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Riesz_function

Riesz function

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
{rm Riesz}(x) = -sum_{k=1}^infty frac{(-x)^k}{(k-1)! zeta(2k)}
If we set F(x) = frac12 {rm Riesz}(4 pi^2 x) we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero. If
frac{x}{2} coth frac{x}{2} = sum_{n=0}^infty c_n x^n = 1 + frac{1}{12} x^2 - frac{1}{720}x^4 + cdots
then F may be defined as
F(x) = sum_{n=1}^infty frac{x^n}{c_{2n}(n-1)!} = 12x - 720x^2 + 15120x^3 - cdots

The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for x exp(-x) shows that it defines an entire function. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of x. Positive values of x are more interesting and delicate.

Riesz criterion

It can be shown that

operatorname{Riesz}(x) = O(x^e)

for any exponent e larger than 1/2, where this is big O notation; taking values both positive and negative. Riesz showed that the Riemann hypothesis is equivalent to the claim that the above is true for any e larger than 1/4.

Mellin transform of the Riesz function

The Riesz function is related to the Riemann zeta function via its Mellin transform. If we take

{mathbf M}({rm Riesz}(z)) = int_0^infty {rm Riesz(z)} z^s frac{dz}{z}
we see that if Re(s)>-1 then
int_0^1 {rm Riesz}(z) z^s frac{dz}{z}
converges, whereas from the growth condition we have that if Re(s) < -frac{1}{2} then
int_1^infty {rm Riesz}(z) z^s frac{dz}{z}
converges. Putting this together, we see the Mellin transform of the Riesz function is defined on the strip -1 < Re(s) < -frac12. On this strip, we have frac{Gamma(s+1)}{zeta(-2s)} = {mathbf M}({rm Riesz}(z))

From the inverse Mellin transform, we now get an expression for the Riesz function, as

{rm Riesz}(z) = int_{c - i infty}^{c+i infty} frac{Gamma(s+1)}{zeta(-2s)} z^{-s} ds
where c is between minus one and minus one-half. If the Riemann hypothesis is true, we can move the line of integration to any value less than minus one-fourth, and hence we get the equivalence between the fourth-root rate of growth for the Riesz function and the Riemann hypothesis.

G.H. Hardy gave the integral representation of f(x) using Borel resummation as

1-exp(-x)= int_{0}^{infty}dt frac{f(t)}{t}lfloor (frac{x}{t})^{1/2}rfloor .

Calculation of the Riesz function

The Maclaurin series coefficients of F increase in absolute value until they reach their maximum at the 40th term of -1.753 x 1017. By the 109th term they have dropped below one in absolute value. Taking the first 1000 terms suffices to give a very accurate value for F(z) for |z| < 9. However, this would require evaluating a polynomial of degree 1000 either using rational arithmetic with the coefficients of large numerator or denominator, or using floating point computations of over 100 digits. An alternative is to use the inverse Mellin transform defined above and numerically integrate. Neither approach is computationally easy.

Another approach is to use acceleration of convergence. We have

{rm Riesz}(x) = sum_{k=1}^infty frac{(-1)^{k+1}x^k}{(k-1)! zeta(2k)}
Since ζ(2k) approaches one as k grows larger, the terms of this series approach sum_{k=1}^infty frac{(-1)^{k+1}x^k}{(k-1)!} = x exp(-x). Indeed, Riesz noted that: {sum_{n=1}^infty rm Riesz(x/n^2) = x exp(-x)}.

Using Kummer's method for accelerating convergence gives

{rm Riesz}(x) = x exp(-x) - sum_{k=1}^infty left(zeta(2k) -1right) left(frac{(-1)^{k+1}}
{(k-1)! zeta(2k)}right)x^k with an improved rate of convergence.

Continuing this process leads to a new series for the Riesz function with much better convergence properties:

{rm Riesz}(x) = sum_{k=1}^infty frac{(-1)^{k+1}x^k}{(k-1)! zeta(2k)}

sum_{k=1}^infty frac{(-1)^{k+1}x^k}{(k-1)!} left(sum_{n

1}^infty mu(n)n^{-2k}right)
sum_{k=1}^infty sum_{n=1}^infty frac{(-1)^{k+1}left(x/n^2right)^k}{(k-1)!}= x sum_{n=1}^infty frac{mu(n)}{n^2} expleft(-frac{x}{n^2}right)
Here μ is the Möbius mu function, and the rearrangement of terms is justified by absolute convergence. We may now apply Kummer's method again, and write
{rm Riesz}(x) = x left(frac{6}{pi^2} + sum_{n=1}^infty frac{mu(n)}{n^2}left(expleft(-frac{x}{n^2}right) - 1right)right)
the terms of which eventually decrease as the inverse fourth power of 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:

{rm Riesz}'(x) = frac{x} - xleft(sum_{n=1}^infty frac{mu(n)}{n^4} expleft(-frac{x}{n^2}right)right)
which may be rearranged as
{rm Riesz}'(x) = frac{x} + xleft(-frac{90}{pi^4} + sum_{n=1}^infty frac{mu(n)}{n^4} left(1-expleft(-frac{x}{n^2}right)right)right)

Appearance of the Riesz function

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.

References

  • Titchmarsh, E. C., The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986, [Section 14.32]
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