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 \(P(x)\).


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