On the pairs of completely multiplicative functions satisfying some relation

Катаи, И.; Phong, Bui Minh

doi:10.14232/actasm-018-279-1

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…This was proved in [20] by the first author. Later, the result was generalized by Kátai and Phong [17] who proved that if

f,g\colon \mathbb {N}\rightarrow S^1

are completely multiplicative and

\begin{align} \sum _{n\leqslant x}|g(2n+1)-zf(n)|=o(x) \end{align}

for some complex number z , then

f(n)=g(n)=n^{it}

. Since in the function field setting there are two varieties of Archimedean characters, namely

e_{\theta }

and short interval characters ξ, our classification of completely multiplicative functions satisfying (6) (and in fact more generally (7)) in function fields takes a slightly different form.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Correlations of multiplicative functions in function fields

2022

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We develop an approach to study character sums, weighted by a multiplicative function f0pt:Fq[t]→S1$f\colon \mathbb {F}_q[t]\rightarrow S^1$, of the form ∑degfalse(Gfalse)=NG0.33emmonicffalse(Gfalse)χfalse(Gfalse)ξfalse(Gfalse),$$\begin{align*}\hskip7pc \sum _{\substack{\textnormal {deg}(G) = N \\ G \text{ monic}}}f(G)\chi (G)\xi (G), \end{align*}$$where χ is a Dirichlet character and ξ is a short interval character over Fq[t]$\mathbb {F}_q[t]$. We then deduce versions of the Matomäki–Radziwiłł theorem and Tao's two‐point logarithmic Elliott conjecture over function fields Fq[t]$\mathbb {F}_q[t]$, where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin–Shusterman on correlations of the Möbius function for various values of q. Compared with the integer setting, we encounter a different phenomenon, specifically a low characteristic issue in the case that q is a power of 2. As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the existing one in the integer case. In a companion paper, we use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve a “corrected” form of the Erdős discrepancy problem over Fq[t]$\mathbb {F}_q[t]$.

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“…This was proved in [20] by the first author. Later, the result was generalized by Kátai and Phong [17] who proved that if

f,g\colon \mathbb {N}\rightarrow S^1

are completely multiplicative and

\begin{align} \sum _{n\leqslant x}|g(2n+1)-zf(n)|=o(x) \end{align}

for some complex number z , then

f(n)=g(n)=n^{it}

. Since in the function field setting there are two varieties of Archimedean characters, namely

e_{\theta }

and short interval characters ξ, our classification of completely multiplicative functions satisfying (6) (and in fact more generally (7)) in function fields takes a slightly different form.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…This was proved in [20] by the first author. Later, the result was generalized by Kátai and Phong [17] who proved that if 𝑓, g ∶ ℕ → 𝑆 1 are completely multiplicative and…”

mentioning

confidence: 97%

Correlations of multiplicative functions in function fields

2022

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“…Shortly after Klurman's 2017 result became known, Kátai and Phong [2] used his result to prove that if f, g ∈ M *…”

Section: Consequences Of Klurman's Resultsmentioning

confidence: 99%

On the variations of completely multiplicative functions at consecutive arguments

Koninck¹,

Катаи²,

Phong³

2021

Publ. Math. Debrecen

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We focus on the class M * 1 of completely multiplicative functions f whose set of values belong to the unit circle and their related function ∆f (n) := f (n + 1) − f (n). For such functions f , we study the higher iterations ∆ m f (n) for fixed integers m ∈ {2, 3, . . . , 7}, and for each of these we establish an absolute bound for |∆ m f (n)|. We also characterise those triplets of multiplicative functions f, g, h with unusually small gaps between their consecutive values. All our characterisations and bounds are obtained following new results of O. Klurman and A. P. Mangerel in the context of their proof of an old conjecture of Kátai characterising subclasses of M * 1 .

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“…then f (n) = n it for some real number t. This was proved in [16]. Later, the result was generalized by Kátai and Phong [13] who proved that if f, g : N → S 1 are completely multiplicative and…”

Section: Remarksmentioning

confidence: 88%

Correlations of multiplicative functions in function fields

Klurman,

Mangerel,

Teräväinen

2020

Preprint

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We develop an approach to study character sums, weighted by a multiplicative function f : Fq[t] → S 1 , of the formwhere χ is a Dirichlet character and ξ is a short interval character over Fq [t]. We then deduce versions of the Matomäki-Radziwi l l theorem and Tao's two-point logarithmic Elliott conjecture over function fields Fq[t], where q is fixed. The former of these improves on work of Gorodetsky, and the latter extends the work of Sawin-Shusterman on correlations of the Möbius function for various values of q.Compared with the integer setting, we encounter some different phenomena, specifically a low characteristic issue in the case that q is a power of 2, as well as the need for a wider class of "pretentious" functions called Hayes characters.As an application of our results, we give a short proof of the function field version of a conjecture of Kátai on classifying multiplicative functions with small increments, with the classification obtained and the proof being different from the integer case.In a companion paper, we will use these results to characterize the limiting behavior of partial sums of multiplicative functions in function fields and in particular to solve the "corrected" form of the Erdős discrepancy problem over Fq[t]. 1 A function f (G) is called a factorization function if it only depends on the values of deg(P ) and vP (G), where P runs through the irreducible divisors of G, and vP (G) denotes the largest integer k with P k | G.

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On the pairs of completely multiplicative functions satisfying some relation

Cited by 4 publications

References 6 publications

Correlations of multiplicative functions in function fields

Correlations of multiplicative functions in function fields

On the variations of completely multiplicative functions at consecutive arguments

Correlations of multiplicative functions in function fields

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