Looking for infinite series resembling an exponential












3












$begingroup$


I'm looking for some $f(x)$ that has the following property:



$sum_{x=1}^infty f(kx) = r^k$



for some real $0 < r < 1$, and at least for strictly positive integer $k$.



Does such an $f(x)$ exist?



This could also be thought of in terms of some sequence of real numbers $f[n]$.



I posted this at MSE but got no answer, so thought I would try MO.






real-analysis sequences-and-series power-series







share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    I'm looking for some $f(x)$ that has the following property:



    $sum_{x=1}^infty f(kx) = r^k$



    for some real $0 < r < 1$, and at least for strictly positive integer $k$.



    Does such an $f(x)$ exist?



    This could also be thought of in terms of some sequence of real numbers $f[n]$.



    I posted this at MSE but got no answer, so thought I would try MO.






    real-analysis sequences-and-series power-series







    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      I'm looking for some $f(x)$ that has the following property:



      $sum_{x=1}^infty f(kx) = r^k$



      for some real $0 < r < 1$, and at least for strictly positive integer $k$.



      Does such an $f(x)$ exist?



      This could also be thought of in terms of some sequence of real numbers $f[n]$.



      I posted this at MSE but got no answer, so thought I would try MO.






      real-analysis sequences-and-series power-series







      share|cite|improve this question









      $endgroup$




      I'm looking for some $f(x)$ that has the following property:



      $sum_{x=1}^infty f(kx) = r^k$



      for some real $0 < r < 1$, and at least for strictly positive integer $k$.



      Does such an $f(x)$ exist?



      This could also be thought of in terms of some sequence of real numbers $f[n]$.



      I posted this at MSE but got no answer, so thought I would try MO.






      real-analysis sequences-and-series power-series




      real-analysis sequences-and-series power-series






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 3 hours ago









      Mike BattagliaMike Battaglia

      1,036523




      1,036523






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          You can solve this system of equations explicitly in terms of the function
          $$F(z)=sum_{m=1}^{infty} mu(m)z^m=z-z^2-z^3-z^5+z^6+cdots$$
          which can be shown to be transcendental (see the paper "Transcendence of Power Series for Some Number Theoretic Functions", by Coons and Borwein). With this notation your sequence can be succinctly expressed as
          $$f(n)=F(r^n).$$
          Indeed, one can check that
          $$sum_{n=1}^{infty} f(kn)=sum_{n=1}^{infty} F(r^{kn})=sum_{m,n=1}^{infty} mu(m)r^{nmk}=sum_{m=1}^{infty}sum_{d|m}mu(d)r^{mk}=r^k$$
          as desired.






          share|cite|improve this answer









          $endgroup$













            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: "504"
            };
            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',
            autoActivateHeartbeat: false,
            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
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f321432%2flooking-for-infinite-series-resembling-an-exponential%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            4












            $begingroup$

            You can solve this system of equations explicitly in terms of the function
            $$F(z)=sum_{m=1}^{infty} mu(m)z^m=z-z^2-z^3-z^5+z^6+cdots$$
            which can be shown to be transcendental (see the paper "Transcendence of Power Series for Some Number Theoretic Functions", by Coons and Borwein). With this notation your sequence can be succinctly expressed as
            $$f(n)=F(r^n).$$
            Indeed, one can check that
            $$sum_{n=1}^{infty} f(kn)=sum_{n=1}^{infty} F(r^{kn})=sum_{m,n=1}^{infty} mu(m)r^{nmk}=sum_{m=1}^{infty}sum_{d|m}mu(d)r^{mk}=r^k$$
            as desired.






            share|cite|improve this answer









            $endgroup$


















              4












              $begingroup$

              You can solve this system of equations explicitly in terms of the function
              $$F(z)=sum_{m=1}^{infty} mu(m)z^m=z-z^2-z^3-z^5+z^6+cdots$$
              which can be shown to be transcendental (see the paper "Transcendence of Power Series for Some Number Theoretic Functions", by Coons and Borwein). With this notation your sequence can be succinctly expressed as
              $$f(n)=F(r^n).$$
              Indeed, one can check that
              $$sum_{n=1}^{infty} f(kn)=sum_{n=1}^{infty} F(r^{kn})=sum_{m,n=1}^{infty} mu(m)r^{nmk}=sum_{m=1}^{infty}sum_{d|m}mu(d)r^{mk}=r^k$$
              as desired.






              share|cite|improve this answer









              $endgroup$
















                4












                4








                4





                $begingroup$

                You can solve this system of equations explicitly in terms of the function
                $$F(z)=sum_{m=1}^{infty} mu(m)z^m=z-z^2-z^3-z^5+z^6+cdots$$
                which can be shown to be transcendental (see the paper "Transcendence of Power Series for Some Number Theoretic Functions", by Coons and Borwein). With this notation your sequence can be succinctly expressed as
                $$f(n)=F(r^n).$$
                Indeed, one can check that
                $$sum_{n=1}^{infty} f(kn)=sum_{n=1}^{infty} F(r^{kn})=sum_{m,n=1}^{infty} mu(m)r^{nmk}=sum_{m=1}^{infty}sum_{d|m}mu(d)r^{mk}=r^k$$
                as desired.






                share|cite|improve this answer









                $endgroup$



                You can solve this system of equations explicitly in terms of the function
                $$F(z)=sum_{m=1}^{infty} mu(m)z^m=z-z^2-z^3-z^5+z^6+cdots$$
                which can be shown to be transcendental (see the paper "Transcendence of Power Series for Some Number Theoretic Functions", by Coons and Borwein). With this notation your sequence can be succinctly expressed as
                $$f(n)=F(r^n).$$
                Indeed, one can check that
                $$sum_{n=1}^{infty} f(kn)=sum_{n=1}^{infty} F(r^{kn})=sum_{m,n=1}^{infty} mu(m)r^{nmk}=sum_{m=1}^{infty}sum_{d|m}mu(d)r^{mk}=r^k$$
                as desired.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Gjergji ZaimiGjergji Zaimi

                62.8k4163310




                62.8k4163310






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f321432%2flooking-for-infinite-series-resembling-an-exponential%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    Knooppunt Holsloot

                    Image
                    Knooppunt Holsloot Kenmerken Wegen A37 × N34 Type knooppunt klaverblad Portaal     Verkeer & Vervoer Knooppunt Holsloot is een verkeersknooppunt op de kruising van de autosnelweg A37 en de N34 in Drenthe, tussen Emmen en Coevorden. Oorspronkelijk was deze aansluiting een afritconstructie van het type Haarlemmermeeraansluiting, zoals bij Knooppunt Hooipolder. Sinds 2002 is het knooppunt omgebouwd tot een volledig klaverblad. Als noviteit werd hier voor het eerst gebruikgemaakt van dimbare verlichting in plaats van de normale vaste tijden waarop de verlichting aan en uit gaat, wat een grote kostenbesparing opleverde. [1] Dat Holsloot als volwaardig knooppunt is uitgevoerd, waarbij alle verkeersstromen en afbuigingen ononderbroken kunnen doorgaan, is uitzonderlijk. Het betreft hier namelijk een aansluiting tussen een autosnelweg en niet-snelweg. Een vergelijkbare situatie treft men nabij Hoensbroek bij het knooppunt Ten Esschen.     Aa
                    Read more

                    Altaar (religie)

                    Image
                    Deel van de serie over kerkelijk gerei waaronder paramenten en liturgisch vaatwerk gebruikt in de liturgie Liturgisch vaatwerk Vasa sacra Miskelk · Pateen Kelklepeltje Ciborie · Monstrans Pyxis · Custodiale Vasa non sacra Ampullen · Wijwatervat Olievaatje Wierookvat · Ablutievat Paramenten Amict · Albe · Baarkleed · Cingel Tuniek · Dalmatiek · Fanon Kazuifel · Manipel · Stola Gremiale · Benedictievelum Mijter Koorkledij Rochet · Superplie Koorkap · Cappa magna Kovel Kelkgerei Bursa · Kelkvelum Ciborievelum Kerklinnen Corporale · Kelkdoekje Lavabodoekje · Palla Altaardwaal Kerkinterieur Altaar · Ambo Biechtstoel · Communiebank Doksaal · Doopvont Faldistorium Godslamp · Hoogaltaar Heilig Kruisaltaar Katheder · Preekstoel Sedilia Tabernakel · Volksaltaar Liturgische boeken Altaarmissaal Benedictionale · Brevier Evangeliarium · Evangelistarium Graduale · Kyriale Lectionarium Psalter · Rituaal Sacramentarium · Volksmissaal Overige Flambou
                    Read more

                    Gregoriusmis

                    Image
                    Een afbeelding van de Gregoriusmis uit een getijdenboek gemaakt in de zuidelijke Nederlanden tussen 1500 en 1525 Gregoriusmis, Meester van het Akense Altaar, ca. 1500-1525. Collectie Museum Catharijneconvent De Gregoriusmis of de Mis van Sint Gregorius is een motief in katholieke kunst, dat het eerst verscheen in de late Middeleeuwen en dat nog steeds werd gebruikt tijdens de contrareformatie. Paus Gregorius I (ca. 540-604) is afgebeeld, de H. Mis opdragend, terwijl Christus als de Man van Smarten verschijnt op het altaar dat voor Sint Gregorius staat. Volgens de legende had St. Gregorius gebeden om een teken voor een betwijfelaar van de transsubstantiatieleer, waarop de lijdende Christus tijdens het opdragen van de Mis verscheen om hem te overtuigen en waarbij zijn bloed in de altaarkelk liep. Vaak zijn ook de passiewerktuigen afgebeeld. Zie de categorie Mass of Saint Gregory van Wikimedia Commons voor mediabestanden over dit onderwerp. This page is onl
                    Read more