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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. Rodosskiń≠ [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 \(\beta=(4c\theta+2\theta-1-4c)/(6+4c)-\varepsilon\), \(c=6/37\).

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