1915/01/01 by S. Ramanujan · 13 citations
Arts and Humanities · Physics and Astronomy · #Diverse Historical and Scientific Studies #Historical Astronomy and Related Studies #History of Science and Medicine
paper · doi:10.1112/plms/s2_14.1.347
~ § 1. I:πtrodμCti01~and sμ1n1narν of results.II , 2-5.Eleme1ιta1'y 1'esuUs concerning the order of d (N).2. ' ' r h e order of d (N) when the prime divisors of N are known.3. The order of d (N) when the number of prime divisors is known.4-5.The order of d (N) when nothing is known about N. III.6-31.The stl'uctw'e of higμly co께oosi te n"mbers.6. Definition of a highly composite number.7. A table of the first 102 highly composite numbers.8. Pirst standard form of a highly composite number.The index of the largest prime divisor.n.Alternative standard form of a highly composite number.10-24.The indices of the prime divisors in the fip,;t standard form.25-27.The final primes in the second standard form.28.The ratio of two consecutive highly composite numbers.29.The form of d (N) for highly composite values of N. 30.The order of dd (N) for highly composite values of N. 31.Some apparently paradoxical forms of d (N) in the table.IV. 32-38.Superi01' hiηhly composite nmnbers.32.Definition of a superiol' highly composite number.33.The exact form of a superior highly composite number.34.The number of divisors of a superior highly composite number.35.The maximum value of d (N) I NI .~for a given value of x. 36.Consecutive superior highly composite numbers.37.The number of superior highly composite numbers less than a given numbor.A table of the first 50 superior highly composite numbers.38.Superior highly composite numbers and the maximum ol'der of d (N).V. 39-45.Applicαtion to the deter찌찌αtion oj' the m'del' of d (N).39.The maximum order of d (N) , assuming the prime number theorem.40-43.The maximum order of d (N) , assuming the Riemann hy'pothesis.* I am indebted to Mr. Hardy