Narrow Results

We study a residual form of a real analytic Siegel–Eisenstein series, which generates a certain derived functor module occurring in a degenerate principal series representation. We compute its Mellin transforms twisted by various Maass wave forms to get explicit formulas as our results. We apply them to prove meromorphic continuations together with functional equations which are satisfied by those twisted Mellin transforms.

A classical theorem of Ramanujan relates an integral of Dedekind eta-function to a special value of a Dirichlet L-function at s = 2. Ahlgren, Berndt, Yee and Zaharescu have generalized this result [1]. In this paper, we generalize this result to the context of holomorphic cusp forms on the upper half space.

It is well known that the Euler product formula for the Riemann zeta function ζ(s) is still valid for ℜ(s) = 1 and s ≠ 1. In this paper, we extend this result to zeta functions of number fields. In particular, we show that the Dedekind zeta function ζ_k(s) for any algebraic number field k can be written as the Euler product on the line ℜ(s) = 1 except at the point s = 1. As a corollary, we obtain the Euler product formula on the line ℜ(s) = 1 for Dirichlet L-functions L(s, χ) of real characters.

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = -N, obtained by analytic continuation of the Dirichlet series , to special values of zeta integrals Z(s;f,g) = ∫_{x∊[0, ∞)^p} g(x)f(x)^-s dx (Re(s) ≫ 0). We prove a simple relation between ζ(-N;f,g) and Z(-N;f_a, g_a), where for a ∈ ℂ^p, f_a(x) is the shifted polynomial f_a(x) = f(a + x). By direct calculation we prove the product rule for zeta integrals at s = 0, degree(fh) ⋅ Z(0;fh, g) = degree(f) ⋅ Z(0;f, g) + degree(h) ⋅ Z(0;h, g), and deduce the corresponding rule for Dirichlet series at s = 0, degree(fh) ⋅ ζ(0;fh, g) = degree(f) ⋅ ζ(0;f, g)+degree(h)⋅ζ(0;h, g). This last formula generalizes work of Shintani and Chen–Eie.

We consider the classical Wiener–Ikehara Tauberian theorem, with a generalized condition of slow decrease and some additional poles on the boundary of convergence of the Laplace transform. In this generality, we prove the otherwise known asymptotic evaluation of the transformed function, when the usual conditions of the Wiener–Ikehara theorem hold. However, our version also provides an effective error term, not known thus far in this generality. The crux of the proof is a proper, asymptotic variation of the lemmas of Ganelius and Tenenbaum, also constructed for the sake of an effective version of the Wiener–Ikehara theorem.

Let ℤ_m be the group of residue classes modulo m. Let s(m, n) denote the total number of subgroups of the group ℤ_m × ℤ_n, where m and n are arbitrary positive integers. We derive asymptotic formulas for the sum ∑_m,n≤x s(m, n) and for the corresponding sum restricted to gcd(m, n) > 1 which concerns the groups ℤ_m × ℤ_n having rank two.

We generalize the correspondence between Dirichlet series with finitely many poles that satisfy a functional equation and automorphic integrals with log-polynomial sum period functions. In particular, we extend the correspondence to hold for Dirichlet series with finitely many essential singularities. We also study Dirichlet series with infinitely many poles in a vertical strip. For Hecke groups with λ ≥ 2 and some weights, we prove a similar correspondence for these Dirichlet series. For this case, we provide a way to estimate automorphic integrals with infinite log-polynomial periods by automorphic integrals with finite log-polynomial periods.

We study the Dirichlet series $F_b(s)={\sum }_{n=1}^{\infty }{d}_b(n)n^{-s}$, where $d_b(n)$ is the sum of the base-b digits of the integer n, and $G_b(s)={\sum }_{n=1}^{\infty }{S}_b(n)n^{-s}$, where $S_b(n)={\sum }_{m=1}^{n-1}{d}_b(m)$ is the summatory function of $d_b(n)$. We show that $F_b(s)$ and $G_b(s)$ have analytic continuations to the plane $ℂ$ as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions $d_b$ and $S_b$ to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

Let ${\mathbb{Z}}_m$ be the additive group of residue classes modulo $m$. For any positive integers $m$ and $n$, let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups and cyclic subgroups of the group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$, respectively. Define

For a natural number $n$, let $Z_1(n):={\sum }_{\rho }\frac{n^{\rho }}{\rho }$ where the sum runs over the nontrivial zeros of the Riemann zeta function. For a primitive Dirichlet character $\chi$ modulo $q\ge 3$, we define $Z_1(s,\chi ):={\sum }_{n=1}^{\infty }\frac{\chi (n)Z_1(n)}{n^s}$ for $\Re (s)>2$ and obtain the meromorphic continuation of the function $Z_1(s,\chi )$ to the region $\Re (s)>\frac{1}{2}$. Our main result indicates that the poles of $Z_1(s,\chi )$ in the region $\frac{1}{2}<\Re (s)<1$, if they exist, are related to the zeros of many Dirichlet $L$-functions in the same region.

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.

Chains of the CME Group Time and Sales E-mini S&P 500 futures tick prices and their a-b-c-d-increments are studied. A discrete probability distribution based on the Hurwitz zeta function and Dirichlet series is suggested for the price increments. The randomness of the ticks is discussed using the notions of typicalness, chaoticness, and stochasticness introduced by Kolmogorov and Uspenskii and developed by predecessors, them, and pupils. They define randomness in terms of the theory of algorithms.