On an Arithmetic Function (II)

1927 ◽  
Vol s1-2 (2) ◽  
pp. 123-130 ◽  
Author(s):  
A. Oppenheim
Keyword(s):  
Arithmetic Function

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

A Note on Dirichlet Convolutions

1966 ◽  
Vol 9 (4) ◽  
pp. 457-462
Author(s):  
S. L. Segal
Keyword(s):  
Arithmetic Function ◽  
General Theorem ◽  
Möbius Function ◽  
Order Condition ◽  
Mobius Function

In [3] Rubel proved that if h(n) is an arithmetic function such that , L finite, then where μ(n) is the Mobius function. This result was extended to functions other than μ(n) in [4]; however, (as first pointed out to the author by Benjamin Volk), the order condition imposed there is unnecessary; in fact, utilizing the result of [3], the following slightly more general theorem has an almost trivial proof.

Footnote to a Formula of Gioia and Subbarao

1967 ◽  
Vol 10 (5) ◽  
pp. 749-750
Author(s):  
S. L. Segal
Keyword(s):  
Explicit Formula ◽  
Arithmetic Function ◽  
Fixed Integer ◽  
Simple Idea ◽  
Image Position ◽  
Multiplicative Formula

Recently Gioia and Subbarao [2] studied essentially the following problem: If g(n) is an arithmetic function, and , then what is the behaviour of H(a, n) defined for each fixed integer a ≥ 2 by1By using Vaidyanathaswamy′s formula [e.g., 1], they obtain an explicit formula for H(a, n) in case g(n) is positive and completely multiplicative (Formula 2.2 of [2]). However, Vaidyanathaswamy′s formula is unnecessary to the proof of this result, which indeed follows more simply without its use, by exploiting a simple idea used earlier by Subbarao [3] (referred to also in the course of [2]).

scholarly journals Some Identities on the Poly-Genocchi Polynomials and Numbers

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1007 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Lee-Chae Jang
Keyword(s):  
Arithmetic Function ◽  
Bernoulli Polynomials ◽  
Genocchi Polynomials

Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them.

scholarly journals Local-global principles for zero-cycles on homogeneous spaces over arithmetic function fields

2019 ◽  
Vol 372 (8) ◽  
pp. 5263-5286
Author(s):  
J.-L. Colliot-Thélène ◽  
D. Harbater ◽  
J. Hartmann ◽  
D. Krashen ◽  
R. Parimala ◽  
...  
Keyword(s):  
Homogeneous Spaces ◽  
Arithmetic Function ◽  
Function Fields ◽  
Global Principles

On a arithmetic function

1959 ◽  
Vol 63 (3) ◽  
pp. 228-230 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Arithmetic Function

scholarly journals Local-global Galois theory of arithmetic function fields

2019 ◽  
Vol 232 (2) ◽  
pp. 849-882
Author(s):  
David Harbater ◽  
Julia Hartmann ◽  
Daniel Krashen ◽  
Raman Parimala ◽  
Venapally Suresh
Keyword(s):  
Galois Theory ◽  
Arithmetic Function ◽  
Function Fields

Inverse Problems for Partition Functions

2001 ◽  
Vol 53 (4) ◽  
pp. 866-896
Author(s):  
Yifan Yang
Keyword(s):  
Asymptotic Behavior ◽  
Partition Function ◽  
Inverse Problems ◽  
Large Class ◽  
Generating Function ◽  
Riemann Hypothesis ◽  
Arithmetic Function ◽  
Partition Functions ◽  
Class Of Functions ◽  
Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

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