CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt146",{id:"formSmash:upper:j_idt146",widgetVar:"widget_formSmash_upper_j_idt146",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt147_j_idt149",{id:"formSmash:upper:j_idt147:j_idt149",widgetVar:"widget_formSmash_upper_j_idt147_j_idt149",target:"formSmash:upper:j_idt147:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

PRIME POLYNOMIALS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONSPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 164, no 2, 277-295 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2015. Vol. 164, no 2, 277-295 p.
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:kth:diva-161612DOI: 10.1215/00127094-2856728ISI: 000349513200003Scopus ID: 2-s2.0-84922496038OAI: oai:DiVA.org:kth-161612DiVA: diva2:797919
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt434",{id:"formSmash:j_idt434",widgetVar:"widget_formSmash_j_idt434",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt440",{id:"formSmash:j_idt440",widgetVar:"widget_formSmash_j_idt440",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true});
##### Note

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x x(is an element of)] is about x(is an element of)/logx. The second says that the number of primes p < x in the arithmetic progression p equivalent to a (mod d), for d < x(1-delta), is about pi(x)/phi(d), where phi is the Euler totient function. More precisely, for short intervals we prove: Let k be a fixed integer Then pi(q)(I(f,is an element of)) similar to #I(f,is an element of)/k, q -> infinity holds uniformly for all prime powers q, degree k monic polynomials is an element of F-a[t] and is an element of(0)(f, q) <= is an element of, where is an element of(0) is either 1/k, or 2/k if p vertical bar k(k-1), or 3/k if further p = 2 and deg f ' <= 1. Here I (f, is an element of) = {g is an element of F-q[t] vertical bar deg( f - g) <= is an element of deg f }, and pi(q)(I(f, is an element of)) denotes the number of prime polynomials in I (f, 6). We show that this estimation fails in the neglected cases. For arithmetic progressions we prove: let k be a fixed integer. Then pi(q)(k; D,f) similar to pi(q)(k)/phi(D), q -> infinity, holds uniformly for all relatively prime polynomials D, f is an element of F-q[t] satisfying parallel to D parallel to <= q(k)(1-(delta 0)) where delta(0) is either 3/k or 4/k if p = 2 and (f /D)' is a constant. Here pi(q)(k) is the number of degree k prime polynomials and pi(a)(k; D, f) is the number of such polynomials in the arithmetic progression P equivalent to f (mod D). We also generalize these results to arbitrary factorization types.

QC 20150325

Available from: 2015-03-25 Created: 2015-03-13 Last updated: 2017-12-04Bibliographically approved
doi
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CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1197",{id:"formSmash:lower:j_idt1197",widgetVar:"widget_formSmash_lower_j_idt1197",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1198_j_idt1200",{id:"formSmash:lower:j_idt1198:j_idt1200",widgetVar:"widget_formSmash_lower_j_idt1198_j_idt1200",target:"formSmash:lower:j_idt1198:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});