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On zeros of Dirichlet’s \(L\)-functions. (Russian. English summary) Zbl 0063.07326

Summary: In this paper I prove the following two theorems.
Theorem 1. Let \(k\) be a positive integer \(\geq 3\) and let \(L(s,\chi)\) denote any Dirichlet \(L\)-function, which belongs to mod \(k\). There are two positive absolute constants \(\gamma<1\) and \(c_4\) such that \(L(s,\chi)\neq 0\) in the domain
\[ 0<| t|\leq t_0,\;\sigma\geq 1-c_4(\log k)^{-1}, \]
\[ | t|\geq t_0,\;\sigma\geq 1-c_4(\log t)^{-\gamma}, \]
where \(\log t_0= (\log k)^{1/\gamma}\).
Theorem 2. Let \(k\) and \(l\) be two positive integers, \((l,k) = 1\), \(x\) is a positive number \(\geq 2\), \(\pi(x,k,l)\) is the number of primes satisfying the conditions \(p\equiv l\pmod k\), \(p\leq x\), \(\beta_1\) is a zero of a certain function \(L(s,\chi)\), \(E = 0\) or \(1\). There are three absolute positive constants \(c_5\), \(c_8\) and \(\mu>\tfrac 12\) such that \[ \left|\pi(x,k,l)-h^{-1}\sum_{2\leq n\leq x} \frac 1{\log n}-Eh^{-1}\overline{\chi}_1(l)\sum_{2\leq n\leq x}\frac{n^{\beta_1-1}}{\log n}\right| \leq c_8xe^{-c_5(\log x)^\mu}. \]

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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