{"product_id":"arithmetic-functions-and-integer-products-p-d-t-a-elliott-9781461385509","title":"Arithmetic Functions and Integer Products","description":"Every 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.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eAuthor:\u003c\/b\u003e P. D. T. a. Elliott\u003cbr\u003e\u003cb\u003eISBN-10:\u003c\/b\u003e 1461385504\u003cbr\u003e\u003cb\u003eISBN-13:\u003c\/b\u003e 9781461385509\u003cbr\u003e\u003cb\u003ePublisher:\u003c\/b\u003e Springer\u003cbr\u003e\u003cb\u003eLanguage:\u003c\/b\u003e English\u003cbr\u003e\u003cb\u003ePublished:\u003c\/b\u003e 10\/18\/2011\u003cbr\u003e\u003cb\u003ePages:\u003c\/b\u003e 461\u003cbr\u003e\u003cb\u003eFormat:\u003c\/b\u003e Paperback\u003cbr\u003e\u003cb\u003eWeight:\u003c\/b\u003e 1.48lbs\u003cbr\u003e\u003cb\u003eSize:\u003c\/b\u003e 9.21h x 6.14w x 0.97d","brand":"P. D. T. a. Elliott","offers":[{"title":"Paperback","offer_id":46136041079039,"sku":"9781461385509","price":54.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0662\/2982\/9887\/files\/img_475238d6-9fd2-4c41-98b2-b66bc5f1a40c.jpg?v=1726415508","url":"https:\/\/www.whiterainbookhouse.com\/products\/arithmetic-functions-and-integer-products-p-d-t-a-elliott-9781461385509","provider":"WR Book House","version":"1.0","type":"link"}