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2 corrected spelling, corrected example

# AnanlyticAnalytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1

A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in addition $\limsup \frac{\log \lvert a_1 + \ldots + a_n \rvert}{\log n} = 0$, then the series is (conditionally) convergent for $\Re s > 0$. (For example, if $a_n$ coincides with a non-principal Legendre character $\mod p$ when $p \nmid n$, and $a_p$ is set arbitrarily, then the partial sums grow logarithmically and we have convergence for $\Re s > 0$.)

I would like to know whether it is possible to get something just a tiny bit stronger. Is there a completely multiplicative $\mathbb{T}$-valued sequence $a_n$ such that the function $f(s)$ derived from it can be analytically continued to a region that includes the closed half-plane $\Re s \geq 0$? Or to put it another way, must $f(s)$ have a singularity on the imaginary axis?

The question is motivated by the question of whether there exists a completely multiplicative sequence in $\mathbb{T}$ with bounded partial sums, which arises naturally in relation to the Erdos Discrepancy Problem.

If the question is hard, are there any known results in this direction? For example, if we relax the multiplicativity condition?

1

# Ananlytic continuation of Dirichlet series with completely multiplicative coefficients of modulus 1

A sequence $a_n \in \mathbb{T}$ ($n \geq 1$) satisfying $a_{mn} = a_m a_n$ for all $m, n \geq 1$ defines a Dirichlet series $f(s) = \sum_n a_n n^{-s}$, absolutely convergent for $\Re s > 1$. If in addition $\limsup \frac{\log \lvert a_1 + \ldots + a_n \rvert}{\log n} = 0$, then the series is (conditionally) convergent for $\Re s > 0$. (For example, if $a_n$ coincides with a Legendre character $\mod p$ when $p \nmid n$, and $a_p$ is set arbitrarily, then the partial sums grow logarithmically and we have convergence for $\Re s > 0$.)

I would like to know whether it is possible to get something just a tiny bit stronger. Is there a completely multiplicative $\mathbb{T}$-valued sequence $a_n$ such that the function $f(s)$ derived from it can be analytically continued to a region that includes the closed half-plane $\Re s \geq 0$? Or to put it another way, must $f(s)$ have a singularity on the imaginary axis?

The question is motivated by the question of whether there exists a completely multiplicative sequence in $\mathbb{T}$ with bounded partial sums, which arises naturally in relation to the Erdos Discrepancy Problem.

If the question is hard, are there any known results in this direction? For example, if we relax the multiplicativity condition?