In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series
{\rmRiesz}(x)=
infty | |
\sum | |
k=1 |
(-1)k-1xk | |
(k-1)!\zeta(2k) |
=x
infty | |
\sum | |
n=1 |
\mu(n) | \exp\left( | |
n2 |
-x | |
n2 |
\right).
F(x)=
12 | |
\rmRiesz |
(4\pi2x)
x | |
2 |
\coth
x | |
2 |
=
infty | |
\sum | |
n=0 |
cnxn=1+
1 | |
12 |
x2-
1 | |
720 |
x4+ …
F
F(x)=
infty | |
\sum | |
k=1 |
xk | |
c2k(k-1)! |
=12x-720x2+15120x3- …
The values of
\zeta(2k)
x\exp(-x)
F(x)=
infty | |
\sum | |
k=1 |
k\overline{k+1 | |
x |
k
n\overline{k
Bn
x
x
It can be shown that
\operatorname{Riesz}(x)=O(xe) (asx\toinfty)
for any exponent
e
1/2
1/4
The Riesz function is related to the Riemann zeta function via its Mellin transform. If we take
{lM}({\rmRiesz}(z))=
infty | |
\int | |
0 |
{\rmRiesz}(z)zs
dz | |
z |
\Re(s)>-1
1 | |
\int | |
0 |
{\rmRiesz}(z)zs
dz | |
z |
\Re(s)<-
1 | |
2 |
infty | |
\int | |
1 |
{\rmRiesz}(z)zs
dz | |
z |
-1<\Re(s)<-
12 | |
\Gamma(s+1) | |
\zeta(-2s) |
={lM}({\rmRiesz}(z))
From the inverse Mellin transform, we now get an expression for the Riesz function, as
{\rmRiesz}(z)=
c+iinfty | |
\int | |
c-iinfty |
\Gamma(s+1) | |
\zeta(-2s) |
z-sds
The Maclaurin series coefficients of
F
-1.753 x 1017
F(z)
|z|<9
Another approach is to use acceleration of convergence. We have
{\rmRiesz}(x)=
infty | |
\sum | |
k=1 |
(-1)k+1xk | |
(k-1)!\zeta(2k) |
.
\zeta(2k)
infty | |
\sum | |
k=1 |
(-1)k+1xk | |
(k-1)! |
=x\exp(-x)
infty | |
{\sum | |
n=1 |
{\rmRiesz}(x/n2)=x\exp(-x)}.
Using Kummer's method for accelerating convergence gives
{\rmRiesz}(x)=x\exp(-x)-
infty | |
\sum | |
k=1 |
\left(\zeta(2k)-1\right)\left(
(-1)k+1 | |
(k-1)!\zeta(2k) |
\right)xk
Continuing this process leads to a new series for the Riesz function with much better convergence properties:
{\rmRiesz}(x)=
infty | |
\sum | |
k=1 |
(-1)k+1xk | |
(k-1)!\zeta(2k) |
=
infty | |
\sum | |
k=1 |
(-1)k+1xk | |
(k-1)! |
infty | |
\left(\sum | |
n=1 |
\mu(n)n-2k\right)
infty | |
\sum | |
k=1 |
infty | |
\sum | |
n=1 |
(-1)k+1\left(x/n2\right)k | |
(k-1)! |
=x
infty | |
\sum | |
n=1 |
\mu(n) | \exp\left(- | |
n2 |
x | |
n2 |
\right).
\mu
{\rmRiesz}(x)=x\left(
6 | |
\pi2 |
+
infty | |
\sum | |
n=1 |
\mu(n) | \left(\exp\left(- | |
n2 |
x | |
n2 |
\right)-1\right)\right)
n
The above series are absolutely convergent everywhere, and hence may be differentiated term by term, leading to the following expression for the derivative of the Riesz function:
{\rmRiesz}'(x)=
Riesz(x) | |
x |
-
infty | |
x\left(\sum | |
n=1 |
\mu(n) | \exp\left(- | |
n4 |
x | |
n2 |
\right)\right)
{\rmRiesz}'(x)=
Riesz(x) | |
x |
+x\left(-
90 | |
\pi4 |
+
infty | |
\sum | |
n=1 |
\mu(n) | \left(1-\exp\left(- | |
n4 |
x | |
n2 |
\right)\right)\right).
Marek Wolf in[2] assuming the Riemann Hypothesis has shown that for large
x
{\rmRiesz}(x)\simKx1/4\sin\left(\phi-
1 | |
2 |
\gamma1log(x)\right)
where
\gamma1=14.13472514...
K= 7.7750627... x 10-5
\phi=-0.54916...=-31,46447\circ
A plot for the range 0 to 50 is given above. So far as it goes, it does not indicate very rapid growth and perhaps bodes well for the truth of the Riemann hypothesis.
G. H. Hardy and J. E. Littlewood[4] [5] proved, by similar methods, that the Riemann hypothesis is equivalent to the claim that the following will be true for any exponent
e
-1/4
infty | |
\sum | |
k=1 |
(-x)k | |
k!\zeta(2k+1) |
=O(xe) (asx\toinfty).