Lower bound for sum of Hecke eigenvalues

Multi tool use
Multi tool use











up vote
1
down vote

favorite












Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.



How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?



Here is the background:
The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.



In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.



My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...










share|cite|improve this question









New contributor




Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
























    up vote
    1
    down vote

    favorite












    Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.



    How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?



    Here is the background:
    The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.



    In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.



    My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...










    share|cite|improve this question









    New contributor




    Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.



      How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?



      Here is the background:
      The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.



      In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.



      My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...










      share|cite|improve this question









      New contributor




      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      Let $lambda$ be weakly multiplicative, $lambda(n)geq0$, $p$ prime and $S(x)=sum_{nleq x}lambda(n)log(frac{x}{n})$ for real $x$.



      How can I show $S(x)gg left(sum_{pleq sqrt{x/3}}lambda(p)right)^2-left(sum_{pleq sqrt{x/3}}1right)$?



      Here is the background:
      The question is coming from IKS, section 3. $lambda(n)$ are the eigenvalues of a newform $f$ of level $N$. All above sums are chosen such that $(n,N)=1$ or $pnotmid N$, thus giving multiplicativity.



      In Xu, section 3.1 something similar happens. Here the hints $lambda(p)geq1$ and $|lambda(n)|leqsigma_0(n)$ (divisor function) are given.



      My approach so far starts estimating $log(x/n)geq1$ but I could not figure out the strange bound of $sqrt{x/3}$...







      number-theory eigenvalues-eigenvectors estimation modular-forms upper-lower-bounds






      share|cite|improve this question









      New contributor




      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited yesterday





















      New contributor




      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked yesterday









      Nodt Greenish

      266




      266




      New contributor




      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Nodt Greenish is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.



























          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });






          Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.










           

          draft saved


          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2995298%2flower-bound-for-sum-of-hecke-eigenvalues%23new-answer', 'question_page');
          }
          );

          Post as a guest





































          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.










           

          draft saved


          draft discarded


















          Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.













          Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.












          Nodt Greenish is a new contributor. Be nice, and check out our Code of Conduct.















           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2995298%2flower-bound-for-sum-of-hecke-eigenvalues%23new-answer', 'question_page');
          }
          );

          Post as a guest




















































































          UYdKCUbIBU,1vWKtv0mRyQonuQ84IZF 8coMNciKR7cXu,54EHMAgRtKCKuYLtW
          knhR3 JjGdF1KmLwy,d1VFjplQpoiYtAfMe9YJ5pc4G XlR1iX5g6RWrh,E7ma7,Fiffhg 4o0bw,X8cjPO

          Popular posts from this blog

          mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

          How to change which sound is reproduced for terminal bell?

          Can I use Tabulator js library in my java Spring + Thymeleaf project?