An asymptotic formula for this sum

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Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.



$$sum_n leq X, ; n, equiv, a modb logn,$$
where $a$ and $b$ are two coprime integers.
Thanks in advance.




nt.number-theory analytic-number-theory




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    up vote
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    Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.



    $$sum_n leq X, ; n, equiv, a modb logn,$$
    where $a$ and $b$ are two coprime integers.
    Thanks in advance.




    nt.number-theory analytic-number-theory




    share|cite|improve this question























      up vote
      5
      down vote

      favorite









      up vote
      5
      down vote

      favorite











      Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.



      $$sum_n leq X, ; n, equiv, a modb logn,$$
      where $a$ and $b$ are two coprime integers.
      Thanks in advance.




      nt.number-theory analytic-number-theory




      share|cite|improve this question













      Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.



      $$sum_n leq X, ; n, equiv, a modb logn,$$
      where $a$ and $b$ are two coprime integers.
      Thanks in advance.





      nt.number-theory analytic-number-theory





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 6 at 8:37









      Carlo Beenakker

      68.1k8154256




      68.1k8154256









      asked Aug 6 at 8:32









      Khadija Mbarki

      672312




      672312




















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          The sum
          $$F(X)=sum_n leq X, ; n, equiv, a modb logn=sum_p=rm Int,[-a/b]^rm Int,[(x-a)/b]log(a+pb)$$
          can be approximated in the large-$X$ limit by
          $$F_infty(X)=sum_p=1^(X-a)/blog(pb)=fracX-ablog b+logGammaleft(fracX-ab+1right)$$



          Here is a plot of $F(X)$ (gold) and $F_infty(X)$ (blue) for $a=5$, $b=11$.








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            1 Answer
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            1 Answer
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            up vote
            9
            down vote



            accepted










            The sum
            $$F(X)=sum_n leq X, ; n, equiv, a modb logn=sum_p=rm Int,[-a/b]^rm Int,[(x-a)/b]log(a+pb)$$
            can be approximated in the large-$X$ limit by
            $$F_infty(X)=sum_p=1^(X-a)/blog(pb)=fracX-ablog b+logGammaleft(fracX-ab+1right)$$



            Here is a plot of $F(X)$ (gold) and $F_infty(X)$ (blue) for $a=5$, $b=11$.








            share|cite|improve this answer



























              up vote
              9
              down vote



              accepted










              The sum
              $$F(X)=sum_n leq X, ; n, equiv, a modb logn=sum_p=rm Int,[-a/b]^rm Int,[(x-a)/b]log(a+pb)$$
              can be approximated in the large-$X$ limit by
              $$F_infty(X)=sum_p=1^(X-a)/blog(pb)=fracX-ablog b+logGammaleft(fracX-ab+1right)$$



              Here is a plot of $F(X)$ (gold) and $F_infty(X)$ (blue) for $a=5$, $b=11$.








              share|cite|improve this answer

























                up vote
                9
                down vote



                accepted







                up vote
                9
                down vote



                accepted






                The sum
                $$F(X)=sum_n leq X, ; n, equiv, a modb logn=sum_p=rm Int,[-a/b]^rm Int,[(x-a)/b]log(a+pb)$$
                can be approximated in the large-$X$ limit by
                $$F_infty(X)=sum_p=1^(X-a)/blog(pb)=fracX-ablog b+logGammaleft(fracX-ab+1right)$$



                Here is a plot of $F(X)$ (gold) and $F_infty(X)$ (blue) for $a=5$, $b=11$.








                share|cite|improve this answer















                The sum
                $$F(X)=sum_n leq X, ; n, equiv, a modb logn=sum_p=rm Int,[-a/b]^rm Int,[(x-a)/b]log(a+pb)$$
                can be approximated in the large-$X$ limit by
                $$F_infty(X)=sum_p=1^(X-a)/blog(pb)=fracX-ablog b+logGammaleft(fracX-ab+1right)$$



                Here is a plot of $F(X)$ (gold) and $F_infty(X)$ (blue) for $a=5$, $b=11$.









                share|cite|improve this answer















                share|cite|improve this answer



                share|cite|improve this answer








                edited Aug 6 at 8:55


























                answered Aug 6 at 8:48









                Carlo Beenakker

                68.1k8154256




                68.1k8154256






















                     

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