Re: Riemann-Siegel computations, truncation

David Bernier wrote:
David Bernier wrote:
For t>=0, we can write:

Z(t) = (+/-) exp(-i*arg(zeta(1/2+i*t))) * zeta(1/2+i*t)

That way, Z(t) is real-valued, and in practice "+" changes to "-" when,
as t increases, we go through a zero of zeta(1/2+i*t), and
vice versa ("-" changes to "+" when (etc.)). I don't know
how Z(t) behaves when passing through a double zero of
zeta(1/2+i*t); however, as far as I know, no multiple zeros
of t |-> zeta(1/2+i*t) are known.
[...]

Then the simplified formula from the one (35) that I chose is:
Z(t) ~=
2*sum(X=1,206393,cos(t*log(X)-t/2*log(t/(2*Pi))+t/2+Pi/8)/sqrt(X))+0.0010102


Note that vartheta(t) doesn't depend on X, or the integer 'n' in (35).

vartheta(t) is approximated here by: t/2*log(t/(2*Pi)) - t/2 - Pi/8.
So one term in the sum, for integer X, reads:
cos( t*log(X)- vartheta(t) )/sqrt(X) .

"vartheta" gets its name from being another way to write a
theta. See CurlyTheta at Wolfram Reference:
<http://reference.wolfram.com/mathematica/ref/character/CurlyTheta.html>
[...]

The hard work in computing Z(t) is summing all the cosine terms, not
the computation of the higher order asymptotic terms which
are O(t^(-1/4)) and less. The reason is that the asymptotic terms have
a complexity computation which is almost O(1) as t varies; not quite
O(1), because if t is multiplied by 1000, about 3 more digits of
accuracy are needed for t (mod 2*pi) and so on.

If we look at the last 10,001 terms in the sum, we see that
log(206393) ~= 12.2375373972857 and
log(190393) ~= 12.15684563595768

so if t changes by 100, t*log(206393) - t*log(190393) ~= 8.069176132
or about 462 degrees. This is not tiny, but is small compared
to say the variation in t*log(206393) - t*log(3) ~= 1114 radians if
t increases by 100.

So I computed and plotted the truncated sum:
2*sum(X=206393-10000,206393,cos(t*log(X)-t/2*log(t/(2*Pi))+t/2+Pi/8)/sqrt(X))+0.0010102
for t=267653412247 to 267653412346.9 in steps of 0.1 change in t .

There apparently is one local maximum and one local minimum, separated
by about 70 in t.

Having a good, easy to compute approximation for this truncated sum
might be interesting. The graph is at:
< http://www.geocities.com/ezcos/Z_trunc_10K_terms.jpg > .

The first nine digits of floor(t) are omitted. The graph
consists of 1000 data-points.

David Bernier



.

Relevant Pages

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