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pro vyhledávání: '"Schmidt, Maxie"'
Autor:
Merca, Mircea, Schmidt, Maxie D.
Publikováno v:
Amer. Math. Monthly 125 10: 929-933 (2018)
In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this context.
Externí odkaz:
http://arxiv.org/abs/2310.13672
Autor:
Merca, Mircea, Schmidt, Maxie D.
Publikováno v:
Ramanujan J. 49: 87-96 (2019)
In this paper, we investigate decompositions of the partition function $p(n)$ from the additive theory of partitions considering the famous M\"{o}bius function $\mu(n)$ from multiplicative number theory. Some combinatorial interpretations are given i
Externí odkaz:
http://arxiv.org/abs/2310.13658
Autor:
Schmidt, Maxie Dion
This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis explores th
Externí odkaz:
http://arxiv.org/abs/2209.12287
We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a theorem of
Externí odkaz:
http://arxiv.org/abs/2201.11274
Autor:
Schmidt, Maxie Dion
The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly additive function
Externí odkaz:
http://arxiv.org/abs/2102.05842
Autor:
Schmidt, Maxie Dion
A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side terms of this
Externí odkaz:
http://arxiv.org/abs/2004.02976
Akademický článek
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Autor:
Mousavi, Hamed, Schmidt, Maxie D.
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansi
Externí odkaz:
http://arxiv.org/abs/1810.08373
Autor:
Schmidt, Maxie D.
The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a
Externí odkaz:
http://arxiv.org/abs/1809.03933