Changes of sign of error terms related to Euler's function and to divisor functions
The author considers the sign change problem of the error terms R(x) (resp. H(x), \(F_ k(x)\), \(P_ k(x))\) associated with the Euler function \(\phi\) (n) (resp. with \(\phi\) (n)/n, the sum-of-divisors function \(\sigma_ k(n)\), the lattice point counting function in certain 4- dimensional ellipsoids). He proves two main theorems which give lower bounds for the number \(X_ f(x)\) of changes of sign of the real function f (taken to be one of the above error functions). Theorem 1 gives the estimate \[ X_ f(x)\geq const\cdot x+o(x) \] under the condition that \(f(x)=f(n)-C\{x\}+o(1)\) between integers n and \(n+1\) and the mean square integral of f is bounded above by \(Kx+o(x)\) with constants \(C>0\) and \(K\geq 0\). This is applied to \(f=H\), \(F_ k\) \((k<0)\) on the basis of the mean value theorems of Chowla and Walfisz dating back to 1930's. To cover the cases \(f=R\), \(F_ k\) \((k>0)\), \(P_ k\) the author proves Theorem 2: If f is as in Theorem 1 and real functions g,h \((h>0)\) satisfy \[ g(x)=h(x)(f(x)+o(1)), \] then one has the same lower bound for \(X_ g(x)\). If, in addition, the distribution function \(D_ f\) for f exists and is continuous, then one gets the asymptotic identities in the conclusions of Theorems 1 and 2. Using the result of \textit{P. Erdős} and \textit{H. N. Shapiro} [Can. J. Math. 7, 63-75 (1955; Zbl 0067.276)] on the existence and continuity of \(D_ H\), the author gets asymptotic identities for \(X_ H\) and \(X_ R.\) The paper is well written and pleasant to read, with carefully made (historical) remarks. In particular, the author is the first to notice that Sylvester has unconsciously disproved his own conjecture on R(n).
scientific article; zbMATH DE number 3991611