Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$ \int_0^T\Delta^k(t)|\zeta(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N) $$ are given. This complements the results of the paper Ivi\'c-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.

On some upper bounds for the zeta-function and the Dirichlet divisor problem

For Re s = σ > 1, theRiemann zeta function ζ(s)is defined by 

$$ \zeta \left( s \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}} . $$

It follows from the definition that ζ(s) is an analytic function in the half-plane Re s > 1.

The Riemann Zeta-Function

/pdf/the-mean-value-of-the-riemann-zeta-function-10z5k31peo.pdf

The mean-value of the Riemann zeta function

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Spectral Theory of the Riemann Zeta-Function

/pdf/the-distribution-and-moments-of-the-error-term-in-the-xkyz28q3ui.pdf

On some upper bounds for the zeta-function and the Dirichlet divisor problem

References

The Riemann Zeta-Function

The mean-value of the Riemann zeta function

Spectral Theory of the Riemann Zeta-Function

The distribution and moments of the error term in the Dirichlet divisor problem

On the Fourth Power Moment of the Riemann Zeta-Function

Related Papers (5)

On some mean value results for the zeta-function and a divisor problem II

On some mean value results for the zeta-function and a divisor problem

On certain integrals involving the Dirichlet divisor problem

On the error term $$\Delta _{(k,l)}(x)$$ Δ ( k , l ) ( x )

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