The Riemann zeta-function, one of the most important special functions, has a ... Since Riemann's famous hypothesis on the non-trivial zeros of the zeta-function,.
111 REFERENCES 1. TIKHONOV A.N. and AP.SENIN V.YA., Methods of solving ill-posed problems. Moscow, Nauka, 1979. 2. VASIL'BV F.P., Methods of solving extremal problems. E~OSCOW, Nauka, 1981. of Steffensen's method for solving non-linear operator equa3. UL'M S.U., A generalization tions. Zh. vychisl. Mat. mat. Fiz., Vo1.4, No.6, pp.1093-1097, 1964. 4. KHROMOVA L.N., One minimization method with a cubic rate of convergence. Vestn. MGU. Vychisl. Matem. i kibernitika, No.3, pp.52-56, 1980. 5. VASIL'EV F.P., KHROMOVA L.N. and YACHIMOVICH M.D., Iterative regularization of one minimization method of the third order. Vestn. 14GU. Vychisl. matem. i kibernitika, No.1, pp.3136, 1981. 6. VASIL'EV F.P., WROMOVA L.tI. and YACHIMOVICII M.D., Iterative regularization of the minimization method with a high rate of convergence. In: Probl. vychisl. matem. Tr. NIVTS t1GU i VT.5 Budapeshtskovo un-ta im. Loranda. Budapest, pp.24-32, 1982.; 7. VASIL'EV F.P. and KHROMOVA L.N., A method of solving ill-posed extremal problems, based on the principle of iterative regularization. In: Prikl. matem. i mekhan.,Tashkent: Tashkentsk. un-t, 1~0.683, pp.36-41, 1982. 8. VASIL'EV F.P. and KHROMOVA L.N., Methods of higher order for solving operator equations. Dokl. AN SSSR, Vo1.270, No.1, pp.28-31, 1983. 9. OSTROVSKII A.M., The solution of equations and sets of equations. Moscow: Izd-vo inostr. lit., 1963. Translated
U.S.S.R. Comput.Maths.f4ath.Phys., vo1.25,No.2,pp.111-119,198s Printed in Great Britain
by H.Z.
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THE CALCULATION OF THE RIEMANN ZETA-FUNCTION IN THE COMPLEX DOMAIN* A.YU.EREMIN,
I.E. KAPORIN
and M.K. KERIMOV
A new, efficient algorithm is proposed for calculating the Riemann zetafunction with high accuracy when the argument acquires complex values from a fairly large domain in the neighbourhood of the pole. The algorithm is based on using an interpolational Newton series with integral grids of interpolation. The results of extensive numerical experiments are presented which confirm the advantages of the suggested algorithm by comparison with previously well-known methods. 1. Introduction. The Riemann zeta-function, one of the most important special functions, has a rich history. Having emerged from the depths of the theory of numbers, it was mainly applied, until recently, to the analysis of the theory of prime numbers /l-3/. At the present time, however, its area of application has been significantly widened. The zeta-function is now also one of the special functions of mathematical physics (see e.g., /4, ch.2, Sect.ll/, 5, Sect.23.2/). Since Riemann's famous hypothesis on the non-trivial zeros of the zeta-function, which for manyyears stimulated intensive investigations in this area, has still not been definitively solved, it is natural to expect an increase in the flow of literature on the theory of the zeta-function, both in the theory of numbers and in the development of effective calculational algorithms for obtaining the values of the zeta-function and its zeros over fairly wide areas of variation of the argument. There is a vast literature dedicated to zeta-function theory (it is sufficient to mention the fundamental monographs /l, 2/). But there are fewer papers on methods of calculating this function and its zeros. Above all this is due to the great difficulty of carrying out the calculations. With the appearance of powerful computers, the flow of literature about calculation methods increase. Some idea of the work carried out up to 1980 can be obtained from the review article /6/. In the last few years, important publications dedicated to the vase calculations connected The numerical experiments in with zeta-function zeros in the critical zone have appeared. /ll/ establish, for example, that the zeta-function c(s) has exactly 30 OCO 001 a complex whilst these are all simple and distributed O2,
Ilz-xll~2nexp[-2(nn)“l,
Ijzll!-‘E
x.-,)’
z:)“’
is a
(3.3)
OO. that &(I-A)Gq'/2. If ci'/, we can use the well-known ma2 we have E(s) and c(l-s). When
functional
equation
/l,
117
Hence it follows that to achieve the specified absolute accuracy ficient
to choose
Thus, when
the value
lsl-20
and
m>m(e),
e=iO-'
e..R,(s)224.
4. Results of numerical experiments, We shall cite some results of a calculation of Riemann's zeta-function using the proposed algorithm and compare them with those obtained using the algorithm in /17/. To this end we used Stiefel's method to calculate the coefficients x( of the interpolation polynomial of We used two criteria to check the accuracy of the the 249-th degree with twice the accuracy. x0-_r is Euler's constant and, secondly, that coefficients obtained, firstly,
The calculated
values
x( satisfy
the inequalities
To confirm the asymptotic estimate coefficients x0,...,%,, in Table 1.
(2.4) we present
the values
of the first twenty Table
L
“k
erp
t-2
k
(nk)‘l:] I
Xk
sxp-t-2
1
(nk)‘h
I
-1.557179971-995 -9.276122SfX-993 -4.32721X931--008 ;.w&gmg 1:oeoie@57-993 6.949529379-997 4.491929297-997 2.94M99M3--007 1.947271329-997
To determine the accuracy attainable, we compared the values t(s)- calculated by this method - with those tabulated in /14/. Using the proposed algorithms (m=Z'&), these data are fully reproduced to Ims=29.4. For comparison we also used the algorithm in /23/, enabling us to obtain the values Using the proposed algorithm, E(s) accurately when sass. we managed to obtain agreement with the data to an accuracy of 18 places. The comparison shows that the accuracy of the results obtained using the proposed algorithm gradually falls as one moves away from the real axes, which might have been expected, bearing in mind the form of Newton's interpolation series. to compile an appropriate program to compare the above with the algorithm in We had Unfortunately, two errors were made in /17/ when describing the algorithm. /lJ/. In the algorithm given in /17/, the zeta-function is calculated from the formula Y
n-r &+.&‘--$;s+r,A., L-l
where
the coefficients
k-L
A, are determined A,-sB,/
(2h”+‘),
from the recurrence
Ah--A,_,
B +
)r
relations
(s+2k)
(s+2k--i)
(2k+2)
(2k+l)N’
and B, are Bernoulli numbers. To calculate k-’ we must turn to the functions exp(o),In(a), and division costs).sin(u), and to obtain the given term Ah, more than ten multiplication It is easy to calculate operations and more than ten addition operations are required.
118 the number of operations in Homer's method, where 4mi-2 multiplications, 2m+2 divisions and 4m+4 additions are required. When m-249 it takes 0.4 seconds to carry out arithmetic operations in the double-accuracy mode for calculating one value of The values L(r+iy). and [(1+28i) were calculated with an accuracy of f (‘/,+ZSi) -IO-' in the double-accuracy mode for comparison with algorithm (4.1). It takes 0.6-0.65 sacs. to calculate one value when N-llil3 and M=i+l2 The data obtained show that the proposed algorithm can fully compete with the standard.algorithm, based on the Euler-Maclaurin formula. In Table 2 we present the values c(s) at the points s=(2+4k)+(l+51)i, k=O, I, 2, L-0, I. 2, 3, 4, which are accurately calculated using Eq.(4.1) (the first rows) and using the proposed algorithm (m=249) (the second rows). Table
2
d
P+i 2 +
6i
2 +lli 2+
I&
2+
21i
,6+ i 6 + 6i 6 + iii 6 +16i 6 +2li
1.130355703251969 1.150333703254902 0.92666343i703168
0.926663431704244 1.118331971169036 1.116331971191666 1.014294612397563 1.014294612400810 0.791271379213667 0.791271379214376 1.012672882550759 1.012672662332433 0.992926673397664
0.992926873396917 1.064.561940660598 1.004561940661460 1.091661766296991 1.001661766300277 0.992802934366409
0.992802934367 137 IO+ i 10+61 lO+lli 10+ 16i lO+Zli
As can be seen from Table decimal point.
1.ooo759077627751 1.009759077829856
0.999302256327104 0.999302236326606 1.ooO236231411690 l.UOO236231412369 1.600096344305031 1.ooO096544305678 0.999594071263345 0.999594071264907 2, the values
obtained
-0.437530665916 -0.437330865919 +0.156667106310 +0.166667106311 -0.262445736129 -0.202445736129 +0.352992210971 +0.X12992210973 -0.106342706234 -0.108342706236 -0.011644626661 -0.011644626601 +0.012696910224 +0.012696910224 -0.014647449704 -0.014647449704 -0.016662896661 -0.01666289686i -0.012924766046 -0.012924766046 -0.000640126026 -0.ooO640126026 +0.000824796 163 +O.WO 824796183
698i 607i 7551 293i 675i 90% 478 363i 907i 267i 079i 113i 306i 33Oi 327i 359i 8Oli 86%
-0.ooO943465566
974i 964i
-0.000943485566
Wi 289i
n6i 7651 718i 7.291’
+0.000968489879
077i +6.ooO966469679 09li -0.000876543029 63.31' -0.000876543029 64ii
agree to 9 or more places
after the
Two problems arose when using Newton-series sections to calculate the Conclusions. Riemann zeta-function approximately: calculating the coefficients of the series and estimating the accuracy of the approximation. The algorithms and estimates presented above made it As a result a compact and reliable algorithm was obtained, possible to solve both problems. which gives a high accuracy of approximation to b(s) in a fairly large neighbourhood of the point s=-1. and which is capable of competing with well-known methods. REFERENCES 1. TITCHMARSH E.C., The theory of the Riemann zeta-function. Oxford: Clarendon Press, 1951. 2. EDWARDS B.M., Riemann's zeta-function, N.Y.: Acad. Press, 1974. 3. KARATSDBA A.A., Principles of the analytic theory of numbers. Izd. 2-e. Moscow, Nauka, 1983. 4. OLVER F.W.J., Introduction of asymptotics and special functions. N.Y.:L: Acad. Press, 1974. 5. ABRAMOWITE M. and STEGUN I., Eds., Handbook of mathematical functions with formulas, graphs and tables. Bur. Standards. nppl. Math. Ser. Vo1.55, Washington D.C.: U.S. 1964. Nat. 6. KERIMOV M-K., Methods of calculating the Riemann zeta-function and some ofitsgeneraliza1980. tions. Zh. vychisl. Mat. mat.Fiz., Vo1.20, No.6, pp.1580-1597, c(r) on short intervals of a critical straight 7. KARATSUBA A.A., The zeros of the function line. Izv. AN SSSR, Ser. matem., ~01.48, No.3, pp.569-584, 1984. 8. KARKOSCHKA E. and WERNER P., Einige Ausnahmen zur Rosserschen Regel in der Theorie-der Riemannschen Zetafunktion - Computing, ~01.27, No.1, 57-69, 1981. 9. LDNE J. Van De, Riele H.J.J. te, and WINTER D.T., Rigorous high speed separation of zeros Wiskunde, of Riemann's zeta-function. - Math. Centrum Amersterdam. Afdeling Numerieke NW113/81, p.l-35, 1981. 10. BRENT R.P. et al., On the zeros of the Riemann's zeta-function in the critical Strip II. Math. Comput. Vo1.39, No.160, p-681-688, 1982. in the 11. LDNE J. van de, RIELE H.J.J. te., On the zeros of the Riemann's zeta-function critical strip. III. - Math. Comput., ~01.41, ~0.164, p.759-767, 1983.
119 12. LUNE J. van de, RIELE H.J.J. te., Rigorous high speed separation of zeros of Riemann's zeta-function. II. - Math. Centrum Amsterdam, Afdeling Numerieke Wiskunde, NW 126/82, 1982. 13. LIJNEJ. van de, RIELE H.J.J. te, On the zeros of the Riemann's zeta-function in the critical strip. III. - Math. Centrum Amsterdam. Afdeling Numerieke Wiskunde, NW 146/83, p.l-10. 1983. 14. HASELGROVE C.B. and MILLER J.C.P., Tables of the Riemann zeta-function. Cambridge: Cambridge Univ. Press, 1960. 15. FRANSEN A. and WRIGGE S., High-precision values of the gamma function and of some related coefficients. - Math. Comput. Vo1.34, 1~0.150, p-553-566, 1980. 16. FRANSBN A., Addendum and corrigendum to"High-precision values of the gamma function and of of some related coefficients." - Math. Comput. Vo1.37, No.155, p.233-235, 1981. for complex A. and DEPINE R-A., A program for computing the Riemann zeta-function 17. BANUEMS argument. - Comput. Phys. Communs, Vo1.20, No.3, p.441-445, 1980. 18. IBRAGIMOV I-I., Methods of interpolating functions and some of their applications. Moscow: Nauka, 1971. 19. LAVRIK A.F., The main term of the problem of denominators and the step series of the Riemann zeta-functioninthe vicinity of its pole. Tr. Matem. in-ta AN SSSR, Moscow, ~01.142, pp.165-173, 1976. 2O.ISRAILOV M.I., Laurent'sexpansionofthe Riemann zeta-function. Tr. Matem. in-ta AN SSSR, MOSCOW, Vo1.158, pp.98-104, 1981. 21. STANKUS E.P., Remarks on the coefficients of a Laurent series of the Rfemann zeta-functionZap. nauchn. seminarov LOMI AN SSSR, Vo1.121, pp.103-107, 1983. 22. STIEFEL E., Kernel polynomials in linear algebra and their numerical applications. In: U.S. Nat. Bur. Standards, Appl. Math. Ser. Vo1.43, p.l-22, 1958. for the Riemann 23. CODY W-J., HILLSTROM K.E. and THACBER H.C., Jr., Chebyshev approximations zeta-function. - Math. Comput. Vo1.25, No.115, p-537-547, 1971. functions. Vo1.2, Moscow: Nauka, 1974. 24. BATEMAN H. and ERDELYI A., Higher transcendental .
Translated
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.,Vo1.25,No.2,pp.119-124,1985
by H-2.
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ALGORITHMS OF INCREASEDACCURACY FOR SOLVING INTEGRAL EQUATIONS OF THE FIRST KIND* A.E. KOROBOCHKIN,
N.A. MARCHENKO
and A.KH.
PERGAMENT
An algorithm is suggested for solving integral equations of the first kind, based on parametric representations of the unknown functions which are optimal from the information point of view. The accuracy of approximation for the compacta (and of the conditionally compact sets) with respect to the order of magnitude is the same as the Kolmogorov diameter of the respective sets. 1. relation
Consider
the problem
of determining
the unknown
function
z(l),tE[O,T] from the
dz(t)=a(t)+A(t),
(1.1)
where A is a linear, fully-continuous operator, which acts from C[O,T] to LIIO,T],and A(t) is a stationary Gaussian random process with zero mean the correlation operator B. The operator B is symmetric, non-negative definite and kernel. It is assumed below that B is non-degenerate. Then an infinite system (Y,}, k=O, I.... of eigenfunctions and eigenvalues ir exists. All h,>O, and the series T&t-' converges. The case is considered when the values of the function A(t) on the grid with a step are statistically independent. (It is assumed below that the length of the segment equals 1.) [O,Tl The process AC(t) is a linear functional of the generalized random process with a constant spectral density a'. The functional is specified by the function K(z),where ris theparameter:
k=ilN
K(r) This random process is Gaussian spectral density is
lrl
