Riemann Xi function

The Riemann function in the complex plane of numbers . ${\displaystyle \xi }$

In mathematics , the Riemann Xi function is a transform of the Riemann zeta function . Its zeros correspond exclusively to the nontrivial zeros of the zeta function, and in contrast to this, the Xi function is holomorphic on the entire complex plane . In addition, it satisfies a particularly simple functional equation . Bernhard Riemann introduced it in 1859 in the same paper on the prime number distribution in which he also formulated the Riemann Hypothesis , which was later named after him .

definition

The Riemann Xi function ( “small xi” ) is defined as ${\displaystyle \xi }$

${\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\ \Gamma \left({\frac {s}{2}}\right)\zeta (s),}$

where denotes the Riemannian function and the gamma function . The product term on the right in front of the Riemann function eliminates exactly all negative zeros and the singularity of the zeta function at that point . The only zeros of are therefore exactly the nontrivial zeros of the function. ${\displaystyle \zeta }$ ${\displaystyle \zeta }$ ${\displaystyle \Gamma (\cdot )}$ ${\displaystyle \zeta }$ ${\displaystyle s=1}$ ${\displaystyle \xi }$ ${\displaystyle \zeta }$

A variant of the Xi function is usually referred to as ("large Xi") and is derived from the variable transformation (i.e. ): ${\displaystyle \Xi }$ ${\displaystyle \xi }$ ${\displaystyle \textstyle s\mapsto t={\frac {i}{2}}-is}$ ${\displaystyle s={\textstyle {\frac {1}{2}}+it}}$

${\displaystyle \Xi (t)=\xi ({\textstyle {\frac {1}{2}}+it})=-{\frac {t^{2}+{\frac {1}{4}}}{2{\sqrt {\pi }}^{1/2+it}}}\ \Gamma \left({\textstyle {\frac {1}{4}}+{\frac {it}{2}}}\right)\zeta ({\textstyle {\frac {1}{2}}+it}).}$

The Riemann Hypothesis is equivalent to the statement that all zeros of are real . ${\displaystyle \Xi }$

Remarkably, Riemann himself used the letter to designate the function that is now (according to Landau ) designated with ; The reason for this initially confusing symbolism lies in an obvious mistake by Riemann, which, however, has no effect on the statements of his article. ${\displaystyle \xi }$ ${\displaystyle \Xi }$

properties

Special values

The following applies:

${\displaystyle \xi (0)=\xi (1)=-\zeta (0)={\frac {1}{2}}}$

${\displaystyle \xi (1/2)=-\zeta (1/2)\cdot {\frac {\Gamma (1/4)}{8\pi ^{\frac {1}{4}}}}=0,4971207781...}$(Minimum in the real-valued domain, sequence A114720 in OEIS )

${\displaystyle \xi (3)={\frac {3}{2\pi }}\,\zeta (3)}$

${\displaystyle \xi (5)={\frac {15}{2\pi ^{2}}}\,\zeta (5).}$

The following applies to even natural numbers :

${\displaystyle \xi (2n)=(-1)^{n+1}{{B_{2n}2^{2n-1}\pi ^{n}(2n^{2}-n)(n-1)!} \over {(2n)!}}\qquad (n=1,2,3,4,...)}$

where the -th denotes Bernoulli number . Among other things, the following values ​​result from this representation: ${\displaystyle B_{2n}}$ ${\displaystyle 2n}$

${\displaystyle \xi (2)={\frac {\zeta (2)}{\pi }}={\frac {\pi }{6}}}$

${\displaystyle \xi (4)={\frac {6}{\pi ^{2}}}\,\zeta (4)={\frac {\pi ^{2}}{15}}.}$

Functional equation

The Xi-function satisfies the functional equation ( "reflection formula")

${\displaystyle \xi (1-s)=\xi (s)}$

or equivalent for the function: ${\displaystyle \Xi }$

${\displaystyle \Xi (-t)=\Xi (t)}$

${\displaystyle \Xi }$is therefore an even function .

Product presentation

${\displaystyle \xi (s)={\frac {1}{2}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}$

where in the product formula runs over all zeros of . ${\displaystyle \rho }$ ${\displaystyle \xi }$

Relationship to the Riemann-Siegel Z function

It applies

${\displaystyle Z(t)=-{\frac {2\pi ^{1/4}}{(t^{2}+{\frac {1}{4}})\,|\Gamma ({\frac {1}{4}}+{\frac {1}{2}}it)|}}\ \Xi (t).}$

Asymptotic behavior

For real values of true ${\displaystyle s}$

${\displaystyle \textstyle \ln \xi (s)=\Theta ({\frac {1}{2}}s\ln s)\qquad (s\in \mathbb {R} ,s\to \infty )}$

so

${\displaystyle \xi (s)=\Theta ({\sqrt {s}}^{\,s})}$

(where denotes the Landau symbol ). The same applies to real values ​​of ${\displaystyle \Theta }$ ${\displaystyle t}$

${\displaystyle \textstyle \Xi (t)=\xi ({\frac {1}{2}}+it)=O(t^{1/4}e^{-\pi t/4})\qquad (t\in \mathbb {R} ,t\to \infty )}$

Li coefficients

The Xi function is closely related to the so-called Li coefficients ${\displaystyle \xi }$

${\displaystyle \lambda _{n}=\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right],}$

where the sum extends over the zeros of ; because the relationships count ${\displaystyle \rho }$ ${\displaystyle \xi }$

${\displaystyle \lambda _{n}={\frac {1}{(n-1)!}}\ \left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} s^{n}}}\ [s^{n-1}\ln \xi (s)]\right|_{s=1}\qquad (n\geqq 1)}$

and

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\ln \xi \left({\frac {z}{z-1}}\right)=\sum _{n=0}^{\infty }\lambda _{n+1}z^{n}.}$

The lish criterion is the property for all positive ones . It is equivalent to the Riemann Hypothesis . ${\displaystyle \lambda _{n}>0}$ ${\displaystyle n}$

literature

• HM Edwards: Riemann's Zeta Function . Dover Publications, Mineola, NY 2001, ISBN 0-486-41740-9 .
• JC Lagarias: Li coefficients for automorphic L-functions . In: Mathematics . 2004. arxiv : math.MG/0404394 .
• B. Riemann : About the number of prime numbers under a given size . In: Monthly reports of the Royal Prussian Academy of Sciences in Berlin . 1859.
• EC Titchmarsh : The Theory of the Riemann Zeta Function, Second revised (Heath-Brown) edition . Oxford University Press, 1986, ISBN 0-19-853369-1 .

Web links

Wikisource: About the number of prime numbers under a given size  - sources and full texts

• Eric W. Weisstein : Xi Function . In: MathWorld (English).

Individual evidence

1. ↑ Edwards (2001), footnote §1.16 (p. 31)
2. ↑ Edwards (2001) §2.1 (p. 39)
3. ↑ Titchmarsh (1986) §4.17 (p. 89)
4. ↑ Titchmarsh (1986) §2.12 (p. 29)
5. ↑ Titchmarsh (1986) §5.1 (p. 96) & §10.2 (p. 257)
6. ↑ Lagarias (2004)