40Z

Dickson's History of the Theory of Numbers, vol II p. 94 refers to some theorems of Kronecker on linear forms:

...find integers $w$ and $w^\prime$ such that $aw+a^\prime w^\prime$ takes a value as near as possible to $\xi$, where $a$, $a^\prime$ and $\xi$ are given real numbers. In general, consider a system of $p$ equations $$ a_{i,1}w_1+\dots+a_{i,q}w_q=\xi_i\qquad(i=1,\dots,p), $$ with real coefficients...

Dickson cites Kronecker's Werke, which I have, but my German is poor. Hardy and Wright has a chapter on 'Kronecker's Theorem', but not in this generality. I'm actually interested in the case $p=1$ and $\xi_1=1$; one equation with $q$ integer variables.

Can anyone suggest a modern reference in English?

Answer by Stopple for Possible locations for non trivial zeroes lying off the critical line

2012-01-10T20:23:27Z

I believe you're mistaken that $$ \lim_{s\to\rho}\left|\frac{\zeta(s)}{\zeta(1-s)}\right|=1. $$ Write $\zeta(1-s)=\zeta(s)f(s)$ with $f(s)$ as implied by your equation (2). The series expansion for $\zeta(s)$ at $s=\rho$ is $$ \zeta(s)=\zeta^\prime(\rho)(s-\rho)+O(s-\rho)^2. $$ The series expansion for $\zeta(1-s)$ at $s=\rho$ is $$ \zeta(1-s)=\zeta(s)f(s)=\zeta^\prime(\rho)f(\rho)(s-\rho)+O(s-\rho)^2. $$ By standard manipulation of series, $$ \frac{\zeta(s)}{\zeta(1-s)}=\frac{1}{f(\rho)}+O(s-\rho), $$ so the limit should equal $1/f(\rho)$.

Answer by Stopple for Values of Dirichlet L-funcions at natural numbers

2012-01-03T17:39:46Z

By the functional equation, the problem is equivalent to evaluating at negative integers. Writing the $L$-function as a linear combination of Hurwitz zeta functions (the coefficients are merely the values of the character), it is enough to evaluate the Hurwitz zeta functions at negative integers. This is Theorem 12.13 in Apostol's 'Introduction to Analytic Number Theory.'

Answer by Stopple for What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros?

2011-11-12T00:51:16Z

To elaborate a little more, here's some Mathematica code:

ListPlot[Table[{rho = ZetaZero[n];z = N[2 (2 Pi)^(rho - 1) Gamma[1 - rho] Sin[Pi rho/2]]; {Re[z], 
Im[z]}}, {n, 1, 100}], AspectRatio -> Automatic]

Here's the output:

Exceptional zeros and Liouville's $\lambda$ function

2011-09-11T22:14:49Z

This originated from an textbook exercise (recently posted to math.stackexchange http://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function with no success) but I think it has research level implications so please bear with me.

I'm working through the problems in Montgomery and Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for all $k\ge 0, x\ge1 $ $$ \sum_{n<x}\frac{\chi(n)}{n}(1-n/x)^k \ge \sum_{n<x}\frac{\lambda(n)}{n}(1-n/x)^k, $$ where $\lambda$ is Liouville's function. This is elementary. In part b, under the hypothesis that there exists a $k$ such that $$ \sum_{n<x}\frac{\lambda(n)}{n}(1-n/x)^k\ge0 $$ for all $x\ge 1$, (no such $k$ is known to exist), one is to show that for all quadratic $\chi$ and all $\sigma>0$ $$ L(\sigma,\chi)>0. $$ I expect one is meant to use a Mellin transform with Cesaro weighting, S 5.1 in Montgomery and Vaughan. The difficulty is that $\chi(n)/n$ are the Dirichlet series coefficients of $L(s+1,\chi)$, not $L(s,\chi)$. Thus (5.18) gives $$ L(\sigma+1,\chi)>0 $$ for all $\sigma>0$.

Question: Show, as the exercise states, that if there exists $k$ such that for all $x\ge1$ $$ \sum_{n<x}\frac{\lambda(n)}{n}(1-n/x)^k\ge0 $$ then for all quadratic $\chi$ and for $\sigma>0$, $$ L(\sigma,\chi)>0. $$

By a theorem of Landau, the positivity hypothesis for some $k$ and all $x$ would imply the Riemann Hypotheses as well; I don't see how to use this (and I think Landau's theorem is beyond the scope of the book.)

Alternately, the method of part a will show that $$ \sum_{n<x}\chi(n)(1-n/x)^k \ge \sum_{n<x}\lambda(n)(1-n/x)^k $$ so under the hypothesis that there exists a $k$ such that for all $x\ge1$ $$ \sum_{n<x}\lambda(n)(1-n/x)^k\ge0 $$ one gets $L(\sigma,\chi)>0$ for all quadratic $\chi$.

The reason one cares which version of part a is used, is that the numerics for small $k$ and moderate $x$ indicate that positivity is at least plausible for the original part a. It is not plausible for the revised version.

Moreover, several conjectures in the literature were known to imply the Riemann Hypothesis, but have since been disproved. Polya conjectured that $$ \sum_{n<x}\lambda(n)\le0 $$ for $x\ge2$, and Turan conjectured that $$ \sum_{n<x}\frac{\lambda(n)}{n}\ge0 $$ for $x\ge 1$. Both where disproved by Haselgrove, and subsequently Odlyzko disproved the Mertens conjecture by similar methods (more or less the Explicit Formula plus numerical values for low lying zeta zeros.) The same techniques should show, as long as $k$ is not too large, that there exists an $x$ such that $$ \sum_{n<x}\frac{\lambda(n)}{n}(1-n/x)^k<0. $$

Updates: (1) Theorem 1.7 (Landau) in Montgomery and Vaughan, has a stronger version: Satz 454 in Landau's Vorlesungen uber Zahlentheorie. From this one deduces that if there exists a $k$ such that for all $x$ $$ \sum_{n<x}\frac{\lambda(n)}{n}(1-n/x)^k \ge 0, $$ then the Riemann Hypothesis follows. But you can't get this directly from Theorem 1.7, and in any case I don't see that it applies to the problem at hand.

(2) I obtained the original paper of Bateman and Chowla that Montgomery and Vaughan reference. There it is proven that for all $x\ge 1$ $$ \sum_{n<x}\frac{\lambda(n)}{n} \ge 0 $$ if and only if for all quadratic $\chi$ and all $x\ge 1$ $$ \sum_{n<x}\frac{\chi(n)}{n} \ge 0. $$ (The proof generalizes from $k=0$ Cesaro weighting to any $k$.) Bateman and Chowla mention that former inequality and Landau's theorem implies the Riemann Hypothesis, but do not make any claim about exceptional zeros of quadratic character $L$-functions.

NB: Let me reiterate that I have no hopes that such a $k$ exists, as Peter Humphries points out this would contradict the Linear Independence Hypothesis.

Answer by Stopple for Exceptional zeros and Liouville's $\lambda$ function

2011-10-06T19:35:37Z

I am now very dubious that one can deduce anything about the sign of $L(\sigma,\chi)$ for $0<\sigma<1$ from the positivity of $$ \sum_{n<x}\frac{\chi(n)}{n}(1-n/x)^k. $$ Here's why. For simplicity consider the case $k=0$, and $\chi$ odd, i.e. complex quadratic. Bateman and Chowla (in the article M&V cite) point out that $L(1,\chi)>d^{-1/2}$, while Summation by Parts and the Polya-Vinograov inequality bounds the tail of the infinite series by $$ \sum_{x<n}\frac{\chi(n)}{n}<\frac{5}{3}\cdot \frac{d^{1/2}\log(d)}{x}. $$ Thus $$ \sum_{n<x}\frac{\chi(n)}{n}>d^{-1/2}-\frac{5}{3}\cdot \frac{d^{1/2}\log(d)}{x}. $$ The above is positive for $$ x>\frac53 d\log(d),\quad\text{or}\quad d<\frac{3x}{5W(3x/5)}, $$ where $W(x)$ is the Lambert function, that is, the inverse function of $x=w\exp(w)$. Thus assuming one could show that $$ \sum_{n<x}\frac{\lambda(n)}{n}>0 $$ for some very large $x$ (Haselgrove's disproof of Turan's conjecture suggests that $x=\exp(853)$ might be possible), one would be able (if the premise of the problem were correct) to rule out the possibility of an exceptional zero for a very large collection of $d$. For example, with $x=\exp(853)$ one would get all $d<2\cdot 10^{367}.$ A similar argument with fixed $k>0$ would give still more $d$.

Answer by Stopple for Negative values of Riemann zeta function on the critical line.

2011-08-17T21:31:14Z

The reason (1) 'appears' to be true for small $t$ is related to Gram's Law for the zeros of $\zeta(s)$. Edwards' book Riemann's Zeta Function (Dover) has a good explanation starting on p.125. The short version is that the Euler Maclaurin formula for $\zeta(1/2+i t)$ starts with a $+1$, and,

"as long as it is not necessary to use too large a value of $N$, it will be unusual for the smaller terms which follow to combine to overwhelm this advantage on the plus side. As Gram puts it, equilibrium between plus and minus values of Re$\zeta$ will be achieved only very slowly as $t$ increases."

Answer by Stopple for Which conjectures only need the Grand Riemann Hypothesis to become genuine theorems?

2011-07-27T20:45:20Z

For the Grand Riemann Hypothesis (RH for zeros of all automorphic $L$-functions), see the (somewhat technical) answer to

http://mathoverflow.net/questions/2826/equivalent-forms-of-the-grand-riemann-hypothesis

I think the Generalized Riemann Hypothesis (RH for zeros of Dirichlet $L$ functions) has the most significant number theoretic consequences. In addition to those listed at

http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis#Consequences_of_GRH

such as easy primality testing and good bounds on primes in arithmetic progressions, one also gets good lower bounds on class numbers for positive definite binary quadratic forms of discriminant $D$ (or equivalently, rings of integers in complex quadratic fields): for every $\epsilon>0$ there exists an effective constant $C(\epsilon)$ such that the class number $h(d)>C(\epsilon)|D|^{1/2-\epsilon}$.

Answer by Stopple for Numbers in the "Fundamentalis Tabula Arithmetica"

2011-07-15T15:30:31Z

Regarding 318, Vincent Forest Hopper writes on p.75 of "Medieval Number Symbolism" (Dover):

"...various of the Church Fathers began to write figurative interpretations of the biblical texts. They found precedent for giving importance to numbers in the precise directions given for the dimensions of the tabernacle, and in the testimony of the Book of Wisdom that 'God has arranged all things in number and measure.' Early interpretation of scriptural numbers is concerned only with the most prominent of them, such as the 12 springs and 70 palm trees of Elim, and the 318 servants of Abraham..." Something of the attitude of gnosis; that is, of scriptural mysteries hidden from the layman, is to be seen in an interpretation of Barnabas: 'Learn then, my children, concerning all things richly, that Abraham, the first who enjoined circumcision, looking forward in spirit to Jesus, practised that rite having received the mysteries of the three letters. For [the Scripture] saith, 'And Abraham circumcised 10, and 8, and 300 men of his household.' What, then, was the knowledge given to him in this? Learn the 18 first and then the 300. The 10 and 8 are thus denoted. Ten by I and eight by H. You have [the initials of the name of] Jesus. And because the cross was to express the grace [of our redemption] by the letter T, he says also 300. No one else has been admitted by me to a more excellent piece of knowledge than this, but I know that ye are worthy.'"

Added:

The moral here is that the medieval mind set is so very different from what we can imagine that it's hard to guess why they thought individual numbers were significant. For more on 318, see en.wikipedia.org/wiki/Dispute_about_Jesus'_execution_method#Interpretation_as_cross

Answer by Stopple for On meromorphic continuation of zeta function(s) and special values at negative integers

2011-07-14T15:10:02Z

In partial answer to 2), one can in fact get information about special values of $L$-functions via theta series. See, for example, Villegas and Zagier's paper "Square roots of central values of Hecke L-series" in the book Advances in Number Theory. (Disclaimer: I'm no expert,and it's certainly a lot more complex than Euler's approach to special values of $\zeta(s)$.)

What does log convexity mean?

2011-01-03T20:04:48Z

The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ as the unique function $f(x)$ on the positive reals such that $f(1)=1$, $f(x+1)=xf(x)$, and $f$ is logarithmically convex, i.e. $\log(f(x))$ is a convex function.

What meaning or insight do we draw from log convexity? There's two obvious but less than helpful answers. One is that log convexity means exactly what the definition says, no more and no less. The other is the more or less circular one that since the Gamma function is so important, any property that characterizes it is also significant.

The wikipedia article http://en.wikipedia.org/wiki/Logarithmic_convexity point out that "a logarithmically convex function is a convex function, but the converse is not always true" with the counterexample of $f(x)=x^2$. The only logarithmically convex examples in the article come trivially from exponentiating convex functions, and the example $\Gamma(x)$.

Let me say in advance that I'm less interested in the Gamma function than I am in the notion of log convexity, so this question is not a duplicate of

http://mathoverflow.net/questions/23229/importance-of-log-convexity-of-the-gamma-function

A thoughtful answer by Andrey Rekalo to that question, is that functions which can be realized as finite of moments of Borel measures are log convex functions. But I'm more interested in things that are implied by (v. imply) log convexity.

My real motivation is the fact that the Riemann Hypothesis implies that the Hardy function is log convex for sufficiently large $t$. (The Hardy function $Z(t)$ is just $\zeta(1/2+it)$ with the phase taken out, so $Z(t)$ is real valued and $|Z(t)|=|\zeta(1/2+it)|$.) This is in Edward's book 'Riemann's Zeta Function' Section 8.3, in the language that RH $\Rightarrow Z^\prime/Z$ is monotonic. This says that between consecutive real zeros, $-\log|Z(t)|$ is convex.

Any insight would be welcome.

Comment by Stopple

2013-05-22T15:46:49Z

An excellent source for Oresme is Clagett "Nicole Oresme and the Medieval Geometry of Qualities and Motions", (1968) University of Wisconsin Press.

Comment by Stopple

2013-03-28T18:59:11Z

So it's actually Zagier and not Zaiger or Zager?

Comment by Stopple

2013-03-22T22:45:15Z

Why not use $\Gamma[3+i-s]=(2+i-s)\Gamma[2+i-s]$ to cancel a Gamma function in the numerator and denominator?

Comment by Stopple

2013-03-16T17:43:12Z

Well, I've never had any answers to this question: mathoverflow.net/questions/38691/…

Comment by Stopple

2013-02-24T03:14:18Z

The closed form expression for triangular numbers was know 2500 years earlier, in ancient Greece. And was well know in Gauss's time.

Comment by Stopple

2013-02-06T20:34:09Z

The appropriate phrase here is 'Hilbert-Polya Conjecture' en.wikipedia.org/wiki/…

Comment by Stopple

2013-02-01T20:31:04Z

Any potential counterexample would lie on the real axis, and so would be the analog of a Landau-Siegel zero.

Comment by Stopple

2013-01-31T14:12:19Z

For the question of how to compute in general, I'm just asking what's known. For practical computations, I'm using Mathematica, which has implemented already 10^7 zeros, for t< about 5*10^6, where they are all known to lie on the critical line.

Comment by Stopple

2013-01-30T16:18:20Z

@Joro, see edit above. I'm interested in computing for a large number of values, and the question in general.

Comment by Stopple

2013-01-29T00:05:18Z

You might have better luck at crypto.stackexchange.com

Comment by Stopple

2013-01-22T16:30:16Z

@Cam No, I don't. I've since tried to track it down and been unable to. As I recall, the number theory section was just a portion of a report on the state of mathematics generally.

Comment by Stopple

2013-01-17T21:43:05Z

I think by $z\in 1,\rho$ you mean to sum over the (one) pole and all the zeros of $\zeta(s)$. This might be clearer if you separated out the contribution of the pole.

Comment by Stopple

2013-01-02T23:34:50Z

See the exercises at the end of chapter 11 in Ireland and Rosen's "A Classical Introduction to Modern Number Theory"

Comment by Stopple

2013-01-02T19:07:53Z

Can you explain what you mean by "true for primes but failed"?

Comment by Stopple

2012-12-05T16:10:02Z

We could debate whether the derivation above means that the 1988 formula is the heat equation. But regardless I think this answer misses the spirit of the original question, of whether the connection to the heat equation is well known. The word 'heat' does not appear.