Multiplicative Number Theory pp 169-171 | Cite as

# An Average Result

Chapter

## Abstract

We now consider the mean square error in the prime number theorem for arithmetic progressions. Work in this direction was initiated by Barban
for

^{1}, and by Davenport and Halberstam^{2}. Their results were sharpened by Gallagher^{3}, who showed that$$ {\sum\limits_{q \le Q} {\sum\limits_{\scriptstyle a = 1 \hfill \atop\scriptstyle (a,q = 1) \hfill}^q {\left( {\psi \left( {x;q,a} \right) - \frac{x}{{{\o}\left( q \right)}}} \right)} } ^2} \ll xQ\log x $$

(1)

*x*(log*x*)^{ -A }≤ Q ≤ x; here*A >*0 is fixed. This estimate is best possible, for Montgomery^{4}has shown that the left-hand side is ~*Qx*log*x*for*Q*in the stated range. Moreover, Hooley^{5}has shown that (1) can be combined with some of Montgomery’s ideas to give, in a simple way, a very precise asymptotic estimate.## Keywords

Longe Range Number Theory Prime Number Average Result Asymptotic Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Ann Davenport 1980