On the distribution of pseudorandom numbers generated by the linear congruential method. II
Author:
Harald Niederreiter
Journal:
Math. Comp. 28 (1974), 11171132
MSC:
Primary 10K05; Secondary 65C10
MathSciNet review:
0457391
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The discrepancy of a sequence of pseudorandom numbers generated by the linear congruential method is estimated for parts of the period which are somewhat larger than the square root of the modulus. Applications to numerical integration are mentioned.
 [1]
N.
M. Korobov, Trigonometric sums with exponential functions, and the
distribution of the digits in periodic fractions, Mat. Zametki
8 (1970), 641–652 (Russian). MR 0280445
(43 #6165)
 [2]
N.
M. Korobov, The distribution of digits in periodic fractions,
Mat. Sb. (N.S.) 89(131) (1972), 654–670, 672
(Russian). MR
0424660 (54 #12619)
 [3]
L.
Kuipers and H.
Niederreiter, Uniform distribution of sequences,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974.
Pure and Applied Mathematics. MR 0419394
(54 #7415)
 [4]
L.
J. Mordell, On the exponential sum
∑ₓ₌₁^{𝑋}
𝑒𝑥𝑝\(2𝜋𝑖(𝑎𝑥+𝑏𝑔^{𝑥})/𝑝),
Mathematika 19 (1972), 84–87. MR 0318073
(47 #6622)
 [5]
L.
J. Mordell, A new type of exponential series, Quart. J. Math.
Oxford Ser. (2) 23 (1972), 373–374. MR 0319912
(47 #8453)
 [6]
H.
Niederreiter, Methods for estimating discrepancy, Applications
of number theory to numerical analysis (Proc. Sympos., Univ.
Montréal, Montreal, Que., 1971) Academic Press, New York, 1972,
pp. 203–236. MR 0354593
(50 #7071)
 [7]
Harald
Niederreiter, On the distribution of pseudorandom
numbers generated by the linear congruential method, Math. Comp. 26 (1972), 793–795. MR 0326979
(48 #5321), http://dx.doi.org/10.1090/S00255718197203269795
 [8]
H.
Niederreiter, Discrepancy and convex programming, Ann. Mat.
Pura Appl. (4) 93 (1972), 89–97. MR 0389828
(52 #10658)
 [9]
H.
Niederreiter, Application of Diophantine approximations to
numerical integration, Diophantine approximation and its applications
(Proc. Conf., Washington, D.C., 1972) Academic Press, New York, 1973,
pp. 129–199. MR 0357357
(50 #9825)
 [10]
H.
Niederreiter and Walter
Philipp, BerryEsseen bounds and a theorem of Erdős and
Turán on uniform distribution 𝑚𝑜𝑑\1,
Duke Math. J. 40 (1973), 633–649. MR 0337873
(49 #2642)
 [11]
R.
G. Stoneham, On the uniform 𝜖distribution of residues
within the periods of rational fractions with applications to normal
numbers, Acta Arith. 22 (1973), 371–389. MR 0318091
(47 #6640)
 [1]
 N. M. KOROBOV, "Trigonometric sums with exponential functions and the distribution of signs in repeating decimals," Mat. Zametki, v. 8, 1970, pp. 641652, = Math. Notes, v. 8, 1970, pp. 831837. MR 43 #6165. MR 0280445 (43:6165)
 [2]
 N. M. KOROBOV, "On the distribution of digits in periodic fractions," Mat. Sb., v. 89 (131), 1972, pp. 654670 = Math. USSR Sb., v. 18, 1972, pp. 659676. MR 0424660 (54:12619)
 [3]
 L. KUIPERS & H. NIEDERREITER, Uniform Distribution of Sequences, Wiley, New York, 1974. MR 0419394 (54:7415)
 [4]
 L. J. MORDELL, "On the exponential sum ," Mathematika, V. 19, 1972, pp. 8487. MR 0318073 (47:6622)
 [5]
 L. J. MORDELL, "A new type of exponential series," Quart. J. Math. (2), v. 23, 1972, pp. 373374. MR 0319912 (47:8453)
 [6]
 H. NIEDERREITER, "Methods for estimating discrepancy," Applications of Number Theory to Numerical Analysis (edited by S. K. Zaremba), Academic Press, New York, 1972, pp. 203236. MR 0354593 (50:7071)
 [7]
 H. NIEDERREITER, "On the distribution of pseudorandom numbers generated by the linear congruential method," Math. Comp., v. 26, 1972, pp. 793795. MR 0326979 (48:5321)
 [8]
 H. NIEDERREITER, "Discrepancy and convex programming," Ann. Mat. Pura Appl. (4), v. 93, 1972, pp. 8997. MR 0389828 (52:10658)
 [9]
 H. NIEDERREITER, "Application of diophantine approximations to numerical integration," Diophantine Approximation and Its Applications (edited by C. F. Osgood), Academic Press, New York, 1973, pp. 129199. MR 0357357 (50:9825)
 [10]
 H. NIEDERREITER & W. PHILIPP, "BerryEsseen bounds and a theorem of Erdös and Turán on uniform distribution ," Duke Math. J., v. 40, 1973, pp. 633649. MR 0337873 (49:2642)
 [11]
 R. G. STONEHAM, "On the uniform distribution of residues within the periods of rational fractions with applications to normal numbers," Acta Arith., v. 22, 1973, pp. 371389. MR 0318091 (47:6640)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
10K05,
65C10
Retrieve articles in all journals
with MSC:
10K05,
65C10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197404573918
PII:
S 00255718(1974)04573918
Keywords:
Pseudorandom numbers,
discrepancy,
uniform distribution,
trigonometric sums,
numerical integration
Article copyright:
© Copyright 1974
American Mathematical Society
