What does this $asymp$ symbol mean? (subject: analytic number theory)











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I'm reading a survey article by Andrew Granville on analytic number theory.



On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below.



Strange Symbol in analytic number theory



Since it's not defined in the paper, I'm assuming it must be standard notation.



From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.



Can anyone identify the symbol, even better explain what it means and/or provide a reference? Is there a name to speak the symbol?



Thanks in advance.






analysis notation asymptotics analytic-number-theory







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    up vote
    7
    down vote

    favorite
    2












    I'm reading a survey article by Andrew Granville on analytic number theory.



    On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below.



    Strange Symbol in analytic number theory



    Since it's not defined in the paper, I'm assuming it must be standard notation.



    From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.



    Can anyone identify the symbol, even better explain what it means and/or provide a reference? Is there a name to speak the symbol?



    Thanks in advance.






    analysis notation asymptotics analytic-number-theory







    share|cite|improve this question


























      up vote
      7
      down vote

      favorite
      2









      up vote
      7
      down vote

      favorite
      2






      2





      I'm reading a survey article by Andrew Granville on analytic number theory.



      On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below.



      Strange Symbol in analytic number theory



      Since it's not defined in the paper, I'm assuming it must be standard notation.



      From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.



      Can anyone identify the symbol, even better explain what it means and/or provide a reference? Is there a name to speak the symbol?



      Thanks in advance.






      analysis notation asymptotics analytic-number-theory







      share|cite|improve this question















      I'm reading a survey article by Andrew Granville on analytic number theory.



      On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot below.



      Strange Symbol in analytic number theory



      Since it's not defined in the paper, I'm assuming it must be standard notation.



      From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.



      Can anyone identify the symbol, even better explain what it means and/or provide a reference? Is there a name to speak the symbol?



      Thanks in advance.






      analysis notation asymptotics analytic-number-theory




      analysis notation asymptotics analytic-number-theory






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      share|cite|improve this question













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      edited Apr 23 '14 at 11:41

























      asked Apr 22 '14 at 19:24









      Assad Ebrahim

      621621




      621621






















          2 Answers
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          up vote
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          down vote



          accepted










          It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $Theta$" (see p. 448), as in



          $$
          f asymp g iff exists, C,D>0 : C|g| leq |f| leq D|g|,
          $$



          but I read a paper recently where it was instead defined to mean the same thing as $sim$ (as defined here).






          share|cite|improve this answer





















          • Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
            – Assad Ebrahim
            Apr 22 '14 at 20:44








          • 1




            (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
            – Assad Ebrahim
            Apr 22 '14 at 20:56




















          up vote
          4
          down vote













          Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.



          From this the following useful reference table is screenshotted below:



          enter image description here






          share|cite|improve this answer























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            2 Answers
            2






            active

            oldest

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            2 Answers
            2






            active

            oldest

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            active

            oldest

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            active

            oldest

            votes








            up vote
            5
            down vote



            accepted










            It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $Theta$" (see p. 448), as in



            $$
            f asymp g iff exists, C,D>0 : C|g| leq |f| leq D|g|,
            $$



            but I read a paper recently where it was instead defined to mean the same thing as $sim$ (as defined here).






            share|cite|improve this answer





















            • Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
              – Assad Ebrahim
              Apr 22 '14 at 20:44








            • 1




              (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
              – Assad Ebrahim
              Apr 22 '14 at 20:56

















            up vote
            5
            down vote



            accepted










            It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $Theta$" (see p. 448), as in



            $$
            f asymp g iff exists, C,D>0 : C|g| leq |f| leq D|g|,
            $$



            but I read a paper recently where it was instead defined to mean the same thing as $sim$ (as defined here).






            share|cite|improve this answer





















            • Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
              – Assad Ebrahim
              Apr 22 '14 at 20:44








            • 1




              (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
              – Assad Ebrahim
              Apr 22 '14 at 20:56















            up vote
            5
            down vote



            accepted







            up vote
            5
            down vote



            accepted






            It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $Theta$" (see p. 448), as in



            $$
            f asymp g iff exists, C,D>0 : C|g| leq |f| leq D|g|,
            $$



            but I read a paper recently where it was instead defined to mean the same thing as $sim$ (as defined here).






            share|cite|improve this answer












            It can mean different things depending on the context. For instance, in Graham, Knuth, and Patashnik's Concrete Mathematics it's defined to mean the same thing as "Big $Theta$" (see p. 448), as in



            $$
            f asymp g iff exists, C,D>0 : C|g| leq |f| leq D|g|,
            $$



            but I read a paper recently where it was instead defined to mean the same thing as $sim$ (as defined here).







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Apr 22 '14 at 19:54









            Antonio Vargas

            20.6k244111




            20.6k244111












            • Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
              – Assad Ebrahim
              Apr 22 '14 at 20:44








            • 1




              (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
              – Assad Ebrahim
              Apr 22 '14 at 20:56




















            • Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
              – Assad Ebrahim
              Apr 22 '14 at 20:44








            • 1




              (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
              – Assad Ebrahim
              Apr 22 '14 at 20:56


















            Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
            – Assad Ebrahim
            Apr 22 '14 at 20:44






            Yes, I think you're right. Having looked at your LaTeX code, I see the symbol is asymp, and doing a search brings up this useful table on asymptotic notations that confirms your (nicer) expression above. (I'll paste the table into a separate answer below for reference.) Accepting your answer as the correct one. Thanks!
            – Assad Ebrahim
            Apr 22 '14 at 20:44






            1




            1




            (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
            – Assad Ebrahim
            Apr 22 '14 at 20:56






            (+1)... in English: "there exist positive constants C,D such that $f$ can be sandwiched between $g$ scaled appropriately above and below."
            – Assad Ebrahim
            Apr 22 '14 at 20:56












            up vote
            4
            down vote













            Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.



            From this the following useful reference table is screenshotted below:



            enter image description here






            share|cite|improve this answer



























              up vote
              4
              down vote













              Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.



              From this the following useful reference table is screenshotted below:



              enter image description here






              share|cite|improve this answer

























                up vote
                4
                down vote










                up vote
                4
                down vote









                Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.



                From this the following useful reference table is screenshotted below:



                enter image description here






                share|cite|improve this answer














                Searching further on Antonio Vargas's accepted answer above finds an insightful short paper of A.J. Hildebrand on Asymptotic Notation.



                From this the following useful reference table is screenshotted below:



                enter image description here







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited yesterday









                malin

                718620




                718620










                answered Apr 22 '14 at 20:51









                Assad Ebrahim

                621621




                621621






























                     

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