Riemann ’ s and ζ ( s )
نویسنده
چکیده
[This document is http://www.math.umn.edu/ ̃garrett/m/complex/notes 2014-15/09c Riemann and zeta.pdf] 1. Riemann’s explicit formula 2. Analytic continuation and functional equation of ζ(s) 3. Appendix: Perron identity [Riemann 1859] exhibited a precise relationship between primes and zeros of ζ(s). A similar idea applies to any zeta or L-function with analytic continuation, functional equation, and Euler product. It took more than 40 years for [Hadamard 1893], [vonMangoldt 1895], and others to complete Riemann’s sketch of the Explicit Formula relating primes to zeros of the Euler-Riemann zeta function. The idea is that equating the Euler product and Riemann-Hadamard product for zeta allows extraction of an exact formula for a weighted counting of primes in terms of a sum over zeros of zeta. [1] An essential supporting point is meromorphic continuation of ζ(s) via integral representation(s) of ζ(s) in terms of theta function(s). [2] Further, these integral representations give vertical growth estimates, allowing invocation of Hadamard’s theorem on product expansions of entire functions. A key in analytic continuation and functional equation of ζ(s) is the functional equation of theta series, from the Poisson summation formula, from the representability of smooth functions by their Fourier series. Asymptotics of Γ(s) and the functional equation of ζ(s) bound the vertical growth of ζ(s), allowing application of the Hadamard product result.
منابع مشابه
Some Identities for the Riemann Zeta-function
Several identities for the Riemann zeta-function ζ(s) are proved. For example, if s = σ + it and σ > 0, then ∞ −∞ (1 − 2 1−s)ζ(s) s 2 dt = π σ (1 − 2 1−2σ)ζ(2σ). Let as usual ζ(s) = ∞ n=1 n −s (ℜe s > 1) denote the Riemann zeta-function. The motivation for this note is the quest to evaluate explicitly integrals of |ζ(1 2 + it)| 2k , k ∈ N, weighted by suitable functions. In particular, the prob...
متن کاملRiemann ’ s ζ - Function
We provide an overview of the Riemann ζ-function with an eye to a proof of the functional equation that is non-standard and more accessible than those proofs readily available in modern texts. This is s survey note with the goal of a more elegant and transparent presentation of the salient features surrounding the functional equation for the ζ-function and the intimate connections with the Riem...
متن کاملThe Fourth Moment of the Zeta-function
which is valid for any complex s, it follows that ζ(s) has zeros at s = −2,−4, . . . . These zeros are traditionally called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s), of which the smallest ones (in absolute value) are 12 ± 14.134725 . . . i. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical strip” 0 < σ = Re s < 1, and the Riema...
متن کاملA Proof for the Density Hypothesis
The Riemann zeta function ζ(s) is defined by ζ(s) = ∑∞ n=1 1 ns for R(s) > 1 and may be extended to a regular function on the whole complex plane excluding its unique pole at s = 1. The Riemann hypothesis is a conjecture made by Riemann in 1859 asserting that all non-trivial zeros for ζ(s) lie on the line R(s) = 12 , which has a broad application in every branch of mathematics. The density hypo...
متن کاملA 11 Integers 11 B ( 2011 ) Dynamical Zeta Functions
These notes are a rather subjective account of the theory of dynamical zeta functions. They correspond to three lectures presented by the author at the “Numeration” meeting in Leiden in 2010. 1 A Selection of Zeta Functions In its various manifestations, a zeta function ζ(s) is usually a function of a complex variable s ∈ C. We will concentrate on three main types of zeta function, arising in t...
متن کامل