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

Sums of averages of generalized Ramanujan sums

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

An arithmetic function 𝑓 is Leibniz-additive if there is a completely multiplicative function ℎ𝑓 such that 𝑓(𝑚𝑛) = 𝑓(𝑚)ℎ𝑓 (𝑛) + 𝑓(𝑛)ℎ𝑓 (𝑚) for all positive integers 𝑚 and 𝑛. A motiva ...

/pdf/the-arithmetic-derivative-and-leibniz-additive-functions-oucrk75314.pdf

The arithmetic derivative and Leibniz-additive functions

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

Let k and l be non-negative integers. For two completely additive functions f and g, we consider various identities for the Dirichlet convolution of the kth powers of f and the lth powers of g. Furthermore, we derive some asymptotic formulas for sums of convolutions on the natural logarithms.

/pdf/on-the-dirichlet-convolution-of-completely-additive-36wqifspqn.pdf

On the Dirichlet Convolution of Completely Additive Functions

We prove the mean square formula of the Euler--Zagier type double zeta-function 
$\zeta_{2}(s_{1},s_{2})$ and provide an improvement on the $\Omega$ results of Kiuchi, Tanigawa, and Zhai. 
We also calculate the double integral $\int_{2}^{N}\int_{2}^{T}|\zeta_{2}(s_{1},s_{2})|^{2}dt_{1} dt_{2}$ under certain 
conditions.

/pdf/mean-square-formula-for-the-double-zeta-function-165y64k01e.pdf

Mean square formula for the double zeta-function

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.

Makoto Minamide

Papers

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

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

On the Dirichlet Convolution of Completely Additive Functions

Mean square formula for the double zeta-function

On a new circle problem