Transcendental equations satisfied by the individual zeros of Riemann $ζ$, Dirichlet and modular $L$-functions
Guilherme França, André LeClair
备注：
Matches the version to appear in Communications in Number Theory and Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and arXiv:1307.8395 [math.NT]
发表信息：
Communications in Number Theory and Physics, Vol. 9, No. 1 (2015)
We consider the non-trivial zeros of the Riemann $ζ$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $ζ$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function.
The convergence of a sequence of Cauchy sequences is conjectured; which if shown to be true, would prove the Riemann hypothesis by way of LeClair and França's transcendental equation criteria.
Let $(X,L)$ be a polarized variety over a number field. We suppose that $L$ is an hermitian line bundle. Let $M$ be a non compact Riemann Surface and $U\subset M$ be a relatively compact open set. Let $φ:M\to X({\Bbb C})$ be a holomorphic map. For every positive real number $T$, let $A_U(T)$ be the cardinality of the set of $z\in U$ such that $φ(z)\in X(K)$ and $h_L(φ(z))\leq T$. After a revisitation of the proof of the sub exponential bound for $A_U(T)$, obtained by Bombieri and Pila , we show that there are intervals of $T$'s as big as we want for which $A_U(T)$ is upper bounded by a polynom…
In previous papers we investigated basic properties of the determinant $G_{K}(s)$ of the Riemann operator: ${\mathcal R}$ acting on $\bigoplus_{n>1} K_{n}(A)_{\mathbb{C}}$, where $A$ is the integer ring of an algebraic number field $K$. The function $G_{K}(s)$ is defined as the regularized determinant \[ G_{K}(s) = {\rm det} ((s I-\mathcal{R}) | \bigoplus_{n>1} K_{n}(A)_{\mathbb{C}} ) \] with $\mathcal{R} | K_{n}(A)_{\mathbb{C}} = \frac{1-n}{2}$. We showed that $G_{K}(s)^{-1}$ is essentially the so called gamma factors of Dedekind zeta function of $K$. In this paper we study the transcendency …
We show that there are an infinite number of Riemann zeros on the critical line, enumerated by the positive integers $n=1,2,\dotsc$, whose ordinates can be obtained as the solution of a new transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of such zeros already saturates the counting formula for the numbers of zeros on the entire critical strip. These results thus constitute a concrete proposal toward verifying the Riemann hypothesis. We perform numerical analyses of the exact equation, and its asymptotic limit of large ordinate. The starting poi…
While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form $$ \sum_{ρ} R(ρ) x^ρ, $$ where the sum is over the non-trivial zeros $ρ$ of $ζ(s)$, $R(x) \in \overline{\Q}(x) $ is a rational function over algebraic numbers and $x >0$ is a real algebraic number. In particular, we show that the function $$ f(x) = \sum_{ρ} \frac{x^ρ}ρ $$ has infinitely many zeros in $(1, \infty)$, at most one of whi…
The function $S_n (t) = π\left( \frac{3}{2} - {frac} \left( \frac{\vartheta(t)}π \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 πe}\right)}{2 π} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S (t_n)_{} = \arg ζ\left( \frac{1}{2} + i t_n \right)$ when $t=t_n$ is the imaginary part of the n-th zero of $ζ$ on the critical line. If $S(t_n)=S_n(t_n)$ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer $n$ which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is e…
For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli space' of rank $n$ vector bundles with flat connection on $X$. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We ad…
We construct a vector field E from the real and imaginary parts of an entire function xi (z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Gamma and Riemann zeta functions and is known to satisfy the functional identity xi (z) = xi (1-z). E satisfies the conditions for a static electric field. The structure of E in the critical strip is determined by its behavior near the Riemann zeros on the critical line Re (z) = 1/2, where each zero can be assigned a + o…
In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned a…
It is known by a formula of Hasse-Sondow that the Riemann zeta function is given, for any $ s=σ+it \in \mathbb{C}$, by $ \sum_{n=0}^{\infty} \widetilde{A}(n,s)$ where $$ \widetilde{A}(n,s):=\frac{1}{2^{n+1}(1-2^{1-s})} \sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{(k+1)^s} .$$ We prove the following approximate functional equation for the Hasse-Sondow presentation: For $ \vert t \vert = πxy $ and $ 2y \neq (2N-1)π$ then $$ ζ(s)= \sum_{n \leq x } \widetilde{A}(n,s)+\frac{χ(s)}{1-2^{s-1}} \left (\sum_{k \leq y} (2k-1)^{s-1} \right ) +O \left (e^{-ω(x,y) t} \right ), $$ where $ 0 <ω(x,y)$ is a certain t…
For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corre…
The integral $R(t)=π^{-1}(lnζ(\frac{1}{2}+it)+i\vartheta (t))$ of the logarithmic derivative of the Hardy Z function $Z(t)=e^{i\vartheta (t)}ζ(\frac{1}{2}+it)$, where $\vartheta (t)$ is the Riemann-Siegel theta function, and $ζ(t)$ is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions $ν(t)=-ν(1-t)=-{ΔR(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(ΔR(t))^{-1}$ is the derivative of the additive inverse of the reciprocal of the Laplacian of $R(t)$ and $χ(t)=-χ(1-t)=\dotν (t)=-iH(\frac{i}{2}-it)$ where $H(t)=\dot{G} (t)$ has roo…
Let $\mathcal{P}$ be the set of all primes and $ψ(n)=n\prod_{n\in \mathcal{P},p|n}(1+1/p)$ be the Dedekind psi function. We show that the Riemann hypothesis is satisfied if and only if $f(n)=ψ(n)/n-e^γ \log \log n <0$ for all integers $n>n_0=30$ (D), where $γ\approx 0.577$ is Euler's constant. This inequality is equivalent to Robin's inequality that is recovered from (D) by replacing $ψ(n)$ with the sum of divisor function $σ(n)\ge ψ(n)$ and the lower bound by $n_0=5040$. For a square free number, both arithmetical functions $σ$ and $ψ$ are the same. We also prove that any exception to (D) may…
The Riemann zeta function $ζ(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $ζ(s)$ lie on the line ${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${\rm Re}\,s<1…
This paper studies deformations of the well-known $Ξ(s)=Ξ(1-s)$ equation for the Riemann $Ξ$ function satisfied by $Ξ_ρ(s):=\int_{0}^\infty\frac{{\d}t}{t}{t^{\frac{s}{2}}}Ψ(t)e^{-ρ\ln^2(t)}$ where ${Ψ(t)=\sum_{n\in\mathbb{N}^+}e^{-π{n}^2t}}$.
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The present essay aims at investigating whether and how far an algebraic analysis of the Zeta Function and of the Riemann Hypothesis can be carried out. Of course the well-established properties of the Zeta Function, explored in depth in over 150 years of world-wide study, are taken for granted. The chosen approach starts from the recognized necessity of formulating an extension of the Zeta Function which is defined for Re(s) = X < 1. A particular form of extension, based on a Euler-McLaurin integration procedure, is chosen and an algebraic appraisal is made of the conditions required to make …
The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An analytic continuation formula for these hypergeometric functions exists and is used to derive some infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of hypergeometric functions at n - 1. The form might be considered to be a shift operator for the Riemann zeta function which leads to the curious values ζF(0)…
In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $ζ(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $ζ(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an infinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $ζ(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label…
We show that the order three algebraic differential equation over ${\mathbb Q}$ satisfied by the analytic $j$-function defines a non-$\aleph_0$-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila's modular Ax-Lindemann-Weierstrass with derivatives theorem using Seidenberg's embedding theorem and a theorem of Nishioka on the differential equations satisfied by automorphic functions. As a by product of this…
