Truncated sum of divisors bound












2












$begingroup$


I am interested in an upper bound for
$$
sum_{substack{d|N\ d>A}}frac{1}{d^3},$$

in particular, I can get the above to be
$$sum_{substack{d|N\ d>A}}frac{1}{d^3}ll frac{text{exp}left(C frac{log(N)}{loglog(N)}right)}{A^3}$$
for some positive constant $C$. However I would like to do better. I think that the upper bound should be around
$$ frac{log^C(N)}{A^3}$$
for some other positive constant $C.$



We also have that $A<N^{1/2}.$
References are welcome.










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    I am interested in an upper bound for
    $$
    sum_{substack{d|N\ d>A}}frac{1}{d^3},$$

    in particular, I can get the above to be
    $$sum_{substack{d|N\ d>A}}frac{1}{d^3}ll frac{text{exp}left(C frac{log(N)}{loglog(N)}right)}{A^3}$$
    for some positive constant $C$. However I would like to do better. I think that the upper bound should be around
    $$ frac{log^C(N)}{A^3}$$
    for some other positive constant $C.$



    We also have that $A<N^{1/2}.$
    References are welcome.










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I am interested in an upper bound for
      $$
      sum_{substack{d|N\ d>A}}frac{1}{d^3},$$

      in particular, I can get the above to be
      $$sum_{substack{d|N\ d>A}}frac{1}{d^3}ll frac{text{exp}left(C frac{log(N)}{loglog(N)}right)}{A^3}$$
      for some positive constant $C$. However I would like to do better. I think that the upper bound should be around
      $$ frac{log^C(N)}{A^3}$$
      for some other positive constant $C.$



      We also have that $A<N^{1/2}.$
      References are welcome.










      share|cite|improve this question











      $endgroup$




      I am interested in an upper bound for
      $$
      sum_{substack{d|N\ d>A}}frac{1}{d^3},$$

      in particular, I can get the above to be
      $$sum_{substack{d|N\ d>A}}frac{1}{d^3}ll frac{text{exp}left(C frac{log(N)}{loglog(N)}right)}{A^3}$$
      for some positive constant $C$. However I would like to do better. I think that the upper bound should be around
      $$ frac{log^C(N)}{A^3}$$
      for some other positive constant $C.$



      We also have that $A<N^{1/2}.$
      References are welcome.







      number-theory analytic-number-theory divisor-sum divisor-counting-function






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 10 at 23:53







      user164144

















      asked Jan 10 at 23:41









      user164144user164144

      515




      515






















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