On the distribution of numbers related to the divisors of xⁿ − 1

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• 作者：Sai Teja Somu ^{somuteja@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Distribution of coefficients of divisors of xn&minus ; 1xn&minus ; 1 ; Coefficients of divisors ; Coefficients of divisors of xn&minus ; 1xn&minus ; 1 ; Coefficients of cyclotomic polynomials ; Cyclotomic polynomials ; On the distribution of numbers related ; Natural density of coefficients
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：170
• 期：Complete
• 页码：3-9
• 全文大小：234 K

文摘

Let d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630155X&_mathId=si1.gif&_user=111111111&_pii=S0022314X1630155X&_rdoc=1&_issn=0022314X&md5=0d2e315a41a7a5f68577075aa40e96ac" title="Click to view the MathML source">n₁,⋯,n_rdden">de">$n_1,\cdots ,n_r$ be any finite sequence of integers and let S be the set of all natural numbers n   for which there exists a divisor d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630155X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1630155X&_rdoc=1&_issn=0022314X&md5=6462ffd0a4155673621ddcc456c62999">dth="164" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630155X-si2.gif">dden">de">$\mathrm{<font\; color="red">d</font>}(x)=1+{\sum }_{i=1}^{\mathrm{<font\; color="red">d</font>}eg(\mathrm{<font\; color="red">d</font>})}{c}_i{x}^i$ of d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630155X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1630155X&_rdoc=1&_issn=0022314X&md5=78e0849b4c24086734d9dd67e4e5c7b1" title="Click to view the MathML source">xⁿ&minus;1dden">de">$x^n\&<font\; color="red">minus</font>;1$ such that d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630155X&_mathId=si4.gif&_user=111111111&_pii=S0022314X1630155X&_rdoc=1&_issn=0022314X&md5=0c8885b446cdef8f3f3eb03c5733754e" title="Click to view the MathML source">c_i=n_idden">de">$c_i=n_i$ for d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630155X&_mathId=si5.gif&_user=111111111&_pii=S0022314X1630155X&_rdoc=1&_issn=0022314X&md5=2fed658ee1a0a48ba96833b5114bd359" title="Click to view the MathML source">1≤i≤rdden">de">$1\le i\le r$. In this paper we show that the set S has a natural density. Furthermore, we find the value of the natural density of S.