-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS MARCO AYMONE

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Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET

CHARACTERS

MARCO AYMONE

Abstract. We consider the problem of Ω bounds for the partial sums of a modiﬁed character,

i.e., a completely multiplicative function fsuch that f(p) = χ(p) for all but a ﬁnite number

of primes p, where χis a primitive Dirichlet character. We prove that in some special circum-

stances, Pn≤xf(n) = Ω((log x)|S|), where Sis the set of primes pwhere f(p)6=χ(p). This

gives credence to a corrected version of a conjecture of Klurman et al., Trans. Amer. Math.

Soc., 374 (11), 2021, 7967–7990. We also compute the Riesz mean of order kfor large kof a

modiﬁed character, and show that the Diophantine properties of the irrational numbers of the

form log p/ log q, for primes pand q, give information on these averages.

1. Introduction.

A quite simple statement about a completely multiplicative1function, and that was proved

only recently, is that no matter how we choose the values f(p) at primes pon the unit circle, we

will always end up with a completely multiplicative function whose partial sums are unbounded,

i.e.,

lim sup

x→∞ X

n≤x

f(n)

=∞.

This result was proved in 2015 by Tao [8] in the context of the Erd˝os discrepancy problem.

Dirichlet characters2play an important role in the Erd˝os discrepancy Theory: The non-principal

characters χare completely multiplicative and have bounded partial sums. But of course they

are not a counterexample to the result of Tao since they vanish at a ﬁnite subset of primes.

However, they seem to be extremal in discrepancy theory due to the following fact: Up to this

date, the known example of a completely multiplicative fwith |f|= 1 whose partial sums

have the lowest known ﬂuctuation is obtained by adjusting a non-principal character χat the

the primes pwhere χ(p) = 0. For instance (see Borwein-Choi-Coons [2]) we can take the non-

principal character mod 3, say χ3, and deﬁne f(3) = ±1 and f(p) = χ3(p) for all the other

primes. The partial sums up to xof this modiﬁcation of χ3are log x.

1A function f:N→Cis completely multiplicative if f(nm) = f(n)f(m) for all positive integers nand m,

and hence, such functions are determined by its values at primes.

2Dirichlet characters of modulus q, often denoted by χ, are q-periodic completely multiplicative functions

such that χ(n) = 0 whenever gcd(n, q)>1.

arXiv:2210.06153v2 [math.NT] 25 Apr 2023

2 MARCO AYMONE

Going further, we deﬁne:

Deﬁnition 1.1 (Modiﬁed characters).We say that f:N→ {z∈C:|z| ≤ 1}is a modiﬁed

character if fis completely multiplicative, and if there is a Dirichlet character χand a ﬁnite

subset of primes Ssuch that:

(1) For all primes p∈S,f(p)6=χ(p).

(2) For all primes p /∈S,f(p) = χ(p).

In this case we also say that fis a modiﬁcation of χwith modiﬁcation set S.

One can easily show (see [1] for a proof using a Tauberian result) that for a modiﬁed

character we have

(1) X

n≤x

f(n)(log x)|S|.

A very open question concerns Ω bounds for the partial sums of such modiﬁcations.

The ﬁrst treatment to such an Ω bound is due to Borwein-Choi-Coons [2] where they

considered the case |S|= 1 and showed that for a quadratic Dirichlet character χwith prime

modulus q, if we set f(p) = χ(p) for all primes except at q, where f(q) = 1, then the partial

sums

n≤x

f(n) = Ω(log x).

Later, Klurman et al. [6], among other results, proved a stronger form of Chudakov’s con-

jecture [5], and also conjectured that the partial sums of a modiﬁed character fare actually

Ω((log x)|S|). Their Corollary 1.6 states that the partial sums are Ω(log x) if the set of modi-

ﬁcations Sis ﬁnite and for at least one prime p∈S,|f(p)|= 1. Their proof is correct when

the Dirichlet character is primitive, but this statement is not true for all characters. This is

because, if we modify a non-primitive character χat a prime rwhere χ(r) = 0, then there

are circumstances where the modiﬁcation fmight end up in another character with smaller

modulus. Therefore, we restate their conjecture in the following form.

Conjecture 1.1. Let fbe a modiﬁcation of a primitive Dirichlet character χ. Assume that

for each prime pin the set of modiﬁcations Swe have |f(p)|= 1. Then

n≤x

f(n) = Ω((log x)|S|).

The ﬁrst result of this paper gives credence to this conjecture by showing that it is true in

some very special circumstances.

Ω-BOUNDS FOR THE PARTIAL SUMS OF SOME MODIFIED DIRICHLET CHARACTERS 3

Corollary 1.1. Let fbe a modiﬁcation of a primitive Dirichlet character χsuch that χ(−1) =

−1. If for each prime pin the set of modiﬁcations Swe have f(p) = +1, then

n≤x

f(n) = Ω((log x)|S|).

The result above is a direct consequence of the slightly more general Theorem below.

Theorem 1.1. Let fbe a modiﬁcation of a primitive Dirichlet character χsuch that for each

prime pin the set of modiﬁcations S,|f(p)|= 1. Let

T=X

p∈S

f(p)=1

1−X

p∈S

χ(p)=1

Let

N=





max{0, T }, if χ(−1) = −1,

max{0, T −1},if χ(−1) = 1.

Then

n≤x

f(n) = Ω((log x)N).

One of the reasons behind the result above is that the Dirichlet series of f, say F(s), can

be written as

(2) F(s) = Y

p∈S

1−χ(p)

1−f(p)

ps!L(s, χ),

and the proof consists in analyzing the behaviour of F(s) as s→0+, and hence, the functional

equation for L(s, χ) is important here.

1.1. Riesz means and irrationality of the numbers log p/ log q.Another way to measure

the mean behaviour of a sequence (f(n))nis by its Riesz means. The Riesz mean of order 1 is

deﬁned as

R1(x) := X

n≤x

f(n) log(x/n).

After partial summation it is not diﬃculty to see that R1(x) is equal to log xtimes the loga-

rithmic average of Pn≤xf(n):

R1(x) = log x×1

log xZx

1 X

n≤t

f(n)!dt

Therefore, it may regarded as a smooth average of f(n).

Going further, we can deﬁne (see the book of Montgomery and Vaughan [7]) the Riesz mean

of order k:

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-BOUNDSFORTHEPARTIALSUMSOFSOMEMODIFIEDDIRICHLETCHARACTERSMARCOAYMONEAbstract.Weconsidertheproblemofboundsforthepartialsumsofamodiedcharacter,i.e.,acompletelymultiplicativefunctionfsuchthatf(p)=(p)forallbutanitenumberofprimesp,whereisaprimitiveDirichletcharacter.Weprovethatinsomespecialcircum-sta...

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