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A statistical density theorem for $$L$$-functions with applications. (English) Zbl 0185.10901
Let $$N(\alpha, T, q, \chi)$$ be the number of zeros of the $$L$$-function $$L(s,\chi) \pmod q$$ in the rectangle $$\alpha\leq\sigma\leq 1$$, $$| t|\leq T$$, and consider the sum $\sum_{q\leq X}\;\sideset\and{^*}\to\sum_{\chi \bmod q} N(\alpha, T, q, \chi), \tag{1}$ where the summation runs over primitive characters only. The main result is the estimate
$\ll(X^7T^4)^{1-\alpha)/\alpha}\log^{c_1}(X+T)$ for (1) ($$c_1$$ a constant). This is obtained by a combination of a “large sieve” inequality of E. Bombieri [originally Mathematika 12, 201–225 (1965; Zbl 0136.33004), see also P. X. Gallagher [Mathematika 14, 14–20 (1967; Zbl 0163.04401)] and a method of K. A. Rodosskiĭ [Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 3(4), 191–197 (1958; Zbl 0122.05101)].
Arithmetical application: Let $$x\geq 2$$, $$y\geq 2$$, $$y=x^\theta$$, $$0<\theta<1$$, $$\theta$$ fixed, $$A>0$$ arbitrary large but fixed. Then
$\sum_{q\leq x^\beta}\max_{z\leq y}\max_{(a,q)=1}|\psi(x+z,q,q)-\psi(x,q,a)-z/\varphi(q)|\ll y\log^{-A}x,$ where $$\beta=(4c\theta+2\theta-1-4c)/(6+4c)-\varepsilon$$, $$c=6/37$$.

##### MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11N36 Applications of sieve methods
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