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

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