Sign changes of the error term in Weyl’s law for Heisenberg manifolds

Abstract

Let R(T ) be the error term in Weyl's law for the (2l + 1)-dimensional Heisenberg manifold (H l /Γ, g l ). In this paper, several results on the sign changes and odd moments of R(t) are proved. In particular, it is proved that for some sufficiently large constant c, R(t) changes sign in the interval [T, T +c √ T ] for all large T . Moreover, for a small constant c 1 there exist infinitely many subintervals in [T, 2T ] of length c 1 √ T log −5 T such that ±R(t) > c 1 t l−1/4 holds on each of these subintervals.

On the divisor problem with congruence conditions

“…2 dx, which would play an important role in the proof of Theorem 2.3. This type of integral was studied for the error term in the mean square of ζ( 1 2 + it) by Good [2], for the error term in the Dirichlet divisor problem by Jutila [10] and for the error term in Weyl's law for Heisenberg manifold by Tsang and Zhai [16]. Here we follows the approach of Tsang and Zhai [16] and prove the following Lemma 5.2.…”

On the Dirichlet divisor problem in short intervals

“…where ψ(t) = {t} − 1/2 and {t} is the fractional part of t. Actually we can prove (5.7) from (5.9) without using Lemma 3.1, following the approach given in Tsang-Zhai [22]. And then we prove Theorem 2 and the corollary.…”

“…Proof. We consider only the case of the " + " sign, and follow the method of proof of Tsang and Zhai [22]. Since U ≫ T 131/416+ε , the condition H CT 1/4 UL 5 log L implies H ≫ T 235/416+ε .…”

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

“…As was pointed out in [36], |Δ(x)| in the above can be replaced by Δ + (x) and Δ − (x) respectively (see also [37]). Here for any real-valued function g, g + = 1 2 (|g| + g), g − = 1 2 (|g| − g) denote the positive and negative parts of g respectively.…”