On the parity of the number of multiplicative partitions

Zaharescu, Alexandru; Zaki, Mohammad

doi:10.4064/aa145-3-2

Cited by 2 publications

(4 citation statements)

References 14 publications

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“…Since the sequence B n of Bell numbers begins B 0 = 1, B 1 = 1, B 2 = 2, it follows that f (N ) is odd for 2/3 of the squarefree numbers (a set of density 4/π 2 ) and f (N ) is even for 1/3 of them (a set of density 2/π 2 ). The constants 4/π 2 and 2/π 2 improve the lower density bounds claimed in [30], which are obtained by more intricate elementary arguments.…”

Section: Paul Pollacksupporting

confidence: 59%

“…Motivated by unsolved problems on the parity distribution of p(N ) (see, e.g., [24], [2], [4], [22]), Zaharescu and Zaki [30] showed that f (N ) is even a positive proportion of the time (in the sense of asymptotic lower density) and odd a positive proportion of the time. Up to 10 7 , about 57 percent of the values of f (N ) are odd, but the arguments of [30] do not suffice to show that there is a limiting proportion of N for which f (N ) is odd.…”

Section: Introductionmentioning

confidence: 99%

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On the parity of the number of multiplicative partitions and related problems

Pollack

2012

Proc. Amer. Math. Soc.

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Let f (N) be the number of unordered factorizations of N , where a factorization is a way of writing N as a product of integers all larger than 1. For example, the factorizations of 30 are 2 • 3 • 5, 5 • 6, 3 • 10, 2 • 15, 30, so that f (30) = 5. The function f (N), as a multiplicative analogue of the (additive) partition function p(N), was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Turán, and others. Recently, Zaharescu and Zaki showed that f (N) is even a positive proportion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression a mod m, the set of N for which f (N) ≡ a(mod m) possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such N. For the case investigated by Zaharescu and Zaki, we show that f is odd more than 50 percent of the time (in fact, about 57 percent).

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Section: Paul Pollacksupporting

confidence: 59%

Section: Introductionmentioning

confidence: 99%

On the parity of the number of multiplicative partitions and related problems

Pollack

2012

Proc. Amer. Math. Soc.

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“…We note that if Dirichlet series coefficients a n are defined by ζ P M (s) =: ∑ n≥1 a n n −s , it is easy to see that a n counts the number of ways to write n as a product of integers in M , where each ordering of factors is only counted once. When M = N, then these ways of writing n as a product of smaller numbers are known as multiplicative partitions, and have been studied in a number of places in the literature; for example, the interested reader is referred to [2,14,34,43,54]. We wish to study partition zeta functions over special subsets of P and arguments s that lead to interesting relations.…”

Section: Partition-theoretic Zeta Functionsmentioning

confidence: 99%

“…it is easy to see that a n counts the number of ways to write n as a product of integers in M , where each ordering of factors is only counted once. When M = N, then these ways of writing n as a product of smaller numbers are known as multiplicative partitions, and have been studied in a number of places in the literature; for example, the interested reader is referred to [2,14,34,43,54].…”

Section: Partition-theoretic Zeta Functionsmentioning

confidence: 99%

Explorations in the Theory of Partition Zeta Functions

Ono

Rolen

Schneider

2017

Exploring the Riemann Zeta Function

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We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the analytic continuations of these "partition zeta functions", find unusual formulas for the Riemann zeta function, prove identities for multiple zeta values, and see that some of the formulas allow for p-adic interpolation. The second family we study was anticipated by Manin and makes use of modular forms, functions which are intimately related to integer partitions by universal polynomial recurrence relations. We survey recent work on these zeta polynomials, including the proof of their Riemann Hypothesis. The setting: Visions of EulerIn antiquity, storytellers began their narratives by invoking the muse whose divine influence would guide the unfolding imagery. It is fitting, then, that we begin this article by praising the immense curiosity of Euler, whose imagination ranged playfully across almost the entire landscape of modern mathematical thought. Euler made spectacular use of product-sum relations, often arrived at by unexpected avenues, thereby inventing one of the principle archetypes of modern number theory. Among his many profound identities is the product formula for what is now called the Riemann zeta function:With this relation, Euler connected the (at the time) cutting-edge theory of infinite series to the ancient set P of prime numbers. Moreover, in solving the famous "Basel problem" posed a century earlier by Pietro Mengoli (1644), Euler showed us how to compute even powers of π using the zeta function, giving explicit formulas of the shape ζ (2N) = π 2N × rational.It turns out there are other classes of zeta functions, arising from other Eulerian formulas in the universe of partition theory. Much like the set of positive integers, but perhaps even more richly, the set of integer partitions ripples with striking patterns and beautiful number-theoretic phenomena. In fact, the positive integers N are embedded in the integer partitions P in a number of ways: obviously, positive integers themselves represent the set of partitions into one part; less trivially, the prime decompositions of integers are in bijective correspondence with the set of prime partitions, i.e., the partitions into prime parts (if we map the number 1 to the empty partition / 0), as Alladi and Erdős note [1]. We might also identify the divisors of n with the partitions of n into identical parts, and there are many other interesting ways to associate integers to the set of partitions.Euler found another profound product-sum identity, the generating function for the partition function p(n)single-handedly establishing the theory of integer partitions. This formula doesn't look much like the zeta function identity (1); however, generalizing Euler's proofs of these theorems leads to a new class of partition-theoretic zeta functions. It turns out that (1) and (3) both arise as specializations of...

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On the parity of the number of multiplicative partitions

Cited by 2 publications

References 14 publications

On the parity of the number of multiplicative partitions and related problems

On the parity of the number of multiplicative partitions and related problems

Explorations in the Theory of Partition Zeta Functions

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