| Summary: | In this paper, a joint approximation of analytic functions by shifts of Dirichlet <i>L</i>-functions <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>+</mo> <mi>i</mi> <msub> <mi>a</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mi>τ</mi> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>L</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>+</mo> <mi>i</mi> <msub> <mi>a</mi> <mi>r</mi> </msub> <msub> <mi>t</mi> <mi>τ</mi> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>r</mi> </msub> </mrow> </semantics> </math> </inline-formula> are non-zero real algebraic numbers linearly independent over the field <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">Q</mi> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>t</mi> <mi>τ</mi> </msub> </semantics> </math> </inline-formula> is the Gram function, is considered. It is proved that the set of their shifts has a positive lower density.
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