Completely multiplicative functions with sum zero over generalised prime systems
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RESEARCH
Completely multiplicative functions with sum zero over generalised prime systems Ammar Ali Neamah * Correspondence:
[email protected]; [email protected] 1 Department of Mathematics, University of Reading, Whiteknights, Reading, UK 2 Faculty of Computer Science and Mathematics, University of Kufa, Najaf, Iraq
Abstract
CMO functions multiplicative functions f for which ∞ n=1 f (n) = 0. Such functions were first defined and studied by Kahane and Saïas [14]. We generalised these to Beurling prime systems with the aim to investigate the theory of the extended functions and we shall call them CMOP functions. We give some properties and find examples of CMOP functions. In particular, we explore how quickly the partial sum of these classes of functions tends to zero with different generalised prime systems. The findings of this paper may suggest that for all CMOP functions f over N with abscissa 1, we have 1 f (n) = √ . x n≤x n∈N
Keywords: Beurling’s generalized primes, Multiplicative functions Mathematics Subject Classification: 11N80, 11N56
1 Introduction 1.1 Completely multiplicative function with sum zero
A function f : N −→ C is called a completely CMO function if it satisfies the two following conditions: f is a completely multiplicative function and
∞
f (n) = 0.
n=1
Such functions were first introduced by Kahane and Saïas [14]. One motivation for them is to gain further insight into the zeros of Dirichlet series with completely multiplicative coefficients. Namely, the Generalised Riemann Hypothesis as discussed below. They also gave some properties and examples of such functions. For instance, they discussed various examples of CMO functions including f (n) = λ(n) n , where λ(n) is the Liouville function and f (n) = χn(n) , where χ is a non-principal Dirichlet character and α is a zero of Lχ with α α > 0. This study led them to consider the question of how quickly n≤x f (n) can tend to zero. They suggested that it is always √1x and the Generalised Riemann Hypothesis– Riemann Hypothesis (GRH–RH) would follow if their statement is true. This suggestion
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