Exponential Sums

  • Olivier Bordellès
Part of the Universitext book series (UTX)


As can be seen in the Dirichlet divisor problem, most of the great problems in multiplicative number theory require non-trivial estimates of exponential sums. In the early 1920s and 1930s, three different schools of thought investigated this problem. Following the lines of van der Corput, we provide the first criteria based upon the second and third derivatives of the studied function, and we apply them to the Dirichlet divisor problem. Many questions are also developed, such as the exponent pairs, the Vinogradov method, the Vaughan identity and the discrete Hardy–Littlewood method. This chapter could be considered as an analytic equivalent to Chap.  5.


Divisor Problem Large Sieve Diophantine Inequality Convolution Identity Exponent Pair 
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  1. [Bak86]
    Baker RC (1986) The greatest prime factor of the integers in an interval. Acta Arith 47:193–231 MathSciNetzbMATHGoogle Scholar
  2. [Bak94]
    Baker RC (1994) The square-free divisor problem. Q J Math Oxford 45:269–277 zbMATHCrossRefGoogle Scholar
  3. [Bak07]
    Baker RC (2007) Sums of two relatively prime cubes. Acta Arith 129:103–146 MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BBR12]
    Berkane D, Bordellès O, Ramaré O (2012) Explicit upper bounds for the remainder term in the divisor problem. Math Comput 81:1025–1051 zbMATHCrossRefGoogle Scholar
  5. [BH09]
    Baker RC, Harman G (2009) Numbers with a large prime factor II. In: Chen WWL, Gowers WT, Halberstam H, Schmidt WM, Vaughan RC (eds) Analytic number theory, essays in honour of Klaus Roth. Cambridge University Press, Cambridge Google Scholar
  6. [BI86]
    Bombieri E, Iwaniec H (1986) On the order of \(\zeta ( \frac{1}{2}+it )\). Ann Sc Norm Super Pisa, Cl Sci 13:449–472 MathSciNetzbMATHGoogle Scholar
  7. [Bor09]
    Bordellès O (2009) Le problème des diviseurs de Dirichlet. Quadrature 71:21–30 zbMATHCrossRefGoogle Scholar
  8. [Bul98]
    Bullen PS (1998) A dictionary of inequalities. Pitman monographs. Addison-Wesley, Reading zbMATHGoogle Scholar
  9. [CZ98]
    Cao X, Zhai W-G (1998) The distribution of square-free numbers of the form [n c]. J Théor Nr Bordx 10:287–299 MathSciNetzbMATHCrossRefGoogle Scholar
  10. [CZ99]
    Cao X, Zhai W-G (1999) On the number of coprime integer pairs within a circle. Acta Arith 89:163–187 MathSciNetGoogle Scholar
  11. [CZ00]
    Cao X, Zhai W-G (2000) Multiple exponential sums with monomials. Acta Arith 92:195–213 MathSciNetzbMATHGoogle Scholar
  12. [Dra03]
    Dragomir SS (2003) A survey on Cauchy–Bunyakovski–Schwarz type discrete inequalities. JIPAM J Inequal Pure Appl Math 4, Article 63 MathSciNetGoogle Scholar
  13. [FI89]
    Fouvry E, Iwaniec H (1989) Exponential sums with monomials. J Number Theory 33:311–333 MathSciNetzbMATHCrossRefGoogle Scholar
  14. [For02]
    Ford K (2002) Recent progress on the estimation of Weyl sums. In: Modern problems of number theory and its applications; Topical problems Part II, Tula, Russia, 2001, pp 48–66 Google Scholar
  15. [GK91]
    Graham SW, Kolesnik G (1991) Van der Corput’s method of exponential sums. London math. soc. lect. note series, vol 126. Cambridge University Press, Cambridge CrossRefGoogle Scholar
  16. [GR96]
    Granville A, Ramaré O (1996) Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika 45:73–107 CrossRefGoogle Scholar
  17. [Gre88]
    Grekos G (1988) Sur le nombre de points entiers d’une courbe convexe. Bull Sci Math 112:235–254 MathSciNetzbMATHGoogle Scholar
  18. [Har16]
    Hardy GH (1916) On Dirichlet’s divisor problem. Proc Lond Math Soc 15:1–25 zbMATHCrossRefGoogle Scholar
  19. [Har07]
    Harman G (2007) Prime-detecting sieves. London math. soc. monographs. Princeton University Press, Princeton zbMATHGoogle Scholar
  20. [HB82]
    Heath-Brown DR (1982) Prime numbers in short intervals and a generalized Vaughan identity. Can J Math 34:1365–1377 MathSciNetzbMATHCrossRefGoogle Scholar
  21. [HS31]
    Hasse H, Suetuna Z (1931) Ein allgemeines Teilerproblem der Idealtheorie. J Fac Sci, Univ Tokyo, Sect 1A, Math 2:133–154 zbMATHGoogle Scholar
  22. [Hux96]
    Huxley MN (1996) Area, lattice points and exponential sums. Oxford Science Publications, London zbMATHGoogle Scholar
  23. [Hux03]
    Huxley MN (2003) Exponential sums and lattice points III. Proc Lond Math Soc 87:591–609 MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Hux05]
    Huxley MN (2005) Exponential sums and the Riemann zeta function V. Proc Lond Math Soc 90:1–41 MathSciNetzbMATHCrossRefGoogle Scholar
  25. [HW88]
    Huxley MN, Watt N (1988) Exponential sums and the Riemann zeta function. Proc Lond Math Soc 57:1–24 MathSciNetzbMATHCrossRefGoogle Scholar
  26. [IK04]
    Iwaniec H, Kowalski E (2004) Analytic number theory. Colloquium publications, vol 53. Am. Math. Soc., Providence zbMATHGoogle Scholar
  27. [IM88]
    Iwaniec H, Mozzochi CJ (1988) On the divisor and circle problems. J Number Theory 29:60–93 MathSciNetzbMATHCrossRefGoogle Scholar
  28. [Ivi85]
    Ivić A (1985) The Riemann zeta-function. Theory and applications. Wiley, New York. 2nd edition: Dover, 2003 Google Scholar
  29. [Kar71]
    Karacuba AA (1971) Estimates for trigonometric sums by Vinogradov’s method, and some applications. Proc Steklov Inst Math 112:251–265 Google Scholar
  30. [Kol85]
    Kolesnik G (1985) On the method of exponent pairs. Acta Arith 45:115–143 MathSciNetzbMATHGoogle Scholar
  31. [KRW07]
    Kowalski E, Robert O, Wu J (2007) Small gaps in coefficients of L-functions and \(\mathfrak{B}\)-free numbers in short intervals. Rev Mat Iberoam 23:281–326 MathSciNetzbMATHCrossRefGoogle Scholar
  32. [Liu93]
    Liu H-Q (1993) The greatest prime factor of the integers in an interval. Acta Arith 65:301–328 MathSciNetzbMATHGoogle Scholar
  33. [Liu94]
    Liu H-Q (1994) The distribution of 4-full numbers. Acta Arith 67:165–176 MathSciNetzbMATHGoogle Scholar
  34. [Liu95]
    Liu H-Q (1995) Divisor problems of 4 and 3 dimensions. Acta Arith 73:249–269 MathSciNetzbMATHGoogle Scholar
  35. [LW99]
    Liu H-Q, Wu J (1999) Numbers with a large prime factor. Acta Arith 89:163–187 MathSciNetzbMATHGoogle Scholar
  36. [Mon94]
    Montgomery HL (1994) Ten lectures on the interface between analytic number theory and harmonic analysis. CMBS, vol 84. Amer. Math. Monthly zbMATHGoogle Scholar
  37. [Mor58]
    Mordell LJ (1958) On the Kusmin–Landau inequality for exponential sums. Acta Arith 4:3–9 MathSciNetzbMATHGoogle Scholar
  38. [MV81]
    Montgomery HL, Vaughan RC (1981) The distribution of squarefree numbers. In: Halberstam H, Hooley C (eds) Recent progress in analytic number theory, vol. I. Academic Press, San Diego Google Scholar
  39. [MV07]
    Montgomery HL, Vaughan RC (2007) Multiplicative number theory Vol. I. Classical theory. Cambridge studies in advanced mathematics, vol 97 Google Scholar
  40. [NP04]
    Niculescu CP, Persson L-E (2003/2004) Old and new on the Hermite–Hadamard inequality. Real Anal Exch 29:663–685 MathSciNetGoogle Scholar
  41. [Pet98]
    Pétermann Y-FS (1998) On an estimate of Walfisz and Saltykov for an error term related to the Euler function. J Théor Nr Bordx 10:203–236 zbMATHCrossRefGoogle Scholar
  42. [PW97]
    Pétermann Y-FS, Wu J (1997) On the sum of exponential divisors of an integer. Acta Math Hung 77:159–175 zbMATHCrossRefGoogle Scholar
  43. [Ram69]
    Ramachandra K (1969) A note on numbers with a large prime factor. J Lond Math Soc 1:303–306 MathSciNetzbMATHCrossRefGoogle Scholar
  44. [RS01]
    Rivat J, Sargos P (2001) Nombres premiers de la forme [n c]. Can J Math 53:190–209 MathSciNetCrossRefGoogle Scholar
  45. [RS03]
    Robert O, Sargos P (2003) A third derivative test for mean values of exponential sums with application to lattice point problems. Acta Arith 106:27–39 MathSciNetzbMATHCrossRefGoogle Scholar
  46. [RS06]
    Robert O, Sargos P (2006) Three-dimensional exponential sums with monomials. J Reine Angew Math 591:1–20 MathSciNetzbMATHCrossRefGoogle Scholar
  47. [SW00]
    Sargos P, Wu J (2000) Multiple exponential sums with monomials and their applications in number theory. Acta Math Hung 88:333–354 MathSciNetCrossRefGoogle Scholar
  48. [Ton56]
    Tong KC (1956) On divisor problem III. Acta Math Sin 6:515–541 Google Scholar
  49. [Vaa85]
    Vaaler J (1985) Some extremal functions in Fourier analysis. Bull Am Math Soc 12:183–216 MathSciNetzbMATHCrossRefGoogle Scholar
  50. [Vin54]
    Vinogradov IM (1954) The method of trigonometric sums in the theory of numbers. Interscience, New York Google Scholar
  51. [Vor03]
    Voronoï G (1903) Sur un problème du calcul des fonctions asymptotiques. J Reine Angew Math 126:241–282 zbMATHGoogle Scholar
  52. [Vor04]
    Voronoï G (1904) Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann Sci Éc Norm Super 21:207–268 zbMATHGoogle Scholar
  53. [Wal63]
    Walfisz A (1963) Weylsche Exponentialsummen in der Neueren Zahlentheorie. VEB, Berlin zbMATHGoogle Scholar
  54. [Wu93]
    Wu J (1993) Nombres \(\mathcal{B}\)-libres dans les petits intervalles. Acta Arith 65:97–98 MathSciNetzbMATHGoogle Scholar
  55. [Wu02]
    Wu J (2002) On the primitive circle problem. Monatshefte Math 135:69–81 zbMATHCrossRefGoogle Scholar
  56. [Zha99]
    Zhai W-G (1999) On sums and differences of two coprime kth powers. Acta Arith 91:233–248 MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Olivier Bordellès
    • 1
  1. 1.AiguilheFrance

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