By Krishnaswami Alladi (auth.), K. Alladi, P. D. T. A. Elliott, A. Granville, G. Tenebaum (eds.)

This quantity incorporates a choice of papers in Analytic and hassle-free quantity conception in reminiscence of Professor Paul Erdös, one of many maximum mathematicians of this century. Written by way of many best researchers, the papers take care of the latest advances in a wide selection of themes, together with arithmetical features, leading numbers, the Riemann zeta functionality, probabilistic quantity conception, houses of integer sequences, modular varieties, walls, and q-series. *Audience:* Researchers and scholars of quantity idea, research, combinatorics and modular kinds will locate this quantity to be stimulating.

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**Sample text**

Let S1 (n) denote the number of partitions ofn into parts differing by :::3 such that consecutive multiples of 3 cannot occur as parts. Then T(n) = S(n) = S1(n). In [3) a two parameter generalization of Theorem 8 was obtained and we describe this now. Consider three colors a, b and c = ab, where a and b are primary colors and c = ab 34 ALLADI is a secondary color. The integer 1 occurs only in the primary colors whereas each integer n 2:: 2 occurs in all three colors. The symbols an, bn and Cn represent the integer n in colors a, band c respectively.

Manufactured in The Netherlands. A Small Maximal Sidon Set lMRE Z. hu 127, H-1364 Hungary To the memory of Paul Erdos, my master and mentor Received February 5, 1997; Accepted March 19, 1997 Abstract. We construct a Sidon set A c [1, N] which has «(NlogN) 113 elements and which is maximal in the sense that the inclusion of any other integer from [ 1, N] destroys the Sidon property. Key words: Sidon set 1991 Mathematics Subject Classification: Primary-11B75, Secondary--{)5Bl0, 11Bl3, 11B34 A set A of integers is a Sidon set, if all the sums a+ a', a, a' E A are distinct.

B; (mod q). Equation (1) is equivalent to (2) and - m - (bu du+v d - d w- + hv q bw) ' -m. (3) Since2-2q ::5 m-(bu+hv-bw) ::5 N+q-2,theintegerm'satisfies-l ::5 m' ::5 M+l. In (2) and (3), w must be different from u and v, but u and v may coincide. If they are all distinct, then the probability that (3) holds is P(du + dv- dw = m') = Sf M 3 , 57 A SMALL MAXIMAL SIDON SET where S denotes the number of triplets (x, y, z) such that x + y - z = m', 0 s x, y, z s M - 1. If u = v, then this probability is S' I M 2 , where S' is the number of pairs (x, z) such that 2x- z = m', 0 s x, z s M-I.