Oscillations of a given size of some arithmetic error terms

Abstract

Abstract. A general method of estimating the number of oscillations of a given size of arithmetic error terms is developed. Special attention is paid to the remainder terms in the prime number formula, in the Dirichlet prime number theorem for primes in arithmetic progressions and to the remainder term in the asymptotic formula for the number of square free divisors of an integer.

Smoothing arithmetic error terms: the case of the Euler φ function

“…For instance in [1], [2], [5], [9] and most recently in [3] and [4] the following smoothing operator acting on the linear space X of real valued functions f (x) defined for positive x which are Lebesgue locally integrable and such that 1 0 |f (t)| | log t| N dt t < ∞ for every integer N ≥ 1, played a prominent role:…”

Irrational factor races

“…So we have only one denominator L(2s − 2, χ 2 ) = L(2s − 2, χ 0 ), which does not have any nontrivial real zeros since the Riemann zeta function does not have any. Hence in this case g(s) is regular on the real segment 5 4 − < σ ≤ 3 2 .…”

“…Also, unless at least one of these denominators L(2s −2, χ 2 ) has nontrivial real zeros, the function g(s) will be regular on the real segment 5 4 − < σ ≤ 3 2 . We now return to our particular case, where q = 3.…”

“…Using the above methods, we do not gain any information about the frequency of sign changes of the function S(x; 3, 1) − S(x; 3, 2). A result by Kaczorowski and Wiertelak (see Lemma 3.1 of [5]), allows us to consider this question and further characterize the behavior of the race. Before we state their result, we provide some necessary notation from Kaczorowski and Pintz in [4].…”

Oscillations of the remainder term related to the Euler totient function

“…(1.8) Notice that δ 1 was used also for instance in [5,6,9,11] and most recently in [7] and [8] showing its usefulness in proving results on the distribution of values of various arithmetic error terms. Our first aim is to solve (1.4) explicitly.…”