Strange and pseudo-differentiable functions with applications to prime partitions

Anji Dong; Nicolas Robles; Alexandru Zaharescu; Dirk Zeindler

doi:10.1007/s40993-025-00628-8

Strange and pseudo-differentiable functions with applications to prime partitions

Anji Dong
, Nicolas Robles
, Alexandru Zaharescu
, Dirk Zeindler

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let pPr(n) denote the number of partitions of n into r-full primes. We use the Hardy–Littlewood circle method to find the asymptotic of pPr(n) as n→∞. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.

Original language	English (US)
Article number	52
Journal	Research in Number Theory
Volume	11
Issue number	2
Early online date	Apr 26 2025
DOIs	https://doi.org/10.1007/s40993-025-00628-8
State	Published - Jun 2025

Keywords

Exponential sums
Hardy–Littlewood circle method
Inclusion–exclusion
Pseudo-differentiable functions
Strange functions
Von Mangoldt function
Weights associated to partitions
Zeros of the zeta function

ASJC Scopus subject areas

Algebra and Number Theory

Online availability

10.1007/s40993-025-00628-8License: CC BY

Library availability

Discover UIUC Full Text

Cite this

@article{2a7ce828fc5646e08d0ae0d3c6839b3f,

title = "Strange and pseudo-differentiable functions with applications to prime partitions",

abstract = "Let pPr(n) denote the number of partitions of n into r-full primes. We use the Hardy{\textendash}Littlewood circle method to find the asymptotic of pPr(n) as n→∞. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.",

keywords = "Exponential sums, Hardy{\textendash}Littlewood circle method, Inclusion{\textendash}exclusion, Pseudo-differentiable functions, Strange functions, Von Mangoldt function, Weights associated to partitions, Zeros of the zeta function",

author = "Anji Dong and Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler",

note = "DZ was supported by the Leverhulme Trust Research Project Grant RPG-2021-129, and wishes to acknowledge Madhuparna Das for fruitful discussions during the early stages of the manuscript. The authors are grateful to the referees for making valuable suggestions that have greatly increased the clarity and value of the manuscript.",

year = "2025",

month = jun,

doi = "10.1007/s40993-025-00628-8",

language = "English (US)",

volume = "11",

journal = "Research in Number Theory",

issn = "2363-9555",

publisher = "SpringerOpen",

number = "2",

}

TY - JOUR

T1 - Strange and pseudo-differentiable functions with applications to prime partitions

AU - Dong, Anji

AU - Robles, Nicolas

AU - Zaharescu, Alexandru

AU - Zeindler, Dirk

N1 - DZ was supported by the Leverhulme Trust Research Project Grant RPG-2021-129, and wishes to acknowledge Madhuparna Das for fruitful discussions during the early stages of the manuscript. The authors are grateful to the referees for making valuable suggestions that have greatly increased the clarity and value of the manuscript.

PY - 2025/6

Y1 - 2025/6

N2 - Let pPr(n) denote the number of partitions of n into r-full primes. We use the Hardy–Littlewood circle method to find the asymptotic of pPr(n) as n→∞. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.

AB - Let pPr(n) denote the number of partitions of n into r-full primes. We use the Hardy–Littlewood circle method to find the asymptotic of pPr(n) as n→∞. This extends previous results in the literature of partitions into primes. We also show an analogue result involving convolutions of von Mangoldt functions and the zeros of the Riemann zeta-function. To handle the resulting non-principal major arcs we introduce the definition of strange functions and pseudo-differentiability.

KW - Exponential sums

KW - Hardy–Littlewood circle method

KW - Inclusion–exclusion

KW - Pseudo-differentiable functions

KW - Strange functions

KW - Von Mangoldt function

KW - Weights associated to partitions

KW - Zeros of the zeta function

UR - https://www.scopus.com/pages/publications/105003644258

UR - https://www.scopus.com/pages/publications/105003644258#tab=citedBy

U2 - 10.1007/s40993-025-00628-8

DO - 10.1007/s40993-025-00628-8

M3 - Article

AN - SCOPUS:105003644258

SN - 2363-9555

VL - 11

JO - Research in Number Theory

JF - Research in Number Theory

IS - 2

M1 - 52

ER -

Strange and pseudo-differentiable functions with applications to prime partitions

Abstract

Keywords

ASJC Scopus subject areas

Online availability

Library availability

Related links

Fingerprint

Cite this