{"product_id":"arithmetic-functions-and-integer-products-9781461385509","title":"Arithmetic Functions and Integer Products","description":"\u003cp\u003e • Author(s): P. D. T. A. Elliott\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Springer\u003cbr\u003e • BISAC: Number Theory\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eEvery positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei =   I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non- negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x . Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func- tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239\/240 of this series, under the title Probabilistic Number Theory.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45284153819287,"sku":"9781461385509","price":3639.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9781461385509.webp?v=1769280337","url":"https:\/\/atlanticbooks.com\/products\/arithmetic-functions-and-integer-products-9781461385509","provider":"Atlantic Books","version":"1.0","type":"link"}