## Sign changes of $$E(T)$$, $$\Delta(x)$$, and $$P(x)$$.(English)Zbl 0810.11046

Let as usual ($$\gamma$$ is Euler’s constant) \begin{aligned} E(T) &= \int_ 0^ m \Bigl| \zeta \bigl( {\textstyle {1\over 2}}+ it\bigr)\Bigr|^ 2 dt- T(\log (T/2\pi)+ 2\gamma-1)\\ \text{and} \Delta(x) &= \sum_{n\leq x} d(n)- x(\log x+ 2\gamma-1), \quad P(x)= \sum_{n\leq x} r(n)- \pi x\end{aligned} denote the error terms in the divisor and circle problem. The authors first prove that, if $$f(t)$$ is any real-valued function satisfying $$| f(t)|\leq c_ 1 t^{1/4}$$, the function $$E(t)+ f(t)$$ changes sign at least once in the interval $$[T, T+c_ 2 \sqrt{T}]$$ for $$T\geq T_ 0$$ and suitable $$c_ 1,c_ 2>0$$. In particular, there exist $$t_ 1,t_ 2\in [T, T+c_ 2 \sqrt{T}]$$ such that $$E(t_ 1)\geq c_ 1 t^{1/4}$$ and $$E(t_ 2)\leq -c_ 1 t_ 2^{1/4}$$ (the last statement is contained in Theorem 3 of the reviewer’s work [Acta Arith. 56, 135–159 (1990; Zbl 0659.10053)]).
In Theorem 2 they show that, for $$\delta>0$$ small and $$T\geq T_ 0(\delta)$$, there are at least $$c_ 4\delta \sqrt{T}\log^ 5 T$$ disjoint subintervals of length $$c_ 5 \delta \sqrt{T} \log^{-5} T$$ in $$[T,2T]$$ such that $$| E(t)|> (c_ 6- \delta) t^{1/4}$$ whenever $$t$$ lies in any of these intervals (this disproves a conjecture of H. J. J. te Riele and the reviewer [Math. Comput. 56, 303–328 (1991; Zbl 0714.11051)] on the zeros of $$E(t)$$).
Finally in Theorem 3 it is shown that the conclusions of the preceding theorems hold with $$E(t)$$ replaced by $$\Delta(x)$$ or $$P(x)$$.

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11N37 Asymptotic results on arithmetic functions

### Citations:

Zbl 0659.10053; Zbl 0714.11051
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