Large values of certain number-theoretic error terms. (English) Zbl 0659.10053

Ivić, Aleksandar

Acta Arith. 56, No. 2, 135-159 (1990).

Large values of certain error terms in asymptotic formulas are investigated. These include error terms pertaining to Dirichlet series satisfying functional equations of the Chandrasekharan-Narasimhan type \((\Delta (s)\phi (s)=\Delta (r-s)\psi (r-s),\) \(\Delta\) a product of gamma-factors) and E(T), the error term in the mean square formula for \(| \zeta (+it)|\) [see Ch. 15 of the author’s book “The Riemann zeta-function” (Wiley, New York 1985; Zbl 0556.10026)]. The main result provides \(\Omega_{\pm}\)-results for the above error terms in intervals \([x, x+x^{\alpha}],\) where \(0<\alpha <1\) and \(\alpha\) is precisely defined.
Specializing to \(\Delta_ k(x)\) (the error term for the sum of \(d_ k(n)\), the general divisor function) one obtains that for \(k\geq 2\) fixed there exist two constants \(B,C>0\) such that for \(T\geq T_ 0\) the interval \([T, T+CT^{(k-1)/k}]\) always contains points \(t_ 1\), \(t_ 2\) for which \[ \Delta_ k(t_ 1)>Bt_ 1^{(k-1)/(2k)},\quad \Delta_ k(t_ 2)\quad <\quad -Bt_ 2^{(k-1)/(2k)}. \] As regards E(T), define \(G(T)=\int^{T}_{2}E(t)dt-\pi T\) \((=O(T^{3/4})\) and \(=\Omega_{\pm}(T^{3/4})\). Then there exist constants \(B,C>0\) such that \([T, T+C\sqrt{T}]\) contains \(\tau_ 1,\tau_ 2,\tau_ 3,\tau_ 4\) for which \[ E(\tau_ 1)>B\tau_ 1^{1/4},\quad E(\tau_ 2)<- B\tau_ 2^{1/4},\quad G(\tau_ 3)>B\tau_ 3^{3/4},\quad G(\tau_ 4)<-B\tau_ 4^{3/4}. \] Moreover, if \(u_ n\) is the n-th zero of G(T), then \[ \limsup_{n\to \infty}\frac{\log (u_{n+1}-u_ n)}{\log u_ n}=\frac{1}{2}. \] Several new results on \(A(x)=\sum_{n\leq x}a(n)\) are obtained, where a(n) are the Fourier coefficients of a cusp form for the full modular form. These include the \(\Omega_{\pm}\)-results analogous to the above results for \(\Delta_ 2(x)\), power moments of A(x), and results on the mean square formula for A(x).

Reviewer: A.Ivić

MSC:

11N37	Asymptotic results on arithmetic functions
11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11F11	Holomorphic modular forms of integral weight

Keywords:

Dirichlet series with functional equation; Riemann zeta-function; Fourier coefficients of cusp forms; Omega-estimates; error terms in asymptotic formulas; mean square formula; divisor function; Fourier coefficients; cusp form