Exact formulas for partial sums of the M\"obius function expressed by partial sums weighted by the Liouville lambda function

Publication date: 10 February 2021

Abstract: The Mertens function,

M (x) : = s u m_{n l e q x} m u (n)

, is defined as the summatory function of the classical M"obius function. The Dirichlet inverse function

g (n) : = (o m e g a + 1)^{- 1} (n)

is defined in terms of the shifted strongly additive function

o m e g a (n)

that counts the number of distinct prime factors of

n

without multiplicity. The Dirichlet generating function (DGF) of

g (n)

is

z e t a (s)^{- 1} (1 + P (s))^{- 1}

for

R e (s) > 1

where

P (s) = s u m_{p} p^{- s}

is the prime zeta function. We study the distribution of the unsigned functions

| g (n) |

with DGF

z e t a (2 s)^{- 1} (1 - P (s))^{- 1}

and

C_{O m e g a} (n)

with DGF

(1 - P (s))^{- 1}

for

R e (s) > 1

. We establish formulas for the average order and variance of

l o g C_{O m e g a} (n)

and prove a central limit theorem for the distribution of its values on the integers

n l e q x

as

x i g h t a r r o w i n f t y

. Discrete convolutions of the partial sums of

g (n)

with the prime counting function provide new exact formulas for

M (x)

.

Has companion code repository: https://github.com/maxieds/MertensFunctionComputations

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Arithmetic functions; related numbers; inversion formulas (11A25) Other results on the distribution of values or the characterization of arithmetic functions (11N64) Distribution functions associated with additive and positive multiplicative functions (11N60)

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