Multiplicative functions in short intervals

Abstract

We introduce a general result relating “short averages” of a multiplicative function to “long averages” which are well understood. This result has several consequences. First, for the Möbius function we show that there are cancellations in the sum of $\mu(n)$ in almost all intervals of the form $[x, x + \psi(x)]$ with $\psi(x) \rightarrow \infty$ arbitrarily slowly. This goes beyond what was previously known conditionally on the Density Hypothesis or the stronger Riemann Hypothesis. Second, we settle the long-standing conjecture on the existence of $x^{\varepsilon}$-smooth numbers in intervals of the form $[x, x + c(\varepsilon) \sqrt{x}]$, recovering unconditionally a conditional (on the Riemann Hypothesis) result of Soundararajan. Third, we show that the mean-value of $\lambda(n)\lambda(n+1)$, with $\lambda(n)$ Liouville’s function, is nontrivially bounded in absolute value by $1 – \delta$ for some $\delta > 0$. This settles an old folklore conjecture and constitutes progress towards Chowla’s conjecture. Fourth, we show that a (general) real-valued multiplicative function $f$ has a positive proportion of sign changes if and only if $f$ is negative on at least one integer and nonzero on a positive proportion of the integers. This improves on many previous works and is new already in the case of the Möbius function. We also obtain some additional results on smooth numbers in almost all intervals, and sign changes of multiplicative functions in all intervals of square-root length.\looseness=-1

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  • [BalogSmooth] A. Balog, On the distribution of integers having no large prime factor, Paris: Math. Soc. France, 1987, vol. 147-148.
    @book{BalogSmooth, mrkey = {0891417},
      author = {Balog, A.},
      title = {On the distribution of integers having no large prime factor},
      note = {Journ{é}es arithm{é}tiques de Besan{ç}on (Besan{ç}on, 1985)},
      series = {Astérisque},
      fjournal = {Astérisque},
      volume = {147-148},
      year = {1987},
      pages = {27--31, 343},
      issn = {0303-1179},
      mrclass = {11N25},
      mrnumber = {0891417},
      publisher = {Math. Soc. France},
      address = {Paris},
      mrreviewer = {E. J. Scourfield},
      zblnumber = {0617.10031},
      }
  • [Cassaigne] J. Cassaigne, S. Ferenczi, C. Mauduit, J. Rivat, and A. Sárközy, "On finite pseudorandom binary sequences. III. The Liouville function. I," Acta Arith., vol. 87, iss. 4, pp. 367-390, 1999.
    @article{Cassaigne, mrkey = {1671629},
      author = {Cassaigne, Julien and Ferenczi, S{é}bastien and Mauduit, Christian and Rivat, J{ö}el and S{á}rk{ö}zy, Andr{á}s},
      title = {On finite pseudorandom binary sequences. {III}. {T}he {L}iouville function. {I}},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {87},
      year = {1999},
      number = {4},
      pages = {367--390},
      issn = {0065-1036},
      mrclass = {11K31 (11K45)},
      mrnumber = {1671629},
      mrreviewer = {Peter J. Hellekalek},
      zblnumber = {0920.11053},
      }
  • [Croot] Go to document E. S. Croot III, "On the oscillations of multiplicative functions taking values $\pm 1$," J. Number Theory, vol. 98, iss. 1, pp. 184-194, 2003.
    @article{Croot, mrkey = {1950444},
      author = {Croot, III, Ernest S.},
      title = {On the oscillations of multiplicative functions taking values {$\pm 1$}},
      journal = {J. Number Theory},
      fjournal = {Journal of Number Theory},
      volume = {98},
      year = {2003},
      number = {1},
      pages = {184--194},
      issn = {0022-314X},
      coden = {JNUTA9},
      mrclass = {11N64},
      mrnumber = {1950444},
      mrreviewer = {K. Soundararajan},
      doi = {10.1016/S0022-314X(02)00024-0},
      zblnumber = {1090.11062},
      }
  • [CrootSmooth] Go to document E. Croot, "Smooth numbers in short intervals," Int. J. Number Theory, vol. 3, iss. 1, pp. 159-169, 2007.
    @article{CrootSmooth, mrkey = {2310498},
      author = {Croot, Ernie},
      title = {Smooth numbers in short intervals},
      journal = {Int. J. Number Theory},
      fjournal = {International Journal of Number Theory},
      volume = {3},
      year = {2007},
      number = {1},
      pages = {159--169},
      issn = {1793-0421},
      mrclass = {11N25},
      mrnumber = {2310498},
      mrreviewer = {Andreas J. Weingartner},
      doi = {10.1142/S1793042107000833},
      zblnumber = {1115.11057},
      }
  • [ElliottPaper] P. D. T. A. Elliott, "On the correlation of multiplicative functions," Notas Soc. Mat. Chile, vol. 11, iss. 1, pp. 1-11, 1992.
    @article{ElliottPaper, mrkey = {1292619},
      author = {Elliott, P. D. T. A.},
      title = {On the correlation of multiplicative functions},
      journal = {Notas Soc. Mat. Chile},
      fjournal = {Notas de la Sociedad de Matemática de Chile},
      volume = {11},
      year = {1992},
      number = {1},
      pages = {1--11},
      issn = {0716-1298},
      mrclass = {11K65 (11K31 11N60 11T23)},
      mrnumber = {1292619},
      mrreviewer = {Claudia A. Spiro-Silverman},
      zblnumber = {0810.11055},
      }
  • [Elliott] Go to document P. D. T. A. Elliott, Duality in Analytic Number Theory, Cambridge: Cambridge Univ. Press, 1997, vol. 122.
    @book{Elliott, mrkey = {1433800},
      author = {Elliott, P. D. T. A.},
      title = {Duality in Analytic Number Theory},
      series = {Cambridge Tracts in Math.},
      volume = {122},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {1997},
      pages = {xviii+341},
      isbn = {0-521-56088-8},
      mrclass = {11K65 (11K70 11N64)},
      mrnumber = {1433800},
      mrreviewer = {D. R. Heath-Brown},
      doi = {10.1017/CBO9780511983405},
      zblnumber = {0887.11002},
      }
  • [ElsGun] Go to document C. Elsholtz and D. S. Gunderson, "Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations," Funct. Approx. Comment. Math., vol. 52, iss. 2, pp. 263-281, 2015.
    @article{ElsGun, mrkey = {3358320},
      author = {Elsholtz, Christian and Gunderson, David S.},
      title = {Congruence properties of multiplicative functions on sumsets and monochromatic solutions of linear equations},
      journal = {Funct. Approx. Comment. Math.},
      fjournal = {Uniwersytet im. Adama Mickiewicza w Poznaniu. Wydzia\l Matematyki i Informatyki. Functiones et Approximatio Commentarii Mathematici},
      volume = {52},
      year = {2015},
      number = {2},
      pages = {263--281},
      issn = {0208-6573},
      mrclass = {11B30 (05D10 11N64)},
      mrnumber = {3358320},
      doi = {10.7169/facm/2015.52.2.6},
      }
  • [Opera] J. Friedlander and H. Iwaniec, Opera de Cribro, Providence, RI: Amer. Math. Soc., 2010, vol. 57.
    @book{Opera, mrkey = {2647984},
      author = {Friedlander, John and Iwaniec, Henryk},
      title = {Opera de Cribro},
      series = {Amer. Math. Soc. Colloq. Publ.},
      volume = {57},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2010},
      pages = {xx+527},
      isbn = {978-0-8218-4970-5},
      mrclass = {11N35 (11-02 11N05 11N13 11N25 11N36)},
      mrnumber = {2647984},
      mrreviewer = {D. R. Heath-Brown},
      zblnumber = {1226.11099},
      }
  • [FriedlanderGranville] Go to document J. B. Friedlander and A. Granville, "Smoothing “smooth” numbers," Philos. Trans. Roy. Soc. London Ser. A, vol. 345, iss. 1676, pp. 339-347, 1993.
    @article{FriedlanderGranville, mrkey = {1253496},
      author = {Friedlander, John B. and Granville, Andrew},
      title = {Smoothing ``smooth'' numbers},
      journal = {Philos. Trans. Roy. Soc. London Ser. A},
      fjournal = {Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical Sciences and Engineering},
      volume = {345},
      year = {1993},
      number = {1676},
      pages = {339--347},
      issn = {0962-8428},
      mrclass = {11N25},
      mrnumber = {1253496},
      mrreviewer = {G{é}rald Tenenbaum},
      doi = {10.1098/rsta.1993.0133},
      zblnumber = {0795.11041},
      }
  • [GhoshSarnak] Go to document A. Ghosh and P. Sarnak, "Real zeros of holomorphic Hecke cusp forms," J. Eur. Math. Soc. $($JEMS$)$, vol. 14, iss. 2, pp. 465-487, 2012.
    @article{GhoshSarnak, mrkey = {2881302},
      author = {Ghosh, Amit and Sarnak, Peter},
      title = {Real zeros of holomorphic {H}ecke cusp forms},
      journal = {J. Eur. Math. Soc. $($JEMS$)$},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {14},
      year = {2012},
      number = {2},
      pages = {465--487},
      issn = {1435-9855},
      mrclass = {11F11 (11F30)},
      mrnumber = {2881302},
      mrreviewer = {Paul M. Jenkins},
      doi = {10.4171/JEMS/308},
      zblnumber = {1287.11054},
      }
  • [GranvilleSurvey] A. Granville, "Smooth numbers: computational number theory and beyond," in Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Cambridge: Cambridge Univ. Press, 2008, vol. 44, pp. 267-323.
    @incollection{GranvilleSurvey, mrkey = {2467549},
      author = {Granville, Andrew},
      title = {Smooth numbers: computational number theory and beyond},
      booktitle = {Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography},
      series = {Math. Sci. Res. Inst. Publ.},
      volume = {44},
      pages = {267--323},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {2008},
      mrclass = {11Y35 (11P05 11R47 11Y16 11Y60 11Y70)},
      mrnumber = {2467549},
      mrreviewer = {Ra{ú}l Dur{á}n D{\'ı}az},
      zblnumber = {1230.11157},
      }
  • [GS03] Go to document A. Granville and K. Soundararajan, "Decay of mean values of multiplicative functions," Canad. J. Math., vol. 55, iss. 6, pp. 1191-1230, 2003.
    @article{GS03, mrkey = {2016245},
      author = {Granville, Andrew and Soundararajan, K.},
      title = {Decay of mean values of multiplicative functions},
      journal = {Canad. J. Math.},
      fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      volume = {55},
      year = {2003},
      number = {6},
      pages = {1191--1230},
      issn = {0008-414X},
      coden = {CJMAAB},
      mrclass = {11N56 (11N37 11N60)},
      mrnumber = {2016245},
      mrreviewer = {G. Greaves},
      doi = {10.4153/CJM-2003-047-0},
      zblnumber = {1047.11093},
      }
  • [Hafner] J. Hafner, On smooth numbers in short intervals under the Riemann hypothesis, 1993.
    @misc{Hafner,
      author = {Hafner, J.},
      title = {On smooth numbers in short intervals under the {R}iemann hypothesis},
      note = {preprint},
      year = {1993},
      }
  • [Harman06] G. Harman, Prime-Detecting Sieves, Princeton, NJ: Princeton Univ. Press, 2007, vol. 33.
    @book{Harman06, mrkey = {2331072},
      author = {Harman, Glyn},
      title = {Prime-Detecting Sieves},
      series = {London Math. Soc. Monogr. Ser.},
      volume = {33},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {2007},
      pages = {xvi+362},
      isbn = {978-0-691-12437-7},
      mrclass = {11N36 (11N25 11N35)},
      mrnumber = {2331072},
      mrreviewer = {S. W. Graham},
      zblnumber = {1220.11118},
      }
  • [HPW85] G. Harman, J. Pintz, and D. Wolke, "A note on the Möbius and Liouville functions," Studia Sci. Math. Hungar., vol. 20, iss. 1-4, pp. 295-299, 1985.
    @article{HPW85, mrkey = {0886032},
      author = {Harman, G. and Pintz, J. and Wolke, D.},
      title = {A note on the {M}öbius and {L}iouville functions},
      journal = {Studia Sci. Math. Hungar.},
      fjournal = {Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences},
      volume = {20},
      year = {1985},
      number = {1-4},
      pages = {295--299},
      issn = {0081-6906},
      coden = {SSMHAX},
      mrclass = {11N37},
      mrnumber = {0886032},
      mrreviewer = {T. M. Apostol},
      zblnumber = {0544.10041},
      }
  • [Harper] A. Harper, Sharp conditional bounds for moments of the zeta function, 2013.
    @misc{Harper,
      author={Harper, A.},
      TITLE={Sharp conditional bounds for moments of the zeta function},
      ARXIV={1305.4618},
      YEAR={2013},
     }
  • [HildebrandReview] A. Hildebrand, Math. Reviews.
    @misc{HildebrandReview,
      author = {Hildebrand, Adolf},
      title = {Math. Reviews},
      MRNUMBER={1222182},
      }
  • [Hildebrand] Go to document A. Hildebrand, "Multiplicative functions at consecutive integers," Math. Proc. Cambridge Philos. Soc., vol. 100, iss. 2, pp. 229-236, 1986.
    @article{Hildebrand, mrkey = {0848849},
      author = {Hildebrand, Adolf},
      title = {Multiplicative functions at consecutive integers},
      journal = {Math. Proc. Cambridge Philos. Soc.},
      fjournal = {Mathematical Proceedings of the Cambridge Philosophical Society},
      volume = {100},
      year = {1986},
      number = {2},
      pages = {229--236},
      issn = {0305-0041},
      coden = {MPCPCO},
      mrclass = {11N37},
      mrnumber = {0848849},
      mrreviewer = {T. M. Apostol},
      doi = {10.1017/S0305004100066056},
      zblnumber = {0615.10053},
      }
  • [Ivic03] A. Ivić, The Riemann Zeta-Function. Theory and Applications, Mineola, NY: Dover Publications, 2003.
    @book{Ivic03, mrkey = {1994094},
      author = {Ivi{ć},
      Aleksandar},
      title = {The {R}iemann Zeta-Function. Theory and Applications},
      note = {reprint of the 1985 original [Wiley, New York; \mr{0792089}]},
      publisher = {Dover Publications},
      address = {Mineola, NY},
      year = {2003},
      pages = {xxii+517},
      isbn = {0-486-42813-3},
      mrclass = {11M06 (11M26)},
      mrnumber = {1994094},
      zblnumber = {1034.11046},
      }
  • [IwKo04] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
    @book{IwKo04, mrkey = {2061214},
      author = {Iwaniec, Henryk and Kowalski, Emmanuel},
      title = {Analytic Number Theory},
      series = {Amer. Math. Soc. Colloq. Publ.},
      volume = {53},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2004},
      pages = {xii+615},
      isbn = {0-8218-3633-1},
      mrclass = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
      mrnumber = {2061214},
      mrreviewer = {K. Soundararajan},
      zblnumber = {1059.11001},
      }
  • [Jutila] M. Jutila, "Zero-density estimates for $L$-functions," Acta Arith., vol. 32, iss. 1, pp. 55-62, 1977.
    @article{Jutila, mrkey = {0429790},
      author = {Jutila, Matti},
      title = {Zero-density estimates for {$L$}-functions},
      journal = {Acta Arith.},
      fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      volume = {32},
      year = {1977},
      number = {1},
      pages = {55--62},
      issn = {0065-1036},
      mrclass = {10H10},
      mrnumber = {0429790},
      mrreviewer = {Bruce C. Berndt},
      zblnumber = {0307.10045},
      }
  • [KRW07] Go to document E. Kowalski, O. Robert, and J. Wu, "Small gaps in coefficients of $L$-functions and $\mathfrak{B}$-free numbers in short intervals," Rev. Mat. Iberoam., vol. 23, iss. 1, pp. 281-326, 2007.
    @article{KRW07, mrkey = {2351136},
      author = {Kowalski, Emmanuel and Robert, Olivier and Wu, Jie},
      title = {Small gaps in coefficients of {$L$}-functions and {$\mathfrak{B}$}-free numbers in short intervals},
      journal = {Rev. Mat. Iberoam.},
      fjournal = {Revista Mathemática Iberoamericana},
      volume = {23},
      year = {2007},
      number = {1},
      pages = {281--326},
      issn = {0213-2230},
      mrclass = {11F30 (11F12 11F66 11L15 11N25)},
      mrnumber = {2351136},
      mrreviewer = {D. R. Heath-Brown},
      doi = {10.4171/RMI/496},
      zblnumber = {1246.11099},
      }
  • [LauLiuWu10] Go to document Y. Lau, J. Y. Liu, and J. Wu, "Coefficients of symmetric square $L$-functions," Sci. China Math., vol. 53, iss. 9, pp. 2317-2328, 2010.
    @article{LauLiuWu10, mrkey = {2718828},
      author = {Lau, Yuk-Kam and Liu, Jian Ya and Wu, Jie},
      title = {Coefficients of symmetric square {$L$}-functions},
      journal = {Sci. China Math.},
      fjournal = {Science China. Mathematics},
      volume = {53},
      year = {2010},
      number = {9},
      pages = {2317--2328},
      issn = {1674-7283},
      mrclass = {11F30 (11F66 11N25)},
      mrnumber = {2718828},
      mrreviewer = {Brundaban Sahu},
      doi = {10.1007/s11425-010-4046-z},
      zblnumber = {1267.11047},
      }
  • [Lenstra] H. W. Lenstra Jr., "Elliptic curves and number-theoretic algorithms," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Providence, RI, 1987, pp. 99-120.
    @inproceedings{Lenstra, mrkey = {0934218},
      author = {Lenstra, Jr., H. W.},
      title = {Elliptic curves and number-theoretic algorithms},
      booktitle = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      venue = {{B}erkeley, {C}alif., 1986},
      pages = {99--120},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1987},
      mrclass = {11Y11 (11G99 11Y05)},
      mrnumber = {0934218},
      mrreviewer = {Samuel S. Wagstaff, Jr.},
      zblnumber = {0686.14039},
      }
  • [LesterMatomakiRadziwill] S. Lester, K. Matomäki, and M. Radziwill, Small scale distribution of zeros and mass of modular forms, 2015.
    @misc{LesterMatomakiRadziwill,
      author = {Lester, S. and Matomäki, K. and Radziwi{\l\l},
      M.},
      title = {Small scale distribution of zeros and mass of modular forms},
      arxiv = {1501.01292},
      year = {2015},
      }
  • [Lucht] Go to document L. Lucht and F. Tuttas, "Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen," Monatsh. Math., vol. 87, iss. 1, pp. 15-19, 1979.
    @article{Lucht, mrkey = {0528874},
      author = {Lucht, Lutz and Tuttas, Friedemann},
      title = {Aufeinanderfolgende {E}lemente in multiplikativen {Z}ahlenmengen},
      journal = {Monatsh. Math.},
      fjournal = {Monatshefte für Mathematik},
      volume = {87},
      year = {1979},
      number = {1},
      pages = {15--19},
      issn = {0026-9255},
      coden = {MNMTA2},
      mrclass = {10H15 (10K20)},
      mrnumber = {0528874},
      doi = {10.1007/BF01470935},
      zblnumber = {0369.10029},
      }
  • [MatomakiSmooth] Go to document K. Matomäki, Another note on smooth numbers in short intervals, 2015.
    @misc{MatomakiSmooth,
      author = {Matom{ä}ki, Kaisa},
      title = {Another note on smooth numbers in short intervals},
      note = {{\em Int. J. Number Theory},
      published online 26 August 2015},
      url={http://dx.doi.org/10.1142/S1793042116500196},
      YEAR={2015},
     }
  • [MatomakiSmooths2] Go to document K. Matomäki, "A note on smooth numbers in short intervals," Int. J. Number Theory, vol. 6, iss. 5, pp. 1113-1116, 2010.
    @article{MatomakiSmooths2, mrkey = {2679459},
      author = {Matom{ä}ki, Kaisa},
      title = {A note on smooth numbers in short intervals},
      journal = {Int. J. Number Theory},
      fjournal = {International Journal of Number Theory},
      volume = {6},
      year = {2010},
      number = {5},
      pages = {1113--1116},
      issn = {1793-0421},
      mrclass = {11N25},
      mrnumber = {2679459},
      mrreviewer = {Andreas J. Weingartner},
      doi = {10.1142/S1793042110003381},
      zblnumber = {1204.11154}
    }
  • [ShortNote] K. Matomäki and M. Radziwiłl, A note on the Liouville function in short intervals, 2015.
    @misc{ShortNote,
      author = {Matom{ä}ki, Kaisa and Radziwi{\l}l, Maksym},
      title = {A note on the {L}iouville function in short intervals},
      arxiv = {1502.02374},
      year = {2015},
      }
  • [MatRadHecke] Go to document K. Matomäki and M. Radziwiłl, "Sign changes of Hecke eigenvalues," Geom. Funct. Anal., vol. 25, iss. 6, pp. 1937-1955, 2015.
    @article{MatRadHecke, mrkey = {3432161},
      author = {Matom{ä}ki, Kaisa and Radziwi{\l}l, Maksym},
      title = {Sign changes of {H}ecke eigenvalues},
      journal = {Geom. Funct. Anal.},
      fjournal = {Geometric and Functional Analysis},
      volume = {25},
      year = {2015},
      number = {6},
      pages = {1937--1955},
      issn = {1016-443X},
      mrclass = {Preliminary Data},
      mrnumber = {3432161},
      doi = {10.1007/s00039-015-0350-7},
      zblnumber = {06526263},
      }
  • [tao] Go to document K. Matomäki, M. Radziwiłl, and T. Tao, "An averaged form of Chowla’s conjecture," Algebra Number Theory, vol. 9, iss. 9, pp. 2167-2196, 2015.
    @article{tao, mrkey = {3435814},
      author = {Matom{ä}ki, Kaisa and Radziwi{\l}l, Maksym and Tao, Terence},
      title = {An averaged form of {C}howla's conjecture},
      journal = {Algebra Number Theory},
      fjournal = {Algebra \& Number Theory},
      volume = {9},
      year = {2015},
      number = {9},
      pages = {2167--2196},
      issn = {1937-0652},
      mrclass = {11P32},
      mrnumber = {3435814},
      doi = {10.2140/ant.2015.9.2167},
      zblnumber = {06529158},
     }
  • [MontgomeryTenLecture] H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Providence, RI: Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc., 1994, vol. 84.
    @book{MontgomeryTenLecture, mrkey = {1297543},
      author = {Montgomery, Hugh L.},
      title = {Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis},
      series = {CBMS Reg. Conf. Ser. Math.},
      volume = {84},
      publisher = {Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1994},
      pages = {xiv+220},
      isbn = {0-8218-0737-4},
      mrclass = {11-02 (11Kxx 11L07 11Mxx 11Nxx)},
      mrnumber = {1297543},
      mrreviewer = {John B. Friedlander},
      zblnumber = {0814.11001},
      }
  • [RadziwillSoundararajan] Go to document M. Radziwiłl and K. Soundararajan, "Moments and distribution of central $L$-values of quadratic twists of elliptic curves," Invent. Math., vol. 202, iss. 3, pp. 1029-1068, 2015.
    @article{RadziwillSoundararajan, mrkey = {3425386},
      author = {Radziwi{\l}l, Maksym and Soundararajan, K.},
      title = {Moments and distribution of central {$L$}-values of quadratic twists of elliptic curves},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {202},
      year = {2015},
      number = {3},
      pages = {1029--1068},
      issn = {0020-9910},
      mrclass = {Preliminary Data},
      mrnumber = {3425386},
      doi = {10.1007/s00222-015-0582-z},
      zblnumber = {06518187},
      }
  • [Ramachandra] K. Ramachandra, "Some problems of analytic number theory," Acta Arith., vol. 31, iss. 4, pp. 313-324, 1976.
    @article{Ramachandra, mrkey = {0424723},
      author = {Ramachandra, K.},
      title = {Some problems of analytic number theory},
      journal = {Acta Arith.},
      fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
      volume = {31},
      year = {1976},
      number = {4},
      pages = {313--324},
      issn = {0065-1036},
      mrclass = {10H15 (10H25 10H10)},
      mrnumber = {0424723},
      mrreviewer = {W. Schwarz},
      zblnumber = {0291.10034},
      }
  • [SaVa77] Go to document B. Saffari and R. C. Vaughan, "On the fractional parts of $x/n$ and related sequences. II," Ann. Inst. Fourier $($Grenoble$)$, vol. 27, iss. 2, p. v, 1-30, 1977.
    @article{SaVa77, mrkey = {0480388},
      author = {Saffari, B. and Vaughan, R. C.},
      title = {On the fractional parts of {$x/n$} and related sequences. {II}},
      journal = {Ann. Inst. Fourier $($Grenoble$)$},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {27},
      year = {1977},
      number = {2},
      pages = {v, 1--30},
      issn = {0373-0956},
      mrclass = {10H25 (10K05)},
      mrnumber = {0480388},
      mrreviewer = {H. G. Meijer},
      url = {http://www.numdam.org/item?id=AIF_1977__27_2_1_0},
      zblnumber = {0379.10023},
      }
  • [Shiu80] Go to document P. Shiu, "A Brun-Titchmarsh theorem for multiplicative functions," J. Reine Angew. Math., vol. 313, pp. 161-170, 1980.
    @article{Shiu80, mrkey = {0552470},
      author = {Shiu, P.},
      title = {A {B}run-{T}itchmarsh theorem for multiplicative functions},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      volume = {313},
      year = {1980},
      pages = {161--170},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {10H25},
      mrnumber = {0552470},
      mrreviewer = {A. I. Vinogradov},
      doi = {10.1515/crll.1980.313.161},
      zblnumber = {0412.10030},
      }
  • [SoundararajanPre] K. Soundararajan, Smooth numbers in short intervals, 2010.
    @misc{SoundararajanPre,
      author = {Soundararajan, K.},
      title = {Smooth numbers in short intervals},
      arxiv = {1009.1591v1},
      year = {2010},
      }
  • [Tenenbaum] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ. Press, Cambridge, 1995, vol. 46.
    @book{Tenenbaum, mrkey = {1342300},
      author = {Tenenbaum, G{é}rald},
      title = {Introduction to Analytic and Probabilistic Number Theory},
      series = {Cambridge Stud. Adv. Math.},
      volume = {46},
      note = {translated from the second French edition (1995) by C. B. Thomas},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {1995},
      pages = {xvi+448},
      isbn = {0-521-41261-7},
      mrclass = {11-02 (11Kxx 11Mxx 11Nxx)},
      mrnumber = {1342300},
      mrreviewer = {H. G. Diamond},
      zblnumber = {0831.11001},
      }

Authors

Kaisa Matomäki

University of Turku, Turku, Finland

Maksym Radziwiłł

Rutgers University, Piscataway, NJ