An Average Result

  • Harold Davenport
Part of the Graduate Texts in Mathematics book series (GTM, volume 74)


We now consider the mean square error in the prime number theorem for arithmetic progressions. Work in this direction was initiated by Barban1, and by Davenport and Halberstam2. Their results were sharpened by Gallagher3, who showed that
$$ {\sum\limits_{q \le Q} {\sum\limits_{\scriptstyle a = 1 \hfill \atop\scriptstyle (a,q = 1) \hfill}^q {\left( {\psi \left( {x;q,a} \right) - \frac{x}{{{\o}\left( q \right)}}} \right)} } ^2} \ll xQ\log x $$
for x(log x) -A ≤ Q ≤ x; here A > 0 is fixed. This estimate is best possible, for Montgomery4 has shown that the left-hand side is ~ Qx log x for Q in the stated range. Moreover, Hooley5 has shown that (1) can be combined with some of Montgomery’s ideas to give, in a simple way, a very precise asymptotic estimate.


Longe Range Number Theory Prime Number Average Result Asymptotic Estimate 


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  1. 1.
    Dokl. Akad. Nauk UzSSR, 1964, No. 5, 5–7.Google Scholar
  2. 2.
    Michigan Math. J., 13, 485–489 (1966); 15, 505 (1968).Google Scholar
  3. 3.
    Mathematika, 14, 14–20 (1967).Google Scholar
  4. 4.
    Michigan Math. J., 17, 33–39 (1970).Google Scholar
  5. 5.
    J. Reine Angew. Math., 274/275, 206–223 (1975).Google Scholar

Copyright information

© Ann Davenport 1980

Authors and Affiliations

  • Harold Davenport
    • 1
  1. 1.Cambridge UniversityCambridgeEngland

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