On the divisor function and the Riemann zeta-function in short intervals

DOI10.1007/s11139-008-9142-0zbMath1226.11086arXiv0707.1756OpenAlexW2135545673MaRDI QIDQ1037559

Aleksandar Ivić

Publication date: 16 November 2009

Published in: The Ramanujan Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0707.1756

zbMATH Keywords

upper boundsRiemann zeta-functiondivisor functionspower moments in short intervals

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) (zeta (s)) and (L(s, chi)) (11M06)

Related Items (23)

On the distribution of the divisor function and Hecke eigenvalues ⋮ Mean square estimate for relatively short exponential sums involving Fourier coefficients of cusp forms ⋮ The variance of a restricted sum-of-squares function over short intervals in \(\mathbb{F}_q [t\)] ⋮ On some mean value results for the zeta-function and a divisor problem. II. ⋮ Moments and oscillations of exponential sums related to cusp forms ⋮ The variance and correlations of the divisor function in \(\mathbb{F}_q[T\), and Hankel matrices] ⋮ Bounds and behaviour of the quantities $ P(x)$, $ \Delta(x)$ on short intervals ⋮ Sums of divisor functions in \(\mathbb {F}_q[t\) and matrix integrals] ⋮ On the mean square of short exponential sums related to cusp forms ⋮ On the divisor function in short intervals ⋮ On short sums involving Fourier coefficients of Maass forms ⋮ The class of the exceptional sets for a general asymptotic formula ⋮ On fourth and higher moments of short exponential sums related to cusp forms ⋮ Some mean value results related to Hardy's function ⋮ Large Values of the Zeta-Function on the Critical Line ⋮ On the Dirichlet divisor problem in short intervals ⋮ Circle problem and the spectrum of the Laplace operator on closed 2-manifolds ⋮ Exponential sums related to Maass forms ⋮ On the variance of sums of divisor functions in short intervals ⋮ On the mean square of the divisor function in short intervals ⋮ On certain integrals involving the Dirichlet divisor problem ⋮ Hybrid moments of the Riemann zeta-function ⋮ Resonances and \(\Omega\)-results for exponential sums related to Maass forms for \(\mathrm{SL}(n, \mathbb{Z})\)

Cites Work

• The average number of solutions of the Diophantine equation \(U^2+V^2=W^3\) and related arithmetic functions
• Some remarks on the moments of \(| \zeta(1/2 + it)| \) in short intervals
• Sums of the error term function in the mean square for \(\zeta (s)\)
• Riemann's zeta-function and the divisor problem
• Lectures on a method in the theory of exponential sums
• Lectures on mean values of the Riemann zeta function
• Riemann's zeta-function and the divisor problem. II
• On zeta-functions associated to certain cusp forms. I
• On the higher moments of the error term in the divisor problem
• On the Riemann zeta-function and the divisor problem. II
• The mean-value of the Riemann zeta function
• Three-dimensional exponential sums with monomials
• THE TWELFTH POWER MOMENT OF THE RIEMANN-FUNCTION
• On an asymptotic formula of Srinivasa Ramanujan
• On higher-power moments of Δ (x)
• On the symmetry of the divisor function in almost all short intervals
• On higher-power moments of Δ(x) (II)
• On higher-power moments of E(t)
• On higher-power moments of Δ(x) (III)
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