DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

/pdf/the-riemann-zeta-function-3l59kypwcz.pdf

The Riemann zeta function

The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volume.

Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions

We first report on computations made using the GP/PARI package that show that the error term ∆(x) in the divisor problem is $= \mathscr{M} (x, 4) + O^* (0.35 x^{1/4} \log x)$ when $x$ ranges $[1 081 080, 10^{10} ]$, where $\mathscr{M }(x, 4)$ is a smooth approximation. The remaining part (and in fact most) of the paper is devoted to showing that $|\Delta(x)| \le 0.397 x^{1/2}$ when $x \ge 5 560$ and that $|\Delta(x)| \le 0.764 x^{1/3}\log x$ when $x\ge 9 995$. Several other bounds are also proposed. We use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function. We finally formulate a positivity conjecture concerning

/pdf/explicit-upper-bounds-for-the-remainder-term-in-the-divisor-4cxpumtr44.pdf

Explicit upper bounds for the remainder term in the divisor problem

The average behaviour of the error term in a new kind of the divisor problem

Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann’s zeta function �(s). To do this, we explore properties of some Sheffer’s sequences of polynomials related to the Kontorovich-Lebedev transform.

/pdf/certain-identities-connection-and-explicit-formulas-for-the-y34zm6kia5.pdf

Certain identities, connection and explicit formulas for the Bernoulli and Euler numbers and the Riemann zeta-values

Abstract In this paper, we shall derive representations of the Chowla-Walum type formula for the error term in a divisor problem related to the derivatives of the Riemann zeta-function. As a direct corollary of this formula, we shall consider estimations of this error term.

Representations and evaluations of the error term in a certain divisor problem

In this paper, we shall investigate several properties of integrals defined by $\int_1^{\infty}t^{-\theta}P(t)\log^jtdt$ with a complex variable $\theta$ and a non-negative integer $j$, where $P(x)$ is the error term in the circle problem of Gauss. We shall also study the analytic continuation of several types of the Dirichlet series related with the circle problem, and study a proof of the functional equation of the Dedekind zeta-function associated with the Gaussian number field ${\mathbb{Q}}(\sqrt{-1})$.

/pdf/on-integrals-and-dirichlet-series-obtained-from-the-error-3s43d3kbm5.pdf

On integrals and Dirichlet series obtained from the error term in the circle problem

Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$

/pdf/on-the-k-free-divisor-problem-4b6x0z9e5i.pdf

On the k-free divisor problem

In this paper we consider the Dirichlet series obtained from the error term in the Dirichlet divisor problem. We shall prove their analytic continuations and determine the locations of poles. We shall also discuss the relation between our results and the conjecture of Lau and Tsang on the mean square estimate of the error term.

Dirichlet series obtained from the error term in the Dirichlet divisor problem

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$
 denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$
 ( $\text{Re}\,s>1$
 ) and $L(s,\unicode[STIX]{x1D712}_{4})$
 the Dirichlet $L$
 -function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$
 ( $\text{Re}\,s>1$
 ) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$
 by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$
 $(\text{Re}\,s>1)$
 . This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$
 . In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$
 . As a new problem, we deduce a ‘truncated Voronoi formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$
 . As a direct application, we show the mean square for the error term in our new problem.

Jun Furuya

Papers

Representations and evaluations of the error term in a certain divisor problem

On integrals and Dirichlet series obtained from the error term in the circle problem

On the k-free divisor problem

Dirichlet series obtained from the error term in the Dirichlet divisor problem

On a new circle problem