{"id":189464,"date":"2013-05-31T00:00:00","date_gmt":"2013-06-11T16:55:21","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/sums-of-squares-characterization-and-distribution\/"},"modified":"2016-08-02T06:11:50","modified_gmt":"2016-08-02T13:11:50","slug":"sums-of-squares-characterization-and-distribution","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/sums-of-squares-characterization-and-distribution\/","title":{"rendered":"Sums of squares \u2013 characterization and distribution"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We survey some of the important and classical facts concerning integers that can be written as the sum of (two, three, or four) squares, as well as the number of such representations, emphasizing the connection to multiplicative functions. We include sketches of proofs of the characterizations of such integers and of Landau&#8217;s theorem on the number of integers that can be represented as the sum of two squares. Finally, we discuss the distribution of such integers in short intervals (including a brief description of sieve methods) and speculate on related questions involving lattice points in thin regions of the plane.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We survey some of the important and classical facts concerning integers that can be written as the sum of (two, three, or four) squares, as well as the number of such representations, emphasizing the connection to multiplicative functions. We include sketches of proofs of the characterizations of such integers and of Landau&#8217;s theorem on the [&hellip;]<\/p>\n","protected":false},"featured_media":197669,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[],"msr-video-type":[206954],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-189464","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-video-type-microsoft-research-talks","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/L5f5oZcbews\/","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/189464"}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/189464\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/197669"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=189464"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=189464"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=189464"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=189464"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=189464"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=189464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}