ALBERT

Description: 〈p〉Publication date: Available online 8 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Number Theory〈/p〉 〈p〉Author(s): Jie Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Denote by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="double-struck"〉P〈/mi〉〈/math〉 the set of all primes and by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 the largest prime factor of integer 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉n〈/mi〉〈mo〉⩾〈/mo〉〈mn〉1〈/mn〉〈/math〉 with the convention 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msup〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉. For each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.svg"〉〈mi〉η〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉, let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉c〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉η〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1〈/mn〉〈/math〉 be some constant depending on 〈em〉η〈/em〉 and〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉c〈/mi〉〈mo〉,〈/mo〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉p〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉P〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉:〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈mi〉p〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉+〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉q〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mi〉a〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="0.25em"〉〈/mspace〉〈mrow〉〈mtext〉for some prime 〈/mtext〉〈mi〉q〈/mi〉〈mtext〉 with〈/mtext〉〈/mrow〉〈mspace linebreak="newline"〉〈/mspace〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈/msup〉〈mo linebreak="badbreak" linebreakstyle="after"〉〈〈/mo〉〈mi〉q〈/mi〉〈mo〉⩽〈/mo〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉η〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈msup〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉}〈/mo〉〈mo〉.〈/mo〉〈/math〉〈/span〉 In this paper, under the Elliott-Halberstam conjecture we prove, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si59.svg"〉〈mi〉y〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mo〉∞〈/mo〉〈/math〉,〈span〉〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉c〈/mi〉〈mo〉,〈/mo〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉:〈/mo〉〈mo linebreak="badbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉]〈/mo〉〈mo〉∩〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉c〈/mi〉〈mo〉,〈/mo〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉|〈/mo〉〈mo〉∼〈/mo〉〈mi〉π〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mspace width="1em"〉〈/mspace〉〈mtext〉or〈/mtext〉〈mspace width="1em"〉〈/mspace〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉c〈/mi〉〈mo〉,〈/mo〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈msub〉〈mrow〉〈mo〉≫〈/mo〉〈/mrow〉〈mrow〉〈mi〉a〈/mi〉〈mo〉,〈/mo〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mi〉π〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉〈/span〉 according to values of 〈em〉η〈/em〉. These are complement for some results of Banks-Shparlinski [1], of Wu [12] and of Chen-Wu [2].〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 2 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Number Theory〈/p〉 〈p〉Author(s): Jean Lelis, Diego Marques〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study a question proposed by Marques and Moreira about transcendental entire functions mapping 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="double-struck"〉Q〈/mi〉〈/math〉 into itself. In particular, we prove that there is no a transcendental entire function 〈em〉f〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.svg"〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉Q〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⊆〈/mo〉〈mi mathvariant="double-struck"〉Q〈/mi〉〈/math〉 and the denominator of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉f〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉p〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉q〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉O〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉q〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, for all rational number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si24.svg"〉〈mi〉p〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉q〈/mi〉〈/math〉, with 〈em〉q〈/em〉 sufficiently large.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Number Theory〈/p〉 〈p〉Author(s): Brandon Williams〈/p〉

Description: 〈p〉Publication date: Available online 27 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Number Theory〈/p〉 〈p〉Author(s): John W. Jones, David P. Roberts〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We determine all degree 7 number fields where the absolute value of the discriminant is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mo〉⋅〈/mo〉〈msup〉〈mrow〉〈mn〉10〈/mn〉〈/mrow〉〈mrow〉〈mn〉8〈/mn〉〈/mrow〉〈/msup〉〈/math〉. To accomplish this large-scale computation, we distributed the computation over many servers on the internet through the BOINC network.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 7 December 2018〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Number Theory〈/p〉 〈p〉Author(s): F. Pellarin, R. Perkins〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Twisted 〈em〉A〈/em〉-harmonic sums are partial sums of a class of zeta values introduced by the first author. We prove some new identities for such sums and we deduce properties of analogues of finite zeta values in the framework of the Carlitz module. In the theory of finite multiple zeta values as introduced by Kaneko and Zagier, finite zeta values are all zero and there is no known non-zero finite multiple zeta value. In the Carlitzian setting the phenomenology is different as we can deduce, from our results, the irrationality of certain finite zeta values.〈/p〉〈/div〉