Restricted linear congruences

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• 作者：Khodakhast Bibak^a ; ^{kbibak@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Bruce M. Kapron^a ; ^{bmkapron@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Venkatesh Srinivasan^a ; ^b ; ^{srinivas@uvic.ca" class="auth_mail" title="E-mail the corresponding author} ; Roberto Tauraso^c ; ^{tauraso@mat.uniroma2.it" class="auth_mail" title="E-mail the corresponding author} ; Lá ; szló ; Tó ; th^d ; ^{ltoth@gamma.ttk.pte.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11D79 ; 11P83 ; 11L03 ; 11A25 ; 42A16
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：128-144
• 全文大小：414 K

文摘

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence mmlsi1" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=3d7accee7ffdc07316f2a1f95c6fc05d">mg class="imgLazyJSB inlineImage" height="16" width="217" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302013-si1.gif">mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi>ami>mrow><mrow><mn>1mn>mrow>msub><msub><mrow><mi>xmi>mrow><mrow><mn>1mn>mrow>msub><mo>+mo><mo>⋯mo><mo>+mo><msub><mrow><mi>ami>mrow><mrow><mi>kmi>mrow>msub><msub><mrow><mi>xmi>mrow><mrow><mi>kmi>mrow>msub><mo>≡mo><mi>bmi><mspace width="0.25em">mspace><mo stretchy="false">(mo><mrow><mi mathvariant="normal">modmi>mrow><mspace width="0.25em">mspace><mi>nmi><mo stretchy="false">)mo>math>, with mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=a88d9f985ecd3e3a9d99522d3dbd70e0" title="Click to view the MathML source">gcd⁡(x_i,n)=t_imathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="normal">gcdmi><mo>⁡mo><mo stretchy="false">(mo><msub><mrow><mi>xmi>mrow><mrow><mi>imi>mrow>msub><mo>,mo><mi>nmi><mo stretchy="false">)mo><mo>=mo><msub><mrow><mi>tmi>mrow><mrow><mi>imi>mrow>msub>math> (mmlsi199" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si199.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=336bb509162c0d3a9ac48daf933e13be" title="Click to view the MathML source">1≤i≤kmathContainer hidden">mathCode"><math altimg="si199.gif" overflow="scroll"><mn>1mn><mo>≤mo><mi>imi><mo>≤mo><mi>kmi>math>), where mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si4.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=e1b405fde4fda511ea9382deb41a36c9" title="Click to view the MathML source">a₁,t₁,…,a_k,t_k,b,nmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi>ami>mrow><mrow><mn>1mn>mrow>msub><mo>,mo><msub><mrow><mi>tmi>mrow><mrow><mn>1mn>mrow>msub><mo>,mo><mo>…mo><mo>,mo><msub><mrow><mi>ami>mrow><mrow><mi>kmi>mrow>msub><mo>,mo><msub><mrow><mi>tmi>mrow><mrow><mi>kmi>mrow>msub><mo>,mo><mi>bmi><mo>,mo><mi>nmi>math> (mmlsi16" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si16.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=aa6305b31f5c4009b2fd05ef6a0ac9ec" title="Click to view the MathML source">n≥1mathContainer hidden">mathCode"><math altimg="si16.gif" overflow="scroll"><mi>nmi><mo>≥mo><mn>1mn>math>) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of this kind of congruence was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of mmlsi6" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si6.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=c1809322cbfb0fdbccd5baec4e622656" title="Click to view the MathML source">a_i=t_i=1mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><msub><mrow><mi>ami>mrow><mrow><mi>imi>mrow>msub><mo>=mo><msub><mrow><mi>tmi>mrow><mrow><mi>imi>mrow>msub><mo>=mo><mn>1mn>math>mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si7.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=0547e290b77712403c266d9d64f2a3ac" title="Click to view the MathML source">(1≤i≤k)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mo stretchy="false">(mo><mn>1mn><mo>≤mo><mi>imi><mo>≤mo><mi>kmi><mo stretchy="false">)mo>math>. Since then, this problem has been studied, in several other special cases, in many papers; in particular, Jacobson and Williams [m>Duke Math. J.  m> 39 (1972) 521–527] gave a nice explicit formula for the number of such solutions when mmlsi8" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si8.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=fbb1cce0eb40e3f4b7a65d967de9023f" title="Click to view the MathML source">(a₁,…,a_k)=t_i=1mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll"><mo stretchy="false">(mo><msub><mrow><mi>ami>mrow><mrow><mn>1mn>mrow>msub><mo>,mo><mo>…mo><mo>,mo><msub><mrow><mi>ami>mrow><mrow><mi>kmi>mrow>msub><mo stretchy="false">)mo><mo>=mo><msub><mrow><mi>tmi>mrow><mrow><mi>imi>mrow>msub><mo>=mo><mn>1mn>math>mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302013&_mathId=si7.gif&_user=111111111&_pii=S0022314X16302013&_rdoc=1&_issn=0022314X&md5=0547e290b77712403c266d9d64f2a3ac" title="Click to view the MathML source">(1≤i≤k)mathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mo stretchy="false">(mo><mn>1mn><mo>≤mo><mi>imi><mo>≤mo><mi>kmi><mo stretchy="false">)mo>math>. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.