WP34s: inconsistent behavior for Zeta near zero

[Dieter]

In the General Forum there is a discussion about the Riemann Zeta function and its implementation on various HP calculators. For this purpose the high accuracy 34s served me very well for reliable reference values. But now I realized that arguments very close to zero may be off. The following examples refer to 34s version 3.3 3840 with printer support.

Note: for x close to zero Zeta(x) is approximately –0,5–x. So Zeta(1E-10) is ~ –0,5000000001 and Zeta(1E-20) rounds to –0,5 in SP mode. So far so good.

But then:

Zeta(3 E-30) yields –0,5 (correct)
Zeta(3,3333 E-30) returns –0,5 (correct)
Zeta(3,33333 E-30) returns –0,5000045000450005 (completely off)
Zeta(–3,33333 E-30) returns –0,5000495049504950 (completely off)
Zeta(3,14159 E-30) returns –0,4999984084542908 (completely off)
Zeta(1E-40) throws a –∞ error

One reason for me posting this is ...I came across the same problem when I tried to calculate Zeta for arguments very close to zero by means of the reflection formula. Here Zeta(x) can be expressed in terms of Zeta(1–x) so that x<0 or even x<0,5 can be handled. Maybe this is also the reason for the mentioned strange results on the 34s. I suspect that 1–x here simply rounds to 1, and Zeta(1) is undefined, or ∞ if you prefer.

Dieter