Related papers: On Louchard's Asymptotic Series

We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity…

It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function $\prod_{k=1}^\infty (1+x^k/k)$ converges to a limit $C > 0$. $C$ can be expressed as $C = \exp\Bigl(-\sum_{k = 2}^\infty…

We provide rapidly converging formulae for the Riemann zeta function at odd integers using the Lambert series $\mathscr{L}_q(s) = \sum_{n=1}^\infty n^{s} q^{n}/(1-q^n)$, $s=-(4k\pm 1)$. Our main formula for $\zeta(4k-1)$ converges at rate…

Lindel{\"o}f's hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann's zeta function $\zeta(1/2+it)$ is of order $O(t^{\varepsilon})$ for any $\varepsilon>0$ . It is well known…

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…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their values.

We consider, for even $s$, the secant Dirichlet series $\psi_s (\tau) = \sum_{n = 1}^{\infty} \frac{\sec (\pi n \tau)}{n^s}$, recently introduced and studied by Lal\'{\i}n, Rodrigue and Rogers. In particular, we show, as conjectured and…

We introduce a one-parameter family of series associated to the Riemann $\zeta$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the…

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1\psi_1 summation formula we establish a new Lambert series identity for…

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function on $\left[0 , \frac{\pi}{2} \right]$ may be determined by a finite or infinite sum involving the Riemann…

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_\gamma\,\gamma^{-i\tau}$, where the summation is over the ordinates of the nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function. In…

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently…

An elementary approach for computing the values at negative integers of the Riemann zeta function is presented. The approach is based on a new method for ordering the integers and a new method for summation of divergent series. We show that…

We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the H\"older mean based on the functional equation for the Riemann zeta function. We find a harmonic mean…

The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$.…

Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is…

Let $f_1,\ldots,f_k : \mathbb{N} \rightarrow \mathbb{C}$ be multiplicative functions taking values in the closed unit disc. Using an analytic approach in the spirit of Hal\'{a}sz' mean value theorem, we compute multidimensional averages of…

In this paper, we derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\dfrac{r(n)}{r(n+1)} = {x}{(\ln x)^{-3/4}}(c+o(1)),\ \ x \to +\infty,$$ where $r(n)$ is the number of representations of $n$ as a sum of two…

We study asymptotic spectral properties of the generalized Hilbert $L$-matrix \[ L_{n}(\nu)=\left(\frac{1}{\max(i,j)+\nu}\right)_{i,j=0}^{n-1}, \] for large order $n$. First, for general $\nu\neq0,-1,-2,\dots$, we deduce the asymptotic…