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Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums

Abstract

We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of \(q^n\) as \(\sum _{d|n} f(d)\) for some arithmetic function f. Our new factorization theorems provide analogs to these established expansions generating corresponding sums of the form \(\sum _{d: (d,n)=1} f(d)\) (type I sums) and the Anderson–Apostol sums \(\sum _{d|(m,n)} f(d) g(n/d)\) (type II sums) for any arithmetic functions f and g. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function p(n) to arbitrary arithmetic functions f. We conclude the last section of the article by directly expanding new formulas for an arithmetic function g by the type II sums using discrete, and discrete time, Fourier transforms (DFT and DTFT) for functions over inputs of greatest common divisors.

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Notes

  1. 1.

    We also have a related identity which allows us to interchange the order of summation in the Anderson–Apostol sums of the following form for any natural numbers \(x \ge 1\) and arithmetic functions \(f,g,h: \mathbb {N} \rightarrow {{\mathbb {C}}}\):

    $$\begin{aligned} \sum _{d=1}^x f(d) \sum _{r|(d,x)} g(r) h\left( \frac{d}{r}\right) = \sum _{r|x} g(r) \sum _{d=1}^{x/r} h(d) f\left( \gcd (x,r) d\right) . \end{aligned}$$

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Appendix A: notation and conventions in the article

Appendix A: notation and conventions in the article

Symbol Definition
\({a_k(f, g; n)}\) Discrete Fourier coefficients of the periodic divisor sums \(s_k(f, g; n)\) defined on Sect. 4.3 and as symbol \(s_{k}(f; g; n)\) in this glossary. The precise definition of these sums is given by \(a_k(f, g; n) = \sum \nolimits _{d|(k, n)} g(d) f(n/d) \frac{d}{k}\)
\(a_{k,\ell }\) Sequence of coefficients that are defined explicitly on Sect. 4.3 in the discrete Fourier series expansion of the type II sums \(L_{f,g,k}(x)\). These coefficients are implicitly defined by Definition 4.13 by the sums \(L_{f,g,k}(n) = \sum \nolimits _{\ell =0}^{k-1} a_{k,\ell } \cdot e(\ell n/k)\), where e(x) is the shorthand for the complex exponential terms in the exponential sums we define in the article
\(\lceil x \rceil \) The ceiling function \(\lceil x \rceil := x + 1 - \{x\}\) where \(0 \le \{x\} < 1\) denotes the fractional part of \(x \in \mathbb {R}\)
\(\upchi _{1,k}(n)\) The principal Dirichlet character modulo k, i.e., the indicator function of the natural numbers which are relatively prime for \(n,k \ge 1\), \(\upchi _{1,k}(n) = \left[ (n,k)=1\right] _{\delta }\)
\(C_k(n)\) Sequence of nested k-convolutions of an arithmetic function f with itself defined on Sect. 4.2. The precise definition of this sequence is given by \(C_k(n) = {\left\{ \begin{array}{ll} \widehat{f}(n) - \widehat{f}(1)\varepsilon (n), &{} \text { if }k = 1; \\ \sum \nolimits _{d|n} \left( \widehat{f}(d) - \widehat{f}(1) \varepsilon (d)\right) C_{k-1}(n/d), &{} \text { if }k \ge 2, \end{array}\right. }\) where the symbol \(\hat{f}(n)\) is defined in glossary entry \(\hat{f}(n)\)
\([q^n] F(q)\) The coefficient of \(q^n\) in the power series expansion of F(q) about zero
\(c_q(n)\) Ramanujan’s sum, \(c_q(n) := \sum \nolimits _{d|(q,n)} d \mu (q/d)\)
\(D_f(n)\) Function related to the Dirichlet inverse of a function f defined on Sect. 4.2. More precisely, this function is defined by the sum \(D_f(n) := \sum \nolimits _{j=1}^n \frac{{\text {ds}}_{2j}(f; n)}{\hat{f}(1)^{2j+1}}\), where this definition involves the glossary symbols \(\hbox {ds}_j(f; n)\) and \(\hat{f}(n)\). Lemma 4.7 relates this function to the Dirichlet inverse of the function \(\hat{f}(n)\)
d(n) The ordinary divisor function, \(d(n) := \sum _{d|n} 1\)
\(\hbox {ds}_j(f; n)\) Summands in the formula for the Dirichlet inverse of an arithmetic function defined on Sect. 4.2. The precise definition of this function is given by \({{\,\mathrm{ds}\,}}_j(f; n) = {\left\{ \begin{array}{ll} (-1)^{\delta _{n,1}} \widehat{f}(n), &{} \text { if }j = 1; \\ \sum \nolimits _{\begin{array}{c} d|n \\ d>1 \end{array}} \widehat{f}(d) {{\,\mathrm{ds}\,}}_{j-1}\left( f; \frac{n}{d}\right) ,&\text { if }j \ge 2, \end{array}\right. }\) where the fixed function \(\hat{f}\) is defined by glossary symbol \(\hat{f}(n)\)
\(\hbox {DFT}[f](k)\) The discrete Fourier transform (DFT) of f at k. We use this transformation in Sect. 4.3 of the article.
\(\hbox {DTFT}[f](k)\) The discrete time Fourier transform (DTFT) of f at k, also denoted by F[k].
\(\varepsilon (n)\) The multiplicative identity with respect to Dirichlet convolution, \(\varepsilon (n) = \delta _{n,1}\)
e(x) The complex exponential function, \(e(x) := \exp (2\pi \imath \cdot x)\)
\(f *C_{-}(m)\) This notation indicates that the index over which we perform the Dirchlet convolution is given by the dash parameter, \((f *C_{-}(m))(n) := \sum _{d|n} f(d) C_{n/d}(m)\)
\(f *C_k(-)\) This notation indicates that the index over which we perform the Dirchlet convolution is given by the dash parameter, \((f *C_k(-))(n) := \sum _{d|n} f(d) C_k(n/d)\)
\(*; f *g\) The Dirichlet convolution of f and g, \(f *g(n) := \sum \nolimits _{d|n} f(d) g(n/d)\), for \(n \ge 1\). This symbol for the discrete convolution of two arithmetic functions is the only notion of convolution of functions we employ within the article
\(\hat{f}(n)\) A shorthand notation for scaled arithmetic function terms \(\hat{f}(n) := w^n / (w^n-1) f(n)\) for some indeterminate w. The notation is defined
\(f^{-1}(n)\) The Dirichlet inverse of f with respect to convolution defined recursively by \(f^{-1}(n) = -\frac{1}{f(1)} \sum \nolimits _{\begin{array}{c} d|n \\ d>1 \end{array}} f(d) f^{-1}(n/d)\) provided that \(f(1) \ne 0\)
F[k] Discrete Fourier transform coefficients defined
\(\lfloor x \rfloor \) The floor function \(\lfloor x \rfloor := x - \{x\}\) where \(0 \le \{x\} < 1\) denotes the fractional part of \(x \in \mathbb {R}\)
\(f_{\pm }(n)\) For any arithmetic function f, we define \(f_{\pm }(n) = f(n) \left[ n > 1\right] _{\delta } - f(1) \left[ n = 1\right] _{\delta }\), i.e., the function that has identical values as f for all \(n \ge 2\), and whose initial value is \(f_{\pm }(1) := -f(1)\) when \(n = 1\)
\(G_j\) Denotes the interleaved (or generalized) sequence of pentagonal numbers defined explictly by the formula \(G_j := \frac{1}{2} \left\lceil \frac{j}{2} \right\rceil \left\lceil \frac{3j+1}{2} \right\rceil \). The sequence begins as \(\{G_j\}_{j \ge 0} = \{0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, \ldots \}\).
\({\text {Id}}_k(n)\) The power-scaled identity function, \({\text {Id}}_k(n) := n^k\) for \(n \ge 1\)
\(\left[ n=k\right] _{\delta }\) Synonym for \(\delta _{n,k}\) which is one if and only if \(n = k\), and zero otherwise
\(\left[ \mathtt {cond}\right] _{\delta }\) For a boolean-valued cond, \(\left[ \mathtt {cond}\right] _{\delta }\) evaluates to one precisely when cond is true, and zero otherwise.
\(L_{f,g,k}(x)\) The type II Anderson–Apostol sum over the arithmetic functions fg, \(L_{f,g,k}(x) := \sum \nolimits _{d|(k,x)} f(d) g(x/d)\)
\({\text {gcd}}(m, n); (m,n)\) The greatest common divisor of m and n. Both notations for the GCD are used interchangably within the article
\(\mu (n)\) The Möbius function
\(\mu _{n,k}\) Matrix sequence defined on Sect. 3.1. The invertible sequence is an analog to the role of the Möbius function in Möbius inversion. In this case these inversion coefficients are defined such that \(g(n) = \sum _{\begin{array}{c} d=1 \\ (d, n)=1 \end{array}}^n f(d) \quad \iff \quad f(n) = \sum _{d=1}^n g(d+1) \mu _{n,d}.\) See Proposition 3.1 and Sect. 3 for the relation of this sequence (and its inverse) to the factorizations of type I sums
\(\mu _{n,k}^{(-1)}\) Inverse matrix sequence of \(\mu _{n,k}\) defined
M(x) The Mertens function which is the summatory function over \(\mu (n)\), \(M(x) := \sum \nolimits _{n \le x} \mu (n)\)
\(\omega (t)\) Orthogonal polynomial orthogonality weight function defined on Sect. 4.4. This function is related to glossary symbol \(P_k(w, t)\).
\(\phi _k(n)\) Generalized totient function, \(\phi _k(n) := \sum \nolimits _{\begin{array}{c} 1 \le d \le n \\ (d,n) = 1 \end{array}} d^k\)
\(\phi (n)\) Euler’s classical totient function, \(\phi (n) := \sum \nolimits _{\begin{array}{c} 1 \le d \le n \\ (d,n) = 1 \end{array}} 1\)
\(\Phi _n(z)\) The nth cyclotomic polynomial in z defined by \(\Phi _n(z) := \prod \nolimits _{\begin{array}{c} 1 \le k \le n \\ (k,n)=1 \end{array}} (z-e^{2\pi \imath k/n})\).
\(P_k(w, t)\) Type of orthogonal polynomial function defined on Sect. 4.4. It satisfies that \(\sum \nolimits _{k=1}^n \hat{u}_{n,k}(f, w) P_k(w, t) = t^n\)
p(n) The partition function generated by \(p(n) = [q^n] \prod \nolimits _{n \ge 1} (1-q^n)^{-1}\)
\((q; q)_{\infty }\) The infinite q-Pochhammer symbol defined as the product \((q; q)_{\infty } := \prod \nolimits _{n \ge 1} (1-q^n)\) for \(|q| < 1\)
\(\sigma _{\alpha }(n)\) The generalized sum-of-divisors function, \(\sigma _{\alpha }(n) := \sum \nolimits _{d|n} d^{\alpha }\), for any \(n \ge 1\) and \(\alpha \in {\mathbb {C}}\)
\(s_k(f, g; n)\) Shorthand for the periodic (modulo k) divisor sums defined on Sect. 4.3 and expanded by the functions listed in \(a_k(f,g;n)\) of this glossary. The precise expansion and corresponding finite Fourier series expansion of this function is given by \(s_k(f, g; n) = \sum \nolimits _{d|(n, k)} f(d) g(k/d) = \sum \nolimits _{m=1}^{k} a_k(f, g; m) e^{2\pi \imath \cdot mn / k}\).
\(s_{n,k}\) Matrix coefficients in Lambert series type factorizations defined on Sect. 2.2. These coefficients are defined precisely as the coefficients of the generating function \([q^n] (q; q)_{\infty } q^k / (1-q^k)\) for \(k \ge 1\) where \((q;q)_{{\infty }}\) is the infinite q-Pochhammer symbol
\(T_f(x)\) The type I sum over an arithmetic function f, \(T_f(n) := \sum \nolimits _{\begin{array}{c} d \le x \\ (d,x)=1 \end{array}} f(d)\)
\(t_{n,k}\) Matrix sequence involved in the generating function expansions of the type I sums defined on Sect. 2.2 as \(T_{f}(x) = [q^x]\left( \frac{1}{(q; q)_{\infty }} \sum _{n \ge 2} \sum _{k=1}^n t_{n,k} f(k) \cdot q^n + f(1) \cdot q\right) \)
\(t_{n,k}^{(-1)}\) Inverse matrix of the sequence \(t_(n, k)\) that is defined
\(\hat{u}_{n,k}(f, w)\) Matrix coefficients defined in terms of an indeterminate parameter w as \(\hat{u}_{n,k}(f, w) := (w^k-1) \cdot u_{n,k}(f, w)\).
\(u_{n,k}(f, w)\) Matrix sequence defined on Sect. 2.2 in the expansion of the generating functions for the type II sums as \(g(x) = [q^x]\left( \frac{1}{(q; q)_{\infty }} \sum _{n \ge 2} \sum _{k=1}^n u_{n,k}(f, w) \left[ \sum _{m=1}^k L_{f,g,m}(k) w^m\right] \cdot q^n\right) ,\ w \in {\mathbb {C}}.\)
\(u_{n,k}^{(-1)}(f, w)\) Inverse matrix terms of the sequence \(u_{n,k}(f,w)\) defined
\(y_f(n)\) The function \(y_f(n)\) denotes the Dirichlet inverse of the function \(h(n) := f(n) \phi (n) / n^2\) where \(\phi (n)\) is Euler’s totient function and f is any invertible arithmetic function such that \(f(1) \ne 0\). This function is used to express the result in Corollary 4.15. A special case, denoted by y(n), corresponding to the case where \(f(n) \equiv n\) is employed in stating Corollary 4.16 in Sect. 4.3
\(\zeta (s)\) The Riemann zeta function, defined by \(\zeta (s) := \sum \nolimits _{n \ge 1} n^{-s}\) when \(\mathfrak {R}(s) > 1\), and by analytic continuation to the entire complex plane with the exception of a simple pole at \(s = 1\)

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Mousavi, H., Schmidt, M.D. Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums. Ramanujan J 54, 309–341 (2021). https://doi.org/10.1007/s11139-020-00323-5

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Keywords

  • Divisor sum
  • Totient function
  • Matrix factorization
  • Möbius inversion
  • Partition function

Mathematics Subject Classification

  • 11N64
  • 11A25
  • 05A17