# zbMATH — the first resource for mathematics

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
Full Text: