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Local problems with primes. I. (English) Zbl 0677.10028
Let R be the error term in the explicit formula for \(\Psi (x)=\sum_{n\leq x}\Lambda (n)\), \[ \Psi (x)=x-\sum_{| \gamma | \leq T}x^{\rho}/\rho +R(x,T) \] (\(\rho\) \(=\beta +i\gamma\) runs through the nontrivial zeros of the Riemann zeta-function). The author is interested in connections between \(I(N,T)=\int^{2N}_{N+1}| R(x,T)|^ 2 dx\) (in the paper there occurs a further parameter \(\tau\in [T,2T]\). It is not always clear how \(\tau\) depends on x and T) and “local \(L^ 2\)-norm” expressions \[ \int^{b}_{a}| S(\alpha)|^ 2 d\alpha,\quad (S(\alpha)=\sum^{2N}_{n=N+1}\Lambda (n)e(n\alpha)). \] It is shown that, if \(N^{-1} \log^ 3N\leq \xi \leq N^{-(\vartheta +\epsilon)}\) \((0<\vartheta <1)\), then \(I(N,T)=o(N^ 3 T^{-2} \log^{-1}N)\) implies \[ \int^{\xi}_{-\xi}| S(\alpha)|^ 2 d\alpha =N+o(N). \] From this one can deduce, on the assumption \(N^{\vartheta +\epsilon}<H\leq N\), \[ \Psi (x+H)-\Psi (x)\quad \sim \quad H\quad for\quad almost\quad all\quad x. \] This is a hypothetical improvement of M. N. Huxley’s result with \(H>N^{1/6} \ell n^ cN\) [Invent. Math. 15, 164-170 (1972; Zbl 0241.10026)]. In the proof P. X. Gallagher’s Lemma [Invent. Math. 11, 329-339 (1970; Zbl 0219.10048)] and the reviewer’s version of the explicit formula [J. Lond. Math. Soc., II. Ser. 28, 404-416 (1983; Zbl 0533.10037)] are used.
Reviewer: D.Wolke

MSC:
11N05 Distribution of primes
11L40 Estimates on character sums
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