On the distribution of pseudo-random numbers generated by the linear congruential method. II

Author:
Harald Niederreiter

Journal:
Math. Comp. **28** (1974), 1117-1132

MSC:
Primary 10K05; Secondary 65C10

DOI:
https://doi.org/10.1090/S0025-5718-1974-0457391-8

MathSciNet review:
0457391

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The discrepancy of a sequence of pseudo-random 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****[2]**N. M. Korobov,*The distribution of digits in periodic fractions*, Mat. Sb. (N.S.)**89(131)**(1972), 654–670, 672 (Russian). MR**0424660****[3]**L. Kuipers and H. Niederreiter,*Uniform distribution of sequences*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0419394****[4]**L. J. Mordell,*On the exponential sum ∑ₓ₌₁^{𝑋} 𝑒𝑥𝑝(2𝜋𝑖(𝑎𝑥+𝑏𝑔^{𝑥})/𝑝)*, Mathematika**19**(1972), 84–87. MR**0318073**, https://doi.org/10.1112/S0025579300004976**[5]**L. J. Mordell,*A new type of exponential series*, Quart. J. Math. Oxford Ser. (2)**23**(1972), 373–374. MR**0319912**, https://doi.org/10.1093/qmath/23.4.373**[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****[7]**Harald Niederreiter,*On the distribution of pseudo-random numbers generated by the linear congruential method*, Math. Comp.**26**(1972), 793–795. MR**0326979**, https://doi.org/10.1090/S0025-5718-1972-0326979-5**[8]**H. Niederreiter,*Discrepancy and convex programming*, Ann. Mat. Pura Appl. (4)**93**(1972), 89–97. MR**0389828**, https://doi.org/10.1007/BF02412017**[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****[10]**H. Niederreiter and Walter Philipp,*Berry-Esseen bounds and a theorem of Erdős and Turán on uniform distribution 𝑚𝑜𝑑1*, Duke Math. J.**40**(1973), 633–649. MR**0337873****[11]**R. G. Stoneham,*On the uniform 𝜖-distribution of residues within the periods of rational fractions with applications to normal numbers*, Acta Arith.**22**(1973), 371–389. MR**0318091**

Retrieve articles in *Mathematics of Computation*
with MSC:
10K05,
65C10

Retrieve articles in all journals with MSC: 10K05, 65C10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0457391-8

Keywords:
Pseudo-random numbers,
discrepancy,
uniform distribution,
trigonometric sums,
numerical integration

Article copyright:
© Copyright 1974
American Mathematical Society