Zeros of Approximate
Functional
Approximations
By Robert Spira
1. Introduction.
(i)
In this paper we discuss a calculation of zeros of
s»W= Ííi- + xWÍ«h
where
(2)
l/x(s)
= (2x)-2(cos(W2))r(S).
The <7jvr(s)are of interest as approximations
to the Riemann zeta function. Let
s = <r A- it. In [1], it was shown that for t sufficiently large, gi(s) and g2(s) have
their zeros on the critical line a = \. After encountering analytical difficulties in
attempting to extend this theorem to further N, the calculations described below
were undertaken. The results strongly suggest that for N 2: 3 one can expect to
find infinitely many zeros off a = §, so that the theorem proved in [1] appears at
its natural limit. For each N there is a region where g?r(s) behaves similarly to f (s),
and also a region where it behaves similarly to 2f(s). This empirical information
should prove very useful for work along the lines of Rouché's theorem, giving a
condition for the Riemann hypothesis to be true in terms of the location of the
zeros of git(s).
In Section 2, we give the theory of calculating the number of zeros of an analytic
function within a closed curve when the information comes from a finite number of
points on the curve. The theorem requires, for application in our case, an estimate
for | gN (s)\ , and this estimate is obtained. In Section 3, the method used for calculating x(s) is described, a difficulty being the calculation of T(s) for low values
of t. Section 4 contains a discussion of the real function Zs(t) analogous to the Z(t)
of the f-function. In Section 5 the general organization of the calculations is described and Section 6 contains a discussion of the results. There are tables and
figures of the zeros at the end.
2. The Theory of Zero Calculation.
the spacing of points along a curve C
counted correctly the number of zeros
ment at these points.
Let f(z) be analytic inside and on a
R. The governing theorem is that if f(z)
We consider first the problem of how close
must be in order to conclude that one has
inside C by calculating the change of arguclosed rectifiable Jordan curve C in a region
^ 0 on C, then the number of zeros within
C is given by
(3)
Nif, C) =J-.f
Zirl
Ú£ dz =~¿ir [Acarg/(«)].
Jc J(z)
By a set of sequential covering disks of C we mean a finite set of closed disks Ci, C2,
Received December 20, 1965.
41
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42
ROBERT
SPIRA
• • •, Cn , C„+i = Ci with centers cy lying on C whose union lies in R and contains C
and such that the portion of C lying between c¡ and cJ+i lies wholly within Cy.
Theorem
(i)
1. //
Ci, • • • , Cn , Cn+i = Ci is a set of sequential covering disks of C, all of radius h,
(ii) M = l.u.b. \f'(z)\ ,zinR,
(iii) 4hM g min |/(cy)| ,
then
(4)
Nif, C) = ¿X
-j- Ê
kg (/(cy+1)//(cy))].
3=1
Proof. We need only show/(z) ^OonC
and that the change of argument as we
pass from c¡ to cy+i is ^ x/4. Let z be on C and within the disk Cy, so z = c¿ + zx,
| 2i | ^ A. Let / = u A- iv, zi = x-i + ¿2/i • Then, applying Taylor's theorem,
\f(Cj)\ -
|/(2)¡
Ú IfiCj + Zi) -f(Ci)\
^
| uicj
+ zi) — uicj)\
Ú \xiuxiii)
ú[\xi\
since |.Ti| +
> 0 as |/(cy)|
+ yiUyih)\
A- | vicj A- zi) — vici)\
A- I xiVxi¿2) + yivviZi)\
A- I 2/1|].[M + M] S 2V2hM,
12/1| ^ \/2 | zi | S V2h. Hence |/(z)|
à 4/îJî, so/(z) ^ 0 on C.
^
|/(cy)|
- 2y/2hM
To complete the proof, note that by the above argument, all the images of the
points on the curve between Cyand Cy+iwill lie in a circle of radius 2-\/2hM about
ficj). Since, by hypothesis, the center of this circle lies at a distance at least AhM
from the origin, there can be no winding of the curve about the origin for z between
Cj and Cy+i. The possible change in argument will be less if the circle with center
f(Cj) has a central distance to the origin greater than 4hM, and by elementary
analytical geometry, at the closest position the angles of images to the central ray
is at most x/4.
If one uses McLeod's mean value theorem [2], hypothesis (iii) can be weakened
to y/2hM ^ |/(cy)| . According to McLeod's theorem, we can write/(cy + zi)
- f(Cj) = Xi2i/(fi) + X«2i/(&), 0 â Xi, X2; Xi + X2 = 1, and & , £2lying between
Cyand cy + zi. The earlier estimate then reduces to hM.
To apply the theorem, one obtains an a priori estimate for M, and works down
to an h small enough to satisfy áhM g¡ |/(cy)| . If f(z) 9e 0 on C, such an h is
easily seen to exist. Setting uk A- ivk = f(ck), the change of argument in (4) is
easily seen to be arctan [(vkuk+i — ukvk+i)/(vkvk+i A- UkUk+i)]. In some cases, one
may wish to locally compute the derivative and obtain only a consistent picture
rather than a rigorous proof.
In our particular case, we have
(5)
\gir'(a) I ^ É (log n)n°
n=2
For convenience
A- \ x(s) | £ (log «)/"'
we take t ï: 10, 0- > \. With this limitation,
[1], Lemma 2,
(6)
+ | x'(s) | £ n'1.
n=2
1x001 < 1-04(|s\/i2rr))W-°.
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n=l
we have, from Spira
ZEROS OF APPROXIMATE
For an estimate of | x'(s)|
(7)
, we differentiate
x'(s) = x(8)[(*/2)
From Spira, [3] equation
43
FUNCTIONAL APPROXIMATIONS
tan(xs/2)
(14), we have
(2), obtaining
+ log2x - r'(s)/r(s)].
| tan (xs/2)|
á
(1 + e"")/(l
-
e""')
^ 1.02 for I ^ 10. From Schoenfeld [4], we have
(8)
r'(s)/r(«)
and estimating
(9)
= log s - 1/(2«) - l/(12s2) + 6 Í
Jo
as done there, we have
|r'(s)/r(s)|
Putting
together
Pz(x)/(s A- x)A dx,
g | logs | + 1/(2 | s |) + 1/(12 I s ¡2) + 1/(10 | ¿I3).
(6), (7) and (9), we obtain
I x'(«)| è 1.04(1 s |/(2x»1/w
(10)
•[.51x + log2x+
so that
an estimate
| log s | + 1/(2 | s |) + 1/(12 | s\2) + 1/(10 \t |3)]
can be made for
|oat'(s)|
from (5), (6) and (10). Table I
gives these estimates for N. = 2,10,100, t ~ 10, t ~ 100, ^r/glandl
3. The x-Function.
The x-function satisfies
(11)
x(s)x(l
-•)
- 1
so that xfs) can be obtained by calculating
use the Stirling formula (de Bruijn [5]) :
(12)
g a ^ 2.
l/x(l
— s). For the T-function,
we
1 + Zo*.
r(s) ~ Vi2ir)e-'s'-112
where the a* are constants. Near the negative real axis, the error term for (12)
becomes large, so that (12) cannot be used for both t and <r small. By repeatedly
using the functional equation T(s + 1) = sr(s), one obtains
Vx(s + 8) = (2x)"""82(cos (xs/2))r(s
( 13 )
+ 8)
m is2 + 7s\ (s2 + 7s + 6\ (s2 A-Is A- 1Q\ /s2 + 7s+12\
V 4x2 )\
4x2
A
4x2
A
4x2
_1_
)x,8y
so that x(s) can be obtained from x(« + 8) for t and a small. We set £ = the
Table I
Estimates for | Ojv'(s) |
i~
< 1
} á
I x(s) |
I x'GO
I g¿ is)
I flrioC«)
\gíoo(s)
1.
5.91
13.01
80.42
1011.00
10
t~
1 < o- < 2
I Ú «rá 1
.84
5.04
16.64
365.67
42903.64
1.
8.08
17.35
102.12
2036.00
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100
lájg2
.27
2.19
7.30
150.72
16685.40
44
ROBERT SPIRA
factor in brackets in (12), and separate
and imaginary parts:
l/x(s)
= [ey+Ttn cos (6 -
the remaining factors of l/x(«)
into real
xcr/2) + e"-*m cos (B + xa/2)
+ iiev+rtl2sm
(0 -
tto/2)
+ e"'*"2 sin (6 + wa/2))]-S
where
y = (o" - h) log (I s |/(2x))
(15)
B = ¿log(|s|/(2x))
For computation,
í(x/2
we rearrange
- o- - ¿arg s,
- Í + (o- - è)args.
ey+ril2. Since
— arg s) = t arctan
(<r/i)
= <r(l - o-2/(3í2) + <r4/(5¿4) - o-Vilf)
A- ■■■)
we have
(16)
e*+*"2 = exp{(<r
-
i)log(|s|/(2x))
-
<7[<r2/(3i2) -
<r4/(5i4) +
•••]}.
For t very large, it helps accuracy if the quantities 0 ± xu/2 are computed and reduced mod 2x in double precision before computing their single precision sines and
cosines.
For the S factor, we write
m
m
s = i + Y,Ck-iHSk,
k=l
k=l
and use the recursion formulas
(17)
where C
= cos (args)
Ck+i = iak+i/ak)[C*Ck
-
S*Sk]/\ s |,
Sk+i = (ak+1/ak)[C*Sk
+ S*Ck]/\ s \
and S* = sin (args).
The constants
ak+i/ak may be found
in Spira [6]. Table II gives check values for x(s) for a: .5, 1; t: 0, 1, 10, 100.
4. ZN(t), 5D(i). From (1) and (11), it follows that
(18)
gN(s) = x(s)s^(l
- s),
so, on <r = §, if guis)
?¿ 0, we have
(19)
¡7*(| + it)/gNih - it) = xii + it).
Table II
Check values of x(s)
= 1
.5
Re x(«)
0.0
1.0
10.0
100.0
1.0
-.92357 14911
.98891 46004
.99988 53642
Im x(s)
0.0
-.38342 62651
- .14848 53968
- .01514 12839
Re x(s)
.22053 91646
.78528 89724
.25063 86409
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Im x(s)
-.14096 10581
- .10788 77223
- .00348 20602
ZEROS OF APPROXIMATE
li
L
li
li
3
3
90-^
Il
li
h
3
3
Il
li
Il
h
Il
11
3
Il
3
3
3
3
a
Il
îi
3
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3
Il
3
3
3
3
U u
3
3
.
3
"33
3
2
2
2
2222
2
2
3
3
2
3
2
2
2
2
2
2 2
2
2
222
2
2
2 2
2
2 2
2
2
2
2,
2
2 2
22
Ä2
,22
2
'
2 2
2
1
11
2 2
2
2
2
2,
2 2
2
2
2
2
3
2
2
2
2
2
322
3
2
22
2
li
2
2
2
2
li
2
2
2
2
332
2
2
2
333222
U
2
2
2 2
2
2*2
3
StH
,22
,2
2,
2
2
2
2
3
3
2
2
2
2,22
2
2
2,
2
3
333
3
J,
11
1
k
2
2
2
2
2
2
11
Il
2
2
3
2
2
2
3
3
¿
3
2
3
3
3333
Il
3
2
2
3
3
2
2
3
3
2
3
2
3,
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
li
li
0 0 1110
3
3
Il
li
3
3
3
3
3
3
10
100
0 1
3,3
3
3
3
3
3
3
li
.
3
3
3
01111110
0101
45
FUNCTIONAL APPROXIMATIONS
1
1 1
!
2
2
2
2
2
h
3
3
3
t
111
3
2
1 .
I
h
3
3
1
1 ,
2
11
111
2
I
11111
3
:
2
;
111111
%
2
111
2.
1
2
1
11111..
'■11110
10 0
11
1 1
1 1, .
'11000
10
1111.
111
'11111
1111.
11111111
111111.
'11110000
0 0 0 0
0000
111
1 1
1
111.
2
2
1,
111
111
111
2
3
1 1
1 1 .
11,
1 1
1 1 .
Il
111
111
11
11
II
2
'11
11
11
A1
2
3
1 1 .
11
000000
00000.
0
000000000000
111111100
11111
1111
'llOOOOOO-
0000
000
00000000000000
11000000000000.
"00000000000
"ooooooooooo
00000
0 0
0
20
30
Fig. 1. Zeros of gs(s)
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Uo
50
46
ROBERT SPIRA
Define
(20)
ô = ait) = -(argxd
+it))/2,
so
(21)
x(* + it)
= e-2i0.
Now define
(22)
ZH(t) - e%(*
+ Ô).
If gtfih + it) 7e 0, then, from (19), arg^(|
+ it) m —#(mod2x), so ZN(t) is
real for such t, and since it clearly vanishes exactly at the zeros of gaii + it), it
is always real. It is a priori possible that ZN(t) be of one sign on both sides of one
of its zeros. If it changes sign over an interval, then there is a zero within the interval.
As in the case of Z(i) for the f-function (Titchmarsh [7], p. 79), we find
(23)
ZNit)
= 2±C0si»-lt2l0*n)
n=i
n1'2
and
(24)
0 = Imlogrd
+ it/2) - (¿logx)/2
which is easily computed for t > 10 using the asymptotic series for log T. For t < 10
one can use formula (20), which will operate by the method of the previous section
100
Fig. 2. Regions of zeros of gx(s) with given real part
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ZEROS OF APPROXIMATE
47
FUNCTIONAL APPROXIMATIONS
Empirically, on a = Y¿, x(s) starts at 1 for í = 0, and as t increases, xis) first
winds counterclockwise approximately 1^$ revolutions, passing —1 near t = .8 and
1 near t = 3.5 and peaking out around i = 2x, then reversing and winding ever
faster clockwise as t goes to infinity.
5. The Programs. Two programs were written. The first straightforwardly
calculated the number of zeros within a rectangle, and printed out this number along
with the minimum of the functional values calculated and the maximum change of
argument between successive points on the path.
The second program was for use in the critical strip. It calculated A arg gtris)
around a rectangle centered on a = ^ by calculating 2A arg gtt(s) — A arg x(s)
around the right half of the rectangle. That this is correct follows from equation
(18). The values of gx(s) and xis) were saved, so that the values for f7,v+i(s)were
easily calculated. Provision was made for expanding the rectangle and refining the
step size in case unsatisfactory
minima of |<7jv(s)| or unsatisfactory
maxima of
change of argument were obtained.
After the number of zeros within the rectangle was satisfactorily ascertained, a
comparison was attempted with sign changes of ZN(t). The interval along the critical
line could be subdivided several times to attempt to force agreement.
As the computation developed, it became clear that there were many zeros off
the critical line, so that the strict requirements of Theorem 1 did not have to be
applied. Thus, the limits for satisfaction were simply set small, but well above the
Table
III
Selected zeros of gn(s)
N
2
3
3
3
3
4
4
4
4
4
4
4
5
5
5
5
5
6
6
6
6
Real
1.473596
.680126
2.561720
1.403170
.833385
1.736309
2.989619
.586301
.679439
.855779
.820997
.549963
1.516003
1.424444
.750056
.893923
1.088392
.895590
1.200929
.721840
.921079
Imaginary
4.259284
3.437405
8.020892
31.800344
48.882105
6.698773
11.976970
41.877039
49.068871
60.034343
66.058877
88.150047
49.031816
60.013084
76.458282
83.732473
88.219634
49.203126
59.767704
76.643891
88.111376
N
7
7
7
7
8
8
8
8
9
9
9
10
10
10
11
11
12
12
13
14
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Real
1.486354
1.214030
1.163081
.855429
1.057658
1.224559
1.056399
1.287352
1.405330
.739636
1.478086
.826287
.982439
1.161492
1.191464
1.423208
.880716
1.498076
1.195442
.863556
Imaginary
60 .035564
76 .770971
88 .014369
95 . 162091
60 .105830
76 .504505
87 .836911
95 .262405
76 .751423
87 .964248
95 .116721
76 .834502
87 .798674
95 .165571
88 .012637
95 .022729
88 .106968
95 .291697
95 .358673
95 .266764
48
ROBERT SPIRA
roundoff noise, say .05, and the objective of the calculation was changed to obtaining
a reasonably accurate picture of the true situation, with not too great an expenditure
of computing time.
There was considerable cross checking between the programs. For instance,
along the critical line, the program for gN(s) was checked against the independent
programs for ZN(t) and û(t). When roots of a = § were found, computing the
corresponding roots with real part 1 — a gave another overall check.
6. The Zeros.
Approximate
locations
iV: 2(1)100 in the region | < jg
of zeros of gif(s) were calculated
5, 1 ¿ í |
for
200. Fig. 1 gives the results for
t 5¡ 100, N ^ 50. In this figure, the integer 0 signifies a zero with real part strictly
between 5 and 1, and the integer k S; 1 signifies a zero with real part ^ k
and < k A- 1. The column of the integer signifying a zero indicates to which N it
belongs, and if the integer is between t and t A- 1, it indicates a zero with imaginary
part è t and < t + 1. This t is usually meant to be an integer, but in some cases
of uncertainty it was chosen a half integer.
As can be seen, the majority of the zeros lie in descending chains of decreasing
real part. In the portions of the chains where there is a change of the integer from
k to k — 1, if there was uncertainty about whether k or k — 1 was to be chosen,
the larger was taken in almost all cases.
Fig. 2 shows the boundaries of the regions where zeros having given real parts
occur, the integers occurring there having the same significance as in Fig. 1. The
zeros within the critical strip appear to lie outside the t range\/(2irN)
^ t ^ 2xjV
for each N. There is also a second, less obvious, t range free of zeros, corresponding
to where the Riemann-Siegel formula is used, N á (í/2x)1/2 < N A- 1. In this
second region, gtr(s) approximates f(s), while in the first region, g¡f(s) is approximately 2t(s) since 2Zn=iw~" is approximately
f(s) for (2x2V)1/2 ^ t ^ 2x2V,
(Spira [8]), and by the functional equation, the other sum of gx(s) is also approxi-
mately f(s).
Table III gives the zeros for N ;£ 4, t 5= 100, and also the zeros in the region to
the left of the line t = 2xiV.
The computations
were done on an IBM 7040 at the University
of Tennessee
Computing Center, which was aided by grants NSF-G13581 and NSF-GP4046.
Department
of Mathematics
University of Tennessee
Knoxville, Tennessee
1. Robert
Spira,
"Approximate
functional
approximations
and the Riemann
hypothesis,"
Proc. Amer. Math. Soc, v. 17, 1966, pp. 314-317.
2. R. McLeod,
"Mean value theorems
for vector valued functions,"
Proc. Edinburgh
Math.
Soc, v. 14, 1965, pp. 197-209.
3. Robert
Spira,
"Zero-free
regions
of tm(s),"
J. London Math. Soc v. 40, 1965, pp.
677-682. MR 31 #5849.
4. Lowell
Schoenfeld
& R. D. Dixon,
"On the size of the Riemann
zeta-function
at
places symmetric to the point $," Duke Math. J. 33 (1966), 291-292.
5. N. G. de Bruijn,
hoff, Groningen;
6. Robert
Asymptotic
Interscience,
Spira,
Methods in Analysis,
North-Holland,
Amsterdam;
Noord-
New York, 1958. MR 20, #6003.
"Calculation
of the Riemann
zeta function,"
Reviewed
of the Riemann
Zeta Function,
Oxford,
in Math. Comp.,
v. 18, 1964, pp. 519-521.
7. E. C. Titchmarsh,
Theory
at the Clarendon
Press, 1951. MR 13, 741.
8. Robert
Spira,
"Zeros
of sections
of the zeta function.
pp. 542-550.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
I," Math.
Comp., v. 20, 1966,