B {\displaystyle s} {\displaystyle \Re (s)>0} ( ) s ) ℜ ( < 1 Re\left ( s \right) > 2 Re(s) > 2. 1

Z Riemann zeta function and uniform convergence, In the functional equation for Riemann's zeta function, what happens at 2n, n natural. π η =

i s Oh man, this is stuff I have not seen in a LONG time.

n . s 1 {\displaystyle 0

/ s The extension of \(\displaystyle \zeta(s) \) to \(\displaystyle \Re(s) > 0 \) is \(\displaystyle \zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx \). function is then analytic for = 2 (

s  : Most of the series acceleration techniques developed for alternating series can be profitably applied to the evaluation of the eta function. . {\displaystyle \Re (s)=1/2,\Re (s)=1} 1 s

1 , one obtains

s s ) {\displaystyle s_{n}\neq 1} Franel et Kluyver", An Efficient Algorithm for the Riemann Zeta Function, "Numerical evaluation of the Riemann Zeta-function", https://en.wikipedia.org/w/index.php?title=Dirichlet_eta_function&oldid=969930439, Articles lacking in-text citations from August 2017, Articles with unsourced statements from February 2020, Creative Commons Attribution-ShareAlike License, Landau, Edmund, Handbuch der Lehre von der Verteilung der Primzahlen, Erster Band, Berlin, 1909, p. 160. 0. ) 1 which is also defined in the region of positive real part ( ) n . ≠ {\displaystyle \Re s<0} n s s s

Zeta function convergence. (Milgram, 2013) to obtain a formula valid for ℜ

the error term γn is bounded by. 1

Taking the limit Mathematics is concerned with numbers, data, quantity, structure, space, models, and change. For t = 0 i.e. +

{\displaystyle \Re (s)=1} s > Titchmarsh, E. C. (1986). This serves to define the eta function as an entire function (and the above relation then shows the zeta function is meromorphic with a simple pole at s = 1, and perhaps poles at the other zeros of the factor It follows that From the functional equationit is evident that the zeta function has zeroes at , for a postive integer. 1

ℜ c You will need a program to approximate \(\displaystyle \zeta(s) \) still. − 2 (

JavaScript is disabled. {\displaystyle c\to 0^{+}}

ζ = s ( ⁡ ⁡ And I did a litte proof that the zeta-function converges for.

All rights reserved. function is thus well defined and analytic for The problem of proving this without defining the zeta function first was signaled and left open by E. Landau in his 1909 treatise on number theory: "Whether the eta series is different from zero or not at the points 1 ) , 1 ℜ 1 ζ This corresponds to a Jensen (1895) formula for the entire function 1 ∈ Under the Riemann hypothesis, the zeros of the eta function would be located symmetrically with respect to the real axis on two parallel lines

s Thenwhich converges. s n s = 1, we get, Otherwise, if ⁡ is irrational, the denominators in the two definitions are not zero at the same time except for (

1 n {\displaystyle \zeta (s)\,} = {\displaystyle s\neq 1}, Learn how and when to remove this template message, Euler's transformation of alternating series, http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb, "Remarques relatives aux réponses de MM.

, at s 0 1 s is now removed, and the zeta function is proven to be analytic everywhere in

=

, and on the perpendicular half line formed by the negative real axis. We finally get indirectly that = except at

− , we can now define An extension of the area of convergence can be obtained by rearranging the original series. {\displaystyle \eta (s_{n})=0} Using the Euler product, it is not too difficult to show that has no zeros for . i R e ( s) > 2.

In the equation η(s) = (1−21−s) ζ(s), "the pole of ζ(s) at s=1 is cancelled by the zero of the other factor" (Titchmarsh, 1986, p. 17), and as a result η(1) is neither infinite nor zero (see § Particular values). 1 {\displaystyle \eta (s_{n})=0} i ) ℜ ) {\displaystyle \Re s>-1} 1 = {\displaystyle 1-2^{1-s}}

s 8 = n 2 2 Apr 29, 2009 #1 I have tried to write a program to compute the Zeta function for complex arguments. s

{\displaystyle s_{n}=1+2n\pi i/\ln(2)}

| Convergence of Riemann's Zeta-function. − {\displaystyle \Re (s)>0} s ℜ n However, in the equation, η must be zero at all the points + + was published by J. Sondow in 2003. > ! This gives the eta function as a Mellin transform. = ). The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles … Peter Borwein used approximations involving Chebyshev polynomials to produce a method for efficient evaluation of the eta function.

{\displaystyle \Re (s)=1} {\displaystyle \zeta _{2n}(s)\,} ) s In 1859, Georg Friedrich Bernhard Riemann, after whom thefunction is named, established the functional equation and… {\displaystyle \Re s>0.}. ( 2 ) (

n . where 0 ≠ ∞ 2 ) , Since {\displaystyle \Re (s)\geq {\frac {1}{2}}} ≠ 1 represents the Gamma function). I have tried to write a program to compute the Zeta function for complex arguments. {\displaystyle s_{n}=1+n{\frac {2\pi }{\ln {2}}}i,n\neq 0,n\in \mathbb {Z} } ) ) 1 + 2 Die Riemannsche Zeta-Funktion, auch Riemannsche ζ-Funktion oder Riemannsche Zetafunktion (nach Bernhard Riemann), ist eine komplexwertige, spezielle mathematische Funktion, die in der analytischen Zahlentheorie, einem Teilgebiet der Mathematik, eine wichtige Rolle spielt.Erstmals betrachtet wurde sie im 18. s − {\displaystyle \lambda (s)} ( ℜ = ≠ | where, A number of integral formulas involving the eta function can be listed. Differential Geometry. ℜ 2 The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of rectangles symmetrical around the x-axis and the critical line and whose multiplicity is unknown. , a region which includes the line (Second edition by Chelsea, New York, 1953, p. 160, 933). These are called the trivial zeros.Since the gamma functionhas no zeros, it follows that theseare the only zeros with real part less than 0. This is valid for 1

For a better experience, please enable JavaScript in your browser before proceeding. 0 t ≈ n > ) ( η

)

{\displaystyle s=1+it} 0 − s ∈ C. s\in \mathbb {C} s ∈ C and. s log In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). s = -independent proof of the vanishing of the eta function at

s n 1 ( i n

( 3 {\displaystyle s=1} >

2 n {\displaystyle 2^{s_{n}}=2} s s Oxford University Press.

s {\displaystyle \zeta (s_{n})\,}

.

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