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\n We prove that, for\n \n \n \n \n T<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n 000<\/mml:mn>\n <\/mml:mrow>\n T\\ge 2\\,000<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n T<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n Q<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 10<\/mml:mn>\n <\/mml:mrow>\n T\\ge Q\\ge 10<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n \n \u03c3\n \n <\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 0.52<\/mml:mn>\n <\/mml:mrow>\n \\sigma \\ge 0.52<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we have\n \n \n \n \n \n \n \u2211\n \n <\/mml:mo>\n \n q<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n \n \n q<\/mml:mi>\n \n \n \u03c6\n \n <\/mml:mi>\n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n \n \u2211\n \n <\/mml:mo>\n \n \n \u03c7\n \n <\/mml:mi>\n \n mod<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n \n N<\/mml:mi>\n (<\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n ,<\/mml:mo>\n T<\/mml:mi>\n ,<\/mml:mo>\n \n \u03c7\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \n \u2264\n \n <\/mml:mo>\n \n 20<\/mml:mn>\n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n 56<\/mml:mn>\n \n \n Q<\/mml:mi>\n \n 5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n T<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n \n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n log<\/mml:mi>\n \n 5<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n T<\/mml:mi>\n )<\/mml:mo>\n \n +<\/mml:mo>\n \n 32<\/mml:mn>\n \n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n log<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n Q<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n T<\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} \\sum _{q\\le Q}\\mkern -2mu \\frac {q}{\\varphi (q)}\\mkern -6mu \\sum _{\\chi \\operatorname {mod}^* q}\\mkern -12mu N(\\sigma ,T,\\chi ) \\!\\le \\! 20\\bigl (56\\,Q^{5}T^3\\bigr )^{1-\\sigma }\\log ^{5-2\\sigma }(Q^2T) \\!+\\!32\\,Q^2\\log ^2(Q^2T), \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n where\n \n \n \n \n \n \u03c7\n \n <\/mml:mi>\n \n mod<\/mml:mi>\n \n \u2217\n \n <\/mml:mo>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n q<\/mml:mi>\n <\/mml:mrow>\n \\chi \\operatorname {mod}^* q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes a sum over all primitive Dirichlet characters\n \n \n \n \n \u03c7\n \n <\/mml:mi>\n \\chi<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n to the modulus\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Furthermore, we have\n \n \n \n \n N<\/mml:mi>\n (<\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n ,<\/mml:mo>\n T<\/mml:mi>\n ,<\/mml:mo>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n )<\/mml:mo>\n \n \u2264\n \n <\/mml:mo>\n 2<\/mml:mn>\n T<\/mml:mi>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \n \n (<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n 1<\/mml:mn>\n +<\/mml:mo>\n \n 9.8<\/mml:mn>\n \n 2<\/mml:mn>\n T<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n (<\/mml:mo>\n 3<\/mml:mn>\n T<\/mml:mi>\n \n )<\/mml:mo>\n \n 8<\/mml:mn>\n (<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n \u03c3\n \n <\/mml:mi>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msup>\n \n log<\/mml:mi>\n \n 5<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n \u03c3\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n T<\/mml:mi>\n )<\/mml:mo>\n \n \n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n +<\/mml:mo>\n \n 103<\/mml:mn>\n (<\/mml:mo>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n T<\/mml:mi>\n \n )<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:msup>\n .<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} N(\\sigma ,T,\\mathbb {1} )\\le 2T \\log \\biggl (1+\\frac {9.8}{2T}(3T)^{8(1-\\sigma )\/{3}}\\log ^{5-2\\sigma }(T)\\biggr ) \\!+\\!103(\\log T)^2. \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n <\/p>","DOI":"10.1090\/mcom\/2991","type":"journal-article","created":{"date-parts":[[2015,6,3]],"date-time":"2015-06-03T12:45:00Z","timestamp":1433335500000},"page":"325-356","source":"Crossref","is-referenced-by-count":25,"title":["An explicit density estimate for Dirichlet \ud835\udc3f-series"],"prefix":"10.1090","volume":"85","author":[{"given":"O.","family":"Ramar\u00e9","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,6,3]]},"reference":[{"key":"1","series-title":"National Bureau of Standards Applied Mathematics Series, No. 55","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","author":"Abramowitz, Milton","year":"1964"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1142\/S021819670800455X","article-title":"Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices","volume":"18","author":"Bacher, Roland","year":"2008","journal-title":"Internat. 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