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\n Let\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a cofinite Fuchsian group acting on hyperbolic two-space\n \n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {H}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n M<\/mml:mi>\n =<\/mml:mo>\n \n \u0393\n \n <\/mml:mi>\n \n \u2216\n \n <\/mml:mo>\n \n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n M=\\Gamma \\setminus \\mathbb {H}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the corresponding quotient space. For\n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , a closed geodesic of\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , let\n \n \n \n \n l<\/mml:mi>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n l(\\gamma )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denote its length. The prime geodesic counting function\n \n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi _{M}(u)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is defined as the number of\n \n \n \n \n \u0393\n \n <\/mml:mi>\n \\Gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -inconjugate, primitive, closed geodesics\n \n \n \n \n \u03b3\n \n <\/mml:mi>\n \\gamma<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n \n e<\/mml:mi>\n \n l<\/mml:mi>\n (<\/mml:mo>\n \n \u03b3\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2264\n \n <\/mml:mo>\n u<\/mml:mi>\n .<\/mml:mo>\n <\/mml:mrow>\n e^{l(\\gamma )} \\leq u.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n The\n prime geodesic theorem<\/italic>\n states that:\n \n \\[\n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n M<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n 0<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munder>\n li<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n \n s<\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n +<\/mml:mo>\n \n O<\/mml:mi>\n M<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n \n u<\/mml:mi>\n \n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\pi _M(u) = \\sum _{0 \\leq \\lambda _{M,j} \\leq 1\/4} \\operatorname {li}(u^{s_{M,j}}) + O_M \\left (\\frac {u^{3\/4}}{\\log u}\\right ),<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n where\n \n \n \n \n 0<\/mml:mn>\n =<\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ><\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ><\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n <\/mml:mrow>\n 0=\\lambda _{M,0} > \\lambda _{M,1} > \\cdots<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n s<\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n +<\/mml:mo>\n \n \n 1<\/mml:mn>\n 4<\/mml:mn>\n <\/mml:mfrac>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:msqrt>\n <\/mml:mrow>\n s_{M,j} = \\frac {1}{2}+\\sqrt {\\frac {1}{4} - \\lambda _{M,j} }<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n C<\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n C_{M}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the smallest implied constant so that\n \n \\[\n \n \n \n \n |<\/mml:mo>\n \n \n \u03c0\n \n <\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n u<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n \n \u2211\n \n <\/mml:mo>\n \n 0<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:munder>\n li<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n \n s<\/mml:mi>\n \n M<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n \n C<\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n u<\/mml:mi>\n \n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n u<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n for all\u00a0<\/mml:mtext>\n \n u<\/mml:mi>\n ><\/mml:mo>\n 1.<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:mrow>\n \\left |\\pi _{M}(u)-\\sum _{0 \\leq \\lambda _{M,j} \\leq 1\/4} \\operatorname {li}(u^{s_{M,j}})\\right | \\leq C_{M}\\frac {u^{3\/4}}{\\log {u}} \\quad \\text {for all $u > 1.$}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n We call the (absolute) constant\n \n \n \n \n C<\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n C_{M}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the Huber constant.\n <\/p>\n

\n The objective of this paper is to give an effectively computable upper bound of\n \n \n \n \n C<\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n C_{M}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for\n \n \n \n \n P<\/mml:mi>\n S<\/mml:mi>\n L<\/mml:mi>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n PSL(2,\\mathbb {Z})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , showing that\n \n \n \n \n \n C<\/mml:mi>\n \n M<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \u2264\n \n <\/mml:mo>\n 16,607,349,020,658<\/mml:mn>\n \n \u2248\n \n <\/mml:mo>\n exp<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n 30.44086643<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n C_{M} \\leq 16{,}607{,}349{,}020{,}658 \\approx \\exp (30.44086643)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-2010-02430-5","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:28Z","timestamp":1293633568000},"page":"1163-1196","source":"Crossref","is-referenced-by-count":6,"title":["An effective bound for the Huber constant for cofinite Fuchsian groups"],"prefix":"10.1090","volume":"80","author":[{"given":"J.","family":"Friedman","sequence":"first","affiliation":[]},{"given":"J.","family":"Jorgenson","sequence":"additional","affiliation":[]},{"given":"J.","family":"Kramer","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,10,28]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1112\/S1461157000001467","article-title":"A polynomial with Galois group \ud835\udc46\ud835\udc3f\u2082(\ud835\udd3d\u2081\u2086)","volume":"10","author":"Bosman, Johan","year":"2007","journal-title":"LMS J. 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