Riesz Function - Calculation of The Riesz Function

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 -1.753×1017. 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 for . 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

Since ζ(2k) approaches one as k grows larger, the terms of this series approach

. Indeed, Riesz noted that:

Using Kummer's method for accelerating convergence gives

with an improved rate of convergence.

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

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

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:

which may be rearranged as

Marek Wolf in assuming the Riemann Hypthesis has shown that for large x:

where is the imaginary part of the first nontrivial zero of the zeta function, and . It agrees with the general theorems about zeros of the Riesz function proved in 1964 by Herbert Wilf.