A. IVIC AND H. J. J. TE RIELE fJo - AMS · 2018. 11. 16. · 304 A. IVIC AND H. J. J. TE RIELE and (3) E(T) = a_lT^ap(f^^/A)\ (D>0), { vaogiogiogr)3/4;] where f(x) = £l+(g(x)) (resp.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
mathematics of computationvolume 56, number 193january 1991. pages 303-328
ON THE ZEROS OF THE ERROR TERM
FOR THE MEAN SQUARE OF |£(± + it)\
A. IVIC AND H. J. J. TE RIELE
Abstract. Let E(T) denote the error term in the asymptotic formula for
fJo 2
2
dt.
The function E(T) has mean value n. By tn we denote the n\X\ zero of
E(T) - n . Several results concerning tn are obtained, including tn+x - tn <i il
tn . An algorithm is presented to compute the zeros of E(T)-n below a given
bound. For T < 500000, 42010 zeros of E(T)-n were found. Various tables
and figures are given, which present a selection of the computational results.
1. Introduction
Let, as usual, for T > 0
EiT) = [\^l + it)\dt-Tlo%{PJ-(2y-l)T
denote the error term in the asymptotic formula for the mean square of the
Riemann zeta function on the critical line (y is Euler's constant). In view of
F. V. Atkinson's explicit formula for E(T) (see [2] and [11, Chapter 15]) and
its important consequences, this function plays a central role in the theory of
as).It is also of interest to consider E(T) in mean square, and one has
:\
3v/2?rC(3)
This formula is due independently to T. Meurman [16] and Y. Motohashi [17],
who improved the previous error term cf(T ' log" T) of D. R. Heath-Brown
[10]. One consequence of (1) is the omega result E(T) = Q(rl/4) [6], which
was sharpened by Hafner and Ivic [7, 8] to
(2) E(T) = Q+{r(logr)'/4(loglogr)<3+'OB4>/4
(1) f E2(t)dt = CTi/2 +cf(T\o%T) Ic -- „^J-ff, - 10.3047
x exp(-73v/kSglöilöiT)} (B > 0)
Received August 23. 1989; revised December 20, 1989.
1980 Mathematics Subject Classification (1985 Revision). Primary 11M06, 11Y35.Key words and phrases. Riemann zeta function, mean square, zeros, gaps between zeros.
Using the technique of extrapolation (cf., e.g., [23, §3.3]), these two values can
be combined to yield the better approximation (provided that h is sufficiently
small):
(46) /«»/a + i/a-/,)/»,
where (I2 - 7,)/15 is a good approximation of the error in I2. This error is
used in our computations as a (very pessimistic) estimate of the error in / .
A possible alternative to (46) might be a Gauss-Legendre quadrature rule. For
example, some experiments revealed that a 3-point Gauss-Legendre rule would
yield roughly the same accuracy as the above 5-point Simpson rule (which,
effectively, is a 4-point rule since the end point value f(T + h) on [T, T + h]
is used as starting point value on [T + h , T + 2h]). However, in order to get
an estimate of the error in the 3-point Gauss-Legendre rule, we know of no
better way (cf. [23, p. 127]) than to apply a 4-point Gauss-Legendre rule, and
compare the results; this would require four extra function evaluations, since the
/-values needed in the 3-point rule cannot be used in the 4-point rule. This is
our motivation for choosing (46). Professor W. Gautschi has kindly pointed out
the alternative of using the 7-point Gauss-Kronrod formula for estimating the
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
318 A. IVIC AND H. J. J. TE RIELE
error in the 3-point Gauss-Legendre formula. This also requires four additional
points, but is more accurate than the 4-point Gauss-Legendre formula, since
it has maximum degree of exactness. See, e.g., W. Gautschi, Gauss-Kronrod
quadrature—A survey, in "Numerical Methods and Approximation Theory III"
(G. V. Milovanovic, ed.), Faculty of Electronic Engineering, Univ. of Nis, Nis,
1988, pp. 39-66.
5.3. The zero-searching algorithm. Our algorithm proceeds with a step h to
find zeros of the function E (t), i.e., after the search has been completed for
t < T, the interval [T, T + h] is searched (in certain cases combined with a
second search on [T - h, T]). Now and then, small parts of the computations
are repeated with a smaller, and also with a larger step. This is in order to check
whether the step has to be decreased or may be increased, respectively, in view
of the required accuracy.
Let Tj := jh , L := 7(7}) and Ej := En(T¡) ,7 = 0,1,.... Suppose thatthe interval [0, T¡] has already been treated. This implies that 7 and E- are
known, for j = 0, 1, ... , i. We now compute T(Tj+x) from 7(7^) (by means
of (42) and Simpson's rule as described in §5.2) and then E¡+x (with the help
of (40)).If EjEj+x < 0, then by continuity there is at least one zero between T¡ and
Tj x . This zero is found by a rootfinder described at the end of this section.
If EiEi+x > 0 and Ej_]Ei < 0, then we are finished with the interval
[T„TM].If EjEj+x > 0 and E¡_XE¡ > 0, then E¡_x , E¡ and Ei+X have the same sign.
We check whether |£| < \E¡_X\ and |£(.| < |7i/+1|. If so, this means we have a
local extremum; if not, we are finished on [Tj, Tj+l].
In case of a local extremum, we check whether
(47) \E¡\<h(\og^- + 2y
If not, we know that there can be no zero on [T¡_x, T¡+x] because of
(48) ±EK{t) [ \ + it log¿ + 2y)>-(l0g%- + 2y
and the mean value theorem, and we are finished on [T¡, Tj+X].
If we do have a local extremum such that (47) holds, we fit a quadratic
polynomial through the three points (T;_x, E¡_x), (T¡, E/), (Tj+] , E¡+x) and
compute the point Te where this polynomial has its extremum. If h is small
enough, there should be a zero, T = Tg, of E'n(t) very close to T = Tc. This
zero is found with the Newton process. Next, 7(7/) and Eg := En(Te) are
computed, and if EeEj < 0, then there are zeros on [Tix, Te] and [Te, Tj+l],
which are found by the rootfinder described below. This completes the descrip-
tion of our algorithm, apart from the rootfinder.
The rootfinder is designed to find a zero of En(t) on the interval [a, b],
where E (a) and En(b) have different sign. First, the intersection point
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
THE ERROR TERM FOR THE MEAN SQUARE OF |{( ' + it)\ 319
(c, En(c)) of the line through (a, En(a)) and (b, En(b)) and the horizon-
tal axis is found. Next, a quadratic polynomial is fitted through the three points
(a,En(a)), (c,En(c)), (b, En(b)), and its zero on [a,b] is taken as the start-
ing point to find a zero of En(t) on [a, b] with the Newton process.
5.4. Error control. The numerical computations were carried out on the CDC
Cyber 995 computer of SARA (the Academic Computer Centre Amsterdam),
which has a floating-point mantissa of 48 bits, i.e., a machine accuracy of about
14 decimal digits.
Our aim was to compute as many zeros as possible of the function En(t)
on the interval [0,5x10], each with an absolute error of about 10~ . This
means an accuracy of at least 5 decimal digits for the smallest zero, and 10
decimal digits for the largest zero below 5 x 105.
The error in the computation of \Ç(j + it)\ was controlled as follows.
For t G [0, 5 x 10 ] we applied the Euler-Maclaurin formula (43) in single
precision. If we assume \Um n(\ + it)\ < 10" , then it follows (cf. [20, pp.
151-152]) that/<•> \—' i r\A/(2m+2),. . ,n ss (2%) 10 (t + m+l).
This still leaves freedom to choose one of either n or m , given / and A . We
took A = 15 and (for most i) m = 100, so that n « 0.2244(? + 101). The
actual error is dominated by the machine errors in the computation of the terms
j-( I +") m (43) a pessimistic upper bound for this error is 10~14/« , and for
the value of n given above, and for t < 5000, this is less than 5.8 x 10
For / g [5 x 10 ,5x10] we applied the Riemann-Siegel formula (44) with
n = 3, in double precision (i.e., with an accuracy of about 28 decimal digits),
and the result was truncated to single precision. We denote this numerical
approximation of Z(t) by Zd(t). An extensive error analysis for t G [3.5 x7 8
10 , 3.72 x 10 ] is given in [15]. A similar analysis shows that for t G [5 x
10 , 5 x 105] the error is dominated by the inherent error in (44), i.e.,
\Zd(t)-Z(t)\ <0.03ir225 < 1.5 x 10"10 for?e[5x lo\ 5x 105].
In order to get an idea of the actual error, we computed \t,(\ + it)\ by (43) and
compared it with |Zrf(r)|,for t = 4900(0.1)5100. The maximum difference we
found was 3.9 x 10~10 at t = 5067.2 .
Since the function En(t) measures how well the integral I(t) is approximated
by the function t(log ^ + 2y - 1) + n, we can expect a loss of significant digits
when we subtract the two terms for the computation of E (t). Therefore, we
computed the integral I(t, h) in (45) so that its contribution to the total error
in I(t + h) (= 7(/) + I(t, h)) was as small as the machine accuracy allows.
Thus, the number h was chosen such that
(49) V1'/,'5 »■»-"•v ' I(t + h)
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
32!) A. IVIC AND H. J. J. TE RIELE
(Recall that (72 - 7,)/15 is a very pessimistic estimate of the quadrature error
in I(t, h).) Actually, we took h = 0.01 for 0 < t < 500, h = 0.02 for500 < t < 2000 and h = 0.05 for 2000 < t < 500000. Several spot checkswere carried out locally for smaller values of h. To summarize, we estimate
that the number of correct digits in our computation of E (t) varies between at
least 13 decimal digits near t = 0 and about 7 near t = 5 x 105. The absolute
error is about I0~nt(\og¿¡ + 2y - 1) « 5.7 x 10"6 for / = 5 x 105.
In the rootfinder used in the zero-searching algorithm described in §5.3, the
Newton process was iterated to machine precision. Usually, no more than two
Newton iterations were needed for this purpose. The influence of the error in
En(t) on its zeros may be quantified as follows. Suppose that in the neigh-
borhood of a zero t = tQ of En(t) we compute with E%(t) rather than with
En(t), where En(t) = En(t) + e, e being a fixed small number. Then the
Newton process for the computation of t = t0 is given by
fM t¡ Ên(ti)_ t¡ EJt*) eE'n(t') E'n(t') E'n(tl) '
so there is a systematic error e/E'n(tl) in the computation of the zero t = t0.
In particular, when E'n(t) is small for / close to t0 , then the error in this zero
may be large. We found
1max —j—— « 3.015,
r<500000, £„(0=0 \E (t)\
where the maximum is assumed for t = 137538.499969. For e = 10~ , this
means a maximum absolute error in the zeros of En(t) of about 3 x 10~ .
6. Results and conjectures
In this section we present a selection of our computational results. We have
found 42010 zeros of the function En(t) on the interval [0, 500000]. The first
100 of them are listed in Table 1.
For selected values of n , Table 2 compares log tn with log n , and tn with
«log«. The quotient log tj log« is slowly changing, with a global tendency
to decrease. We believe it converges to 1, although very large tn will certainly
have to be computed in order to corroborate this. The quotient tn/n log« first
decreases to 0.8904, and then increases slowly to 1.1180 ; no possible conclu-
sion about a limit is apparent from these data. Perhaps «log« is just a rough
approximation to tn , much as « log « is a rough approximation to pn, the «th
prime.
Data on gaps between consecutive zeros of En(t) are shown in Tables 3, 4, 5
and 6. It appears that the gaps dn := tn-tn_x, « = 2,3,... behave in a very
irregular way. Although we cannot exclude the possibility that k = K(n) = j ,
this seems unlikely to us. In fact, we believe k = \ to hold. Maxima and
minima of the quotient dn/tlJ4{ are presented in Tables 4 and 5, respectively.
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
THE ERROR TERM FOR THE MEAN SQUARE OF |£(i + ¿f)| 321
Table 1
The first 100 zeros of En(t)
tn tn tn
1
2
34
5
67
89
10
11
12
13
14
15
16
17
18
19
20
21
2223
24
25
1
4.
91317
222731
35.40
45
50
51
52
54
56
63
69
73
76
81
8590
95
97
199593
757482
117570545429685444
098708706900884578337567500321610584
514621658642
295421
750880819660010778
178386799939909522
138399065503665198958639
460878
2627
282930
31323334
35
3637
3839
40
41
42
43
44
45
4647
48
49
50
99.048912
99.900646101.331134109.007151116.158343
117.477368119.182848119.584571121.514013
126.086783130.461139136.453527
141.371299
144.418515
149.688528
154.448617159.295786160.333263
160.636660171.712482
179.509721181.205224182.410680182.899197
185.733682
51 190.52 192
53 19954 21155 217
56 22457 22658 22959 23560 23961 245
62 256
63 26264 26765 280
66 28967 290
68 29469 29770 297
71 29872 29973 30874 314
75 316
809257450016
646158
864426647450
290283323460548079172515635323494672571746
343301822499
805140701637.222188
.912620,288651.883251
,880777.919407
652004683833
505614
76 31877 319
78 32179 32680 330
81 33582 33983 34384 349
85 35486 35887 371
88 384
89 39090 39691 39992 402
93 40694 40895 417
96 43097 43498 43999 445
100 448
788055913514
209365203904978187
589281871410370082890794639224371624554495
873869001409.118200
102390212210
737516735190047725
962383927645425963.648250
037348
Table 2
Some data concerning the order of t
n tn log cn/ log n tn/n log n
25
10
2050
100
200
500
10002000
500010000
2000042010
417
40
76185448978
27666174
13807
3931089563
204737499993
757482685444
500321
909522733682037348559572
.863752
.307534257919
200279
343441
805598
656034
2.25021.7849
1.6075
1.44961.33551.32571.29971.2753
1.26351.2542
1.24211.2380
1.2349
1.2326
3.43182.1977
1.7589
1.28370.9496
0.97290.92350.89040.89380.9083
0.9231
0.9724
1.0337
1.1180
c tlJ4, logtn-n-\
,, which in general is
Combined with the data in
For 4 < « < 42010 we observed that dn
close to best possible, in view of (2), (3), and (30)
Table 5, this supports the conjecture that k(tc) = \ (where k is defined in (6)).
The data on gn and gj log tn in Table 3 support (36) for a = 2 . Table 6 gives a
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
322 A. IVIC AND H. J. J. TE RIELE
Table 3
Various data related to the gaps between consecutive zeros
^~j^aäzs: tñh"T.^A W°g'nin :=t.-t.- log¿n/log tn logdn/logn gn
2
5
10
20
50
100
200
5001000
2000
500010000
2000042010
3.5578894.140015
5.1627543.109583
2.8344852.389098
0.0759803.6248240.753268
0.596044
7.98303322.172542
1.240345
1.636594
3.24841.1249
0.86850.3620
0.20960.11320.00240.06900.0096
0.0051
0.0403
0.0741
0.00270.0023
3.39962.1580
2.11751.0609
0.77080.5200
0.01360.5000
0.0850
0.05500.56701.2818
0.0583
0.0615
0.8137
0.4945
0.44350.2612
0.19940.1427-0.3743
0.1625-0.0325
-0.0543
0.1964
0.27180.0176
0.0375
1.83100.8827
0.71290.37870.26630.1891-0.4864
0.2072-0.0410-0.0681
0.2439
0.33650.02170.0463
1.80161.8864
1.71651.5146
1.37731.4164
1.38351.2801
1.2895
1.30671.3173
1.39311.4619
1.5505
1.15510.6566
0.46380.34880.26360.2320
0.20090.16150.14770.13710.1245
0.12220.11950.1182
Table 4
Maxima of djtn 1/4
tn ^^T2
370510
11761321
1322
1472
20744224
4692
4848
5006
6058
823017138
18198
21804
23764
39084
41992
2850
74208475
8520
9708
1436532120
3668538070
3951849552
71699
170654
183304
227502
252647457431
757482136994
462567
277407806973
092619104280
.716667209803
.948268
374558
339822122137
441192
832030147130
378144
958173381229
3.55788924.861362
31.291596
42.08575243.841653
44.285645
54.053035
61.75103076.460074
82.898386
88.99070296.093410
104.276659
123.858798
165.382076
169.425143
186.717169
213.951458261.454651
3.39963.73304.2943
4.54104.5751
4.6155
5.4531
5.64655.7148
5.99336.3746
6.81966.9928
7.5724
8.13898.1900
8.5512
9.545110.0549
frequency distribution of the computed values of dn/tj_x , in classes of length
0.1 . For example, we found 10641 values in the interval [0, 0.1), 818 in the
interval [0.9, 1.0) and 1 value (the largest) in the interval [10.0, 10.1) (cf. the
last entry in Table 4). To summarize: 82% of all values are in [0.0, 1.0), 11%
in [1.0, 2.0), 4% in [2.0, 3.0), 2% in [3.0,4.0) and 1% in [4.0,10.1).
Table 7 presents maximal values of \E (t)\ in intervals of length 25000, and
the location of the adjacent zeros. The computed values of En(t)/t
the order results of E(t) as discussed at the beginning of §1.
1/4 confirm
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
THE ERROR TERM FOR THE MEAN SQUARE OF |Ç($ + it)\ 323
Table 5
Minima of dn/tn1/4
234
5
689
13
143344
159200
301628
10302674
361668418088
118571198727021
U4.7574829.117570
13.545429
17.685444
22.098708
31.88457835.337567
51.65864252.295421
119.584571160.636660
753.427349
978.5595721604.0128273569.014754
6389.01163818818.622459
27076.31467157197.58187070009.242085
110163.040870111649.073447294421.287720
3.5578894.360087
4.427859
4.140015
4.413263
4.1776773.452989
1.144021
0.6367790.4017220.303397
0.2807390.0759800.0636530.062385
0.0380080.037263
0.0311370.0229310.021013
0.0067780.0047890.005105
ääs:3.39962.95222.5481
2.1580
2.1521
1.82091.4531
0.42910.23750.12160.0853
0.05360.01360.0101
0.00810.00430.0032
0.00240.00150.0013
0.00040.00030.0002
Table 6
Frequency distribution of the dn/tj4x-values, in classes of length 0.1