Fabius function

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In mathematics, the Fabius function is a smooth cumulative distribution function ${\displaystyle F:\mathbb {R} \to [0,1]}$ that is strictly increasing on ${\displaystyle [0,1]}$ but nowhere real-analytic. It is the distribution function of the random series ${\displaystyle X=\sum _{n=1}^{\infty }2^{-n}U_{n},}$ where the ${\displaystyle U_{n$ are independent random variables uniformly distributed on ${\displaystyle [0,1]}$ .

Equivalently, it is characterized by ${\displaystyle F(0)=0,\;F(1-x)=1-F(x),\;F'(x)=2F(2x)\,\left(0\leq x\leq {\tfrac {1}{2\right).}$ Although every positive-order derivative of ${\displaystyle F}$ vanishes at both endpoints, ${\displaystyle F}$ is nonconstant on every subinterval of ${\displaystyle [0,1]}$ .^[1]^[2]

The same object occurs, after a change of variables, as Rvachev's up function ${\displaystyle u}$ , an even ${\displaystyle C^{\infty$ bump function supported on ${\displaystyle [-1,1]}$ . Its Fourier transform is an infinite product of sinc functions, or equivalently a product of powers of cosines. The function also has an unusual arithmetic side: every value ${\displaystyle F(a/2^{n})}$ at a dyadic rational is rational, and the numerators, denominators and 2-adic valuations of these values obey strong divisibility laws.^[2]^[3]

The underlying distribution already appeared in work of Børge Jessen and Aurel Wintner in 1935. Jaap Fabius independently introduced it probabilistically in 1966 as an example of a nowhere-analytic smooth function, and V. A. Rvachev independently introduced the associated compactly supported solution of a functional–differential equation.^[4]^[1]^[3] Near zero, ${\displaystyle F}$ is smaller than every power of its argument; its exact rate of decay is governed by the lower branch of the Lambert W function together with a small correction that repeats each time the argument is halved. That correction alters ${\displaystyle F}$ by only about two parts in a million, but it does not fade away as the argument tends to zero, so no asymptotic formula that omits it can be exact.^[5]

Notation#edit

The literature uses several incompatible conventions. In this article, ${\displaystyle F}$ denotes the bounded cumulative distribution function, extended by constants outside ${\displaystyle [0,1]}$ ; ${\displaystyle u}$ denotes Rvachev's compactly supported up function; and ${\displaystyle {\mathcal {F$ denotes the signed global extension satisfying ${\displaystyle {\mathcal {F'(x)=2{\mathcal {F(2x)}$ on the whole real line. Logarithms are natural unless a base is written explicitly, ${\displaystyle s_{2}(n)}$ is the sum of the binary digits of ${\displaystyle n}$ , and ${\displaystyle \nu _{2}(n)}$ is the exponent of 2 in ${\displaystyle n}$ .^[3]

Definition and elementary properties#edit

There is a unique ${\displaystyle C^{\infty$ function ${\displaystyle F:\mathbb {R} \to [0,1]}$ such that ${\displaystyle F(x)=0\quad (x\leq 0),\qquad F(x)=1\quad (x\geq 1),}$ and, for ${\displaystyle 0\leq x\leq 1}$ , ${\displaystyle F(1-x)=1-F(x),}$ while ${\displaystyle F'(x)=2F(2x)}$ for ${\displaystyle 0\leq x\leq 1/2.}$ The constant continuation outside the unit interval is smooth because all positive-order derivatives vanish at the endpoints.^[2]^[3]

By symmetry, the differential equation can be written on the whole unit interval as ${\displaystyle F'(x)={\begin{cases}2F(2x),&0\leq x\leq {\tfrac {1}{2,\\[3pt]2F(2-2x),&{\tfrac {1}{2\leq x\leq 1.\end{cases$ The derivative is positive on ${\displaystyle (0,1)}$ , so ${\displaystyle F}$ is strictly increasing there. In particular, ${\displaystyle F\!\left({\tfrac {1}{2\right)={\tfrac {1}{2,\qquad F'\!\left({\tfrac {1}{2\right)=2,}$ and ${\displaystyle F'(1-x)=F'(x),\qquad F'(x)+F'\!\left({\tfrac {1}{2-x\right)=2\quad \left(0\leq x\leq {\tfrac {1}{2\right).}$ The endpoint flatness is ${\displaystyle F^{(m)}(0)=F^{(m)}(1)=0\qquad (m\geq 1).}$

Integrating the differential equation gives a self-integral equation, which together with the symmetry and the normalization characterizes ${\displaystyle F}$ :^[2] ${\displaystyle F(x)=\int _{0}^{2x}F(t)\,dt\qquad \left(0\leq x\leq {\tfrac {1}{2\right).}$

Probabilistic and convolution constructions#edit

Let ${\displaystyle U_{1},U_{2},\ldots }$ be independent random variables, each uniformly distributed on ${\displaystyle [0,1]}$ . The series ${\displaystyle X=\sum _{n=1}^{\infty }2^{-n}U_{n$ converges almost surely and takes values in ${\displaystyle [0,1]}$ . The Fabius function is ${\displaystyle F(x)=\Pr(X\leq x).}$ The decomposition ${\displaystyle X\ {\stackrel {d}{=\ {\frac {U+X'}{2,}$ where ${\displaystyle U}$ is uniform on ${\displaystyle [0,1]}$ and ${\displaystyle X'}$ is an independent copy of ${\displaystyle X}$ , gives ${\displaystyle F(x)=\int _{0}^{1}F(2x-t)\,dt.}$ For ${\displaystyle 0\leq x\leq {\tfrac {1}{2$ , the constant continuation ${\displaystyle F(t)=0}$ for ${\displaystyle t\leq 0}$ reduces this to the self-integral equation above.^[1]^[2]

The elementary moments of ${\displaystyle X}$ are^[1]^[2] ${\displaystyle \operatorname {E} X={\frac {1}{2,\qquad \operatorname {Var} (X)={\frac {1}{36.}$ Set ${\displaystyle Y=2X-1=\sum _{n=1}^{\infty }2^{-n}V_{n},\qquad V_{n}=2U_{n}-1.}$ The variables ${\displaystyle V_{n$ are independent and uniform on ${\displaystyle [-1,1]}$ . The random variable ${\displaystyle Y}$ has density ${\displaystyle u}$ , and ${\displaystyle F'(x)=2u(2x-1)\qquad (0\leq x\leq 1).}$ Thus ${\displaystyle u}$ is the infinite convolution^[4]^[2] ${\displaystyle u=h_{1}*h_{2}*h_{3}*\cdots ,\qquad h_{n}(t)=2^{n-1}\mathbf {1} _{[-2^{-n},\,2^{-n}]}(t),}$ where ${\displaystyle \mathbf {1} _{A$ denotes the indicator of ${\displaystyle A}$ . This construction accounts for both the compact support and the product formula for the Fourier transform.

Rvachev's up function#edit

The up function associated with ${\displaystyle F}$ is ${\displaystyle u(t)={\begin{cases}F(t+1),&-1\leq t\leq 0,\\F(1-t),&0\leq t\leq 1,\\0,&|t|>1.\end{cases$ Equivalently, ${\displaystyle u(t)=F(1-|t|)\qquad (|t|\leq 1).}$ It is even, nonnegative, positive on ${\displaystyle (-1,1)}$ , satisfies ${\displaystyle u(0)=1}$ , has integral one, and has topological support exactly ${\displaystyle [-1,1]}$ .^[6] Up to a constant factor it is the unique smooth compactly supported solution of the delay differential equation ${\displaystyle u'(t)=2{\bigl (}u(2t+1)-u(2t-1){\bigr )},}$ and the normalization ${\displaystyle \textstyle \int _{\mathbb {R} }u(t)\,dt=1}$ fixes that factor. Every derivative of ${\displaystyle u}$ vanishes at ${\displaystyle t=\pm 1}$ .^[2]^[3]^[7]

Fourier transform#edit

Use the Fourier-transform convention ${\displaystyle {\widehat {u(z)=\int _{\mathbb {R} }u(t)e^{-2\pi izt}\,dt.}$ Then ${\displaystyle {\widehat {u$ is an entire even function with ${\displaystyle {\widehat {u(0)=1}$ , and has the equivalent product representations^[4]^[2] ${\displaystyle {\widehat {u(z)=\prod _{h=0}^{\infty }{\frac {\sin(\pi z/2^{h})}{\pi z/2^{h=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m\right)^{m}=$ Consequently, the zeros of ${\displaystyle {\widehat {u$ are precisely the non-zero integers, and the zero at a non-zero integer ${\displaystyle m}$ has multiplicity ${\displaystyle 1+\nu _{2}(|m|)}$ .

Fourier inversion gives^[2] ${\displaystyle u(x)=\int _{\mathbb {R} }{\widehat {u(t)e^{2\pi itx}\,dt.}$ Because ${\displaystyle u}$ is smooth and compactly supported, its Fourier transform decreases faster than every inverse power on the real axis.

Partitions of unity and Fourier series#edit

The integer translates of ${\displaystyle u}$ form a partition of unity: ${\displaystyle \sum _{k\in \mathbb {Z} }u(x+k)=1.}$ More generally, for every positive integer ${\displaystyle n}$ , ${\displaystyle \sum _{k\in \mathbb {Z} }u\!\left(x+{\frac {k}{n\right)=n.}$ Each sum is locally finite because ${\displaystyle u}$ is compactly supported. These identities also follow from the zeros of ${\displaystyle {\widehat {u$ and the Poisson summation formula.^[2]

The 2-periodic extension of ${\displaystyle u}$ restricted to the interval ${\displaystyle [-1,1]}$ has the absolutely and uniformly convergent cosine series^[2] ${\displaystyle u(x)={\frac {1}{2+\sum _{j=0}^{\infty }{\widehat {u\!\left({\frac {2j+1}{2\right)\cos {\bigl (}(2j+1)\pi x{\bigr )}.}$ Only odd harmonics occur because ${\displaystyle {\widehat {u(k)=0}$ for every nonzero integer ${\displaystyle k}$ .

Fourier–Legendre expansion#edit

Because ${\displaystyle u}$ is even, only even Legendre polynomials occur in its Fourier–Legendre series: ${\displaystyle u(x)=\sum _{n=0}^{\infty }a_{n}P_{2n}(x),\qquad -1\leq x\leq 1,}$ where ${\displaystyle a_{n}={\frac {4n+1}{2\int _{-1}^{1}u(x)P_{2n}(x)\,dx.}$

A formula for the coefficients in terms of reciprocal-power values of ${\displaystyle F}$ is^[8] ${\displaystyle a_{n}=4^{-n}(4n+1)\sum _{k=0}^{n}(-1)^{n+k}{\binom {2n}{n+k{\binom {2n+2k}{2n(2k)!\,2^{\binom {2k+1}{2F\!\left(2^{-2k-1}\right).}$ The series converges absolutely and uniformly on ${\displaystyle [-1,1]}$ , including at the endpoints.

Signed global extension and non-analyticity#edit

There is a unique extension ${\displaystyle {\mathcal {F$ of ${\displaystyle F}$ to the whole real line that satisfies the self-differential equation everywhere: ${\displaystyle {\mathcal {F'(x)=2{\mathcal {F(2x)\qquad (x\in \mathbb {R} ).}$ It is given by ${\displaystyle {\mathcal {F(x)=0}$ for ${\displaystyle x\leq 0}$ , by ${\displaystyle {\mathcal {F(x+1)=1-{\mathcal {F(x)}$ for ${\displaystyle 0\leq x\leq 1}$ , and by ${\displaystyle {\mathcal {F(x+2^{r})=-{\mathcal {F(x)\qquad (0\leq x\leq 2^{r})}$ for every positive integer ${\displaystyle r}$ . Equivalently, it is the locally finite sum ${\displaystyle {\mathcal {F(x)=\sum _{n=0}^{\infty }(-1)^{s_{2}(n)}u(x-2n-1).}$ It agrees with the bounded Fabius function on ${\displaystyle [0,1]}$ , and on the interval ${\displaystyle [2n,2n+2]}$ is a translated copy of ${\displaystyle u}$ with sign ${\displaystyle (-1)^{s_{2}(n)$ . Its sequence of signs is therefore the Thue–Morse sequence.^[2]^[3]^[9]

Repeated differentiation gives ${\displaystyle {\mathcal {F^{(k)}(x)=2^{\binom {k+1}{2{\mathcal {F(2^{k}x)\qquad (k\geq 0).}$ In particular, every positive-order derivative vanishes at every integer.

If ${\displaystyle q=a/2^{n}>0}$ is a positive dyadic number, with ${\displaystyle a}$ a positive integer, then ${\displaystyle {\mathcal {F^{(k)}(q)=2^{\binom {k+1}{2{\mathcal {F(2^{k}q).}$ For ${\displaystyle k>n}$ , the argument ${\displaystyle 2^{k}q}$ is an even integer, so these derivatives vanish. Hence the Taylor series of ${\displaystyle {\mathcal {F$ at a dyadic point is a polynomial of degree at most ${\displaystyle n}$ ; when ${\displaystyle a}$ is odd, the degree is exactly ${\displaystyle n}$ . The function does not agree locally with this polynomial, and the dyadic rationals are dense, so ${\displaystyle F}$ is nowhere real-analytic on ${\displaystyle [0,1]}$ .^[2]

Moments and generating functions#edit

Let ${\displaystyle c_{n}=\int _{-1}^{1}t^{2n}u(t)\,dt=\operatorname {E} (Y^{2n})\qquad (n\geq 0).}$ All odd moments of ${\displaystyle u}$ vanish. The even moments are rational and satisfy^[2]^[3] ${\displaystyle c_{0}=1,\qquad (2n+1)(2^{2n}-1)c_{n}=\sum _{k=0}^{n-1}{\binom {2n+1}{2kc_{k}\quad (n\geq 1).}$ The moment expansion of the Fourier transform is ${\displaystyle {\widehat {u(z)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {c_{n{(2n)!(2\pi z)^{2n}.}$ The moments are also related to Fabius values by^[2]^[3] ${\displaystyle c_{n}=2(2n)!\,2^{\binom {2n+1}{2F\!\left(2^{-2n-1}\right).}$

The moment-generating function of ${\displaystyle X}$ is ${\displaystyle {\begin{aligned}D(z)&=\operatorname {E} (e^{zX})=\sum _{n=0}^{\infty }d_{n}{\frac {z^{n{n!\\&=\prod _{j=1}^{\infty }{\frac {e^{z/2^{j-1$ It obeys the dilation equation ${\displaystyle D(2z)={\frac {e^{z}-1}{zD(z).}$ Consequently, ${\displaystyle d_{0}=1}$ and ${\displaystyle (n+1)(2^{n}-1)d_{n}=\sum _{k=0}^{n-1}{\binom {n+1}{kd_{k}.}$ for all ${\displaystyle n\geq 1}$ . The coefficients are the raw moments ${\displaystyle d_{n}=\operatorname {E} (X^{n})}$ and satisfy the identity ${\displaystyle d_{n}=n!\,2^{\binom {n}{2F(2^{-n}),}$ which yields the values ${\displaystyle F(2^{-n})}$ in exact rational arithmetic.^[3]

Dyadic values and exact computation#edit

Every value of ${\displaystyle F}$ at a dyadic rational is rational. A closed formula can be stated using the even moments ${\displaystyle c_{k$ . For integers ${\displaystyle n\geq 0}$ and ${\displaystyle 0\leq a\leq 2^{n$ , ${\displaystyle {\begin{aligned}F\!\left({\frac {a}{2^{n\right)={}&{\frac {2^{-{\binom {n+1}{2{n!\sum _{h=0}^{a-1}(-1)^{s_{2}(h)}\sum _{k=0}^&#1$ The appearance of the binary digit sum ${\displaystyle s_{2}(h)}$ reflects the Thue–Morse signs of the global extension.^[3]

Values at reciprocal powers of two^[10]^[11]
${\displaystyle n}$	${\displaystyle F(2^{-n})}$	${\displaystyle d_{n}=n!\,2^{\binom {n}{2F(2^{-n})}$
0	${\displaystyle 1}$	${\displaystyle 1}$
1	${\displaystyle {\tfrac {1}{2$	${\displaystyle {\tfrac {1}{2$
2	${\displaystyle {\tfrac {5}{72$	${\displaystyle {\tfrac {5}{18$
3	${\displaystyle {\tfrac {1}{288$	${\displaystyle {\tfrac {1}{6$
4	${\displaystyle {\tfrac {143}{2073600$	${\displaystyle {\tfrac {143}{1350$
5	${\displaystyle {\tfrac {19}{33177600$	${\displaystyle {\tfrac {19}{270$
6	${\displaystyle {\tfrac {1153}{561842749440$	${\displaystyle {\tfrac {1153}{23814$
7	${\displaystyle {\tfrac {583}{179789679820800$	${\displaystyle {\tfrac {583}{17010$

There is also an efficient recursive evaluator:^[3]^[12] given ${\displaystyle 0<x\leq 1}$ , choose the unique integer ${\displaystyle n\geq 0}$ such that ${\displaystyle 2^{-n}\leq x<2^{-n+1},}$ and put ${\displaystyle y=x-2^{-n$ . Then ${\displaystyle F(x)=-F(y)+\sum _{k=0}^{n}{\frac {2^{\binom {k+1}{2F(2^{k-n})}{k!\,y^{k}.}$ For a dyadic input, replacing ${\displaystyle x}$ by ${\displaystyle y}$ removes its leading nonzero binary digit, so the iteration terminates after as many steps as there are 1s in the binary numerator, and every step is exact rational arithmetic.^[3]^[13]

Asymptotic behaviour near zero#edit

The flatness of the function at zero implies ${\displaystyle F(x)=o(x^{N})\qquad (x\to 0^{+})}$ for every fixed ${\displaystyle N>0}$ , and more precisely ${\displaystyle \log F(x)\sim -{\frac {\log ^{2}x}{2\log 2.}$ A sharp description requires both the Lambert W function and a logarithmically periodic correction.^[5]

Sharp Lambert-W form#edit

Set ${\displaystyle L=\log 2}$ and ${\displaystyle C_{0}={\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2{12,}$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant and ${\displaystyle \gamma _{1$ is the first Stieltjes constant, normalized by

${\displaystyle \zeta (s)={\frac {1}{s-1+\gamma -\gamma _{1}(s-1)+\cdots }$

For sufficiently small positive ${\displaystyle x}$ , set ${\displaystyle W=W_{-1}(-Lx),\qquad \lambda (x)=-{\frac {W}{L,}$ where ${\displaystyle W_{-1$ is the lower real branch of the Lambert ${\displaystyle W}$ function; equivalently, ${\displaystyle \lambda (x)}$ is the large positive solution of ${\displaystyle \lambda \,2^{-\lambda }=x}$ . The sharp main term is^[5] ${\displaystyle M(x)=-{\frac {7L}{12-{\frac {1}{2\log(\pi x)+{\frac {C_{0}-W-{\tfrac {1}{2W^{2{L+\Psi \!{\bigl (}\lambda (x){\bigr )},}$ where ${\displaystyle \Psi }$ is the one-periodic function described below, and, as ${\displaystyle x\to 0^{+$ , ${\displaystyle \log F(x)=M(x)+O\!\left({\frac {1}{-\log x\right).}$

The periodic correction#edit

The function ${\displaystyle \Psi }$ is real-valued, smooth and one-periodic, and is normalized to have mean zero over a period. With ${\displaystyle \chi _{k}=2\pi ik/L}$ , its absolutely convergent Fourier series is^[5] ${\displaystyle \Psi (t)=\sum _{k\in \mathbb {Z} \setminus \{0\\left(-{\frac {\Gamma (-\chi _{k})\zeta (1-\chi _{k})}{L\right)e^{2\pi ikt},}$ where ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle \zeta }$ is the Riemann zeta function. The coefficient of index ${\displaystyle k}$ is nonzero for every ${\displaystyle k\neq 0}$ , because the gamma function has no zeros and the zeta function has no zeros on the line ${\displaystyle \operatorname {Re} (s)=1}$ , on which ${\displaystyle 1-\chi _{k$ lies. Hence ${\displaystyle \Psi }$ is nonconstant, and deleting ${\displaystyle \Psi (\lambda (x))}$ from ${\displaystyle M(x)}$ leaves a bounded oscillation in ${\displaystyle \log F(x)}$ that the ${\displaystyle O(1/(-\log x))}$ remainder cannot absorb. The oscillation is small: ${\displaystyle \Psi }$ has amplitude about ${\displaystyle 2.1\times 10^{-6$ .^[5]

Expanded elementary form#edit

The Lambert phase can itself be expanded in powers of ${\displaystyle -\log x}$ and ${\displaystyle \log(-\log x)}$ . Writing ${\displaystyle \ell =\log x}$ , ${\displaystyle a=\log(-\ell )}$ and ${\displaystyle b=\log L}$ , the sharp form becomes^[5] ${\displaystyle {\begin{aligned}\log F(x)=-{\frac {\ell ^{2{2L+{\frac {\ell a}{L&-\left({\frac {1}{2+{\frac {1+b}{L\right)\ell -{\frac {a^{2{2L+{\frac {ba}{L$ At this precision, the periodic term is evaluated at the exact phase ${\displaystyle \lambda (x)}$ .

For reciprocal powers of two, let ${\displaystyle \lambda _{n}=\lambda (2^{-n})}$ , so that ${\displaystyle \lambda _{n}-\log _{2}\lambda _{n}=n}$ , and set ${\displaystyle C_{\mathrm {sharp} }={\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2{12L-{\frac {7L}{12-{\frac {\log \pi }{2.}$ Then^[5] ${\displaystyle {\begin{aligned}\log F(2^{-n})={}&-{\frac {L}{2n^{2}-n\log n+\left(1+{\frac {L}{2\right)n-{\frac {\log ^{2}n}{2L+C_{\mathrm {sharp} }\\&-&#12$ The oscillation is periodic in the exact Lambert phase, not in the integer ${\displaystyle n}$ itself.

Origin of the logarithmic oscillation#edit

For ${\displaystyle s>0}$ the moment-generating function has the product ${\displaystyle D(-s)=\prod _{j=1}^{\infty }{\frac {1-e^{-s/2^{j{s/2^{j,}$ whose logarithm has an exact dyadic-scale decomposition^[5] ${\displaystyle \log D(-s)=-{\frac {\log ^{2}s}{2L+{\frac {1}{2\log s+R(\log _{2}s)+E(s),}$ with ${\displaystyle R}$ one-periodic and ${\displaystyle |E(s)|\leq e^{-s}/(1-e^{-s})^{2$ . Rescaling by a factor of 2 therefore contributes an additive period-one term in the logarithmic variable, and saddle-point inversion carries the centred form of that term to ${\displaystyle \Psi (\lambda (x))}$ in the asymptotic of ${\displaystyle F(x)}$ .

Full saddle-point expansion#edit

There are continuous one-periodic functions ${\displaystyle P_{1},P_{2},\ldots }$ such that, for every integer ${\displaystyle N\geq 1}$ ,^[5] ${\displaystyle \log F(x)=M(x)+\sum _{j=1}^{N-1}{\frac {P_{j}(\lambda (x))}{\lambda (x)^{j+O\!\left(\lambda (x)^{-N}\right),}$ where ${\displaystyle M}$ is the sharp main term. The first correction is ${\displaystyle P_{1}(t)={\frac {1}{24+{\frac {1}{2L-{\frac {\Psi ''(t)+(\Psi '(t))^{2{2L^{2.}$

History#edit

Arias de Reyna's historical survey records at least six independent appearances of the function or an equivalent one.^[3]

In 1935, Jessen and Wintner considered the associated distribution and obtained its smoothness through the infinite Fourier product.^[4]
In 1966, Fabius gave the probabilistic construction as an example of a smooth nowhere-analytic function.^[1]
In 1971, V. A. Rvachev introduced the compactly supported solution now called the up function through its functional–differential equation; the work is recorded in a monograph by V. L. Rvachev and V. A. Rvachev, and a later survey by V. A. Rvachev developed a broader theory of compactly supported solutions of such equations.^[3]^[7]
G. Kh. Kirov and G. A. Totkov gave another independent construction in 1981.^[3]
Arias de Reyna developed the up function systematically in 1982, proving uniqueness, the probability interpretation, partition-of-unity identities, derivative formulas, rationality at dyadic points and exact computation formulas. An English translation appeared in 2017.^[2]
R. Schnabl gave a further independent construction in 1985.^[3]

Arias de Reyna's 2018 paper developed the arithmetic of the dyadic values, including the exact 2-adic valuation, denominator bounds and a terminating exact evaluator.^[3]^[14] Haugland gave another explicit treatment of exact evaluation at arbitrary dyadic arguments.^[13]

References#edit

^ a b c d e Fabius, J. (1966). "A probabilistic example of a nowhere analytic C^∞-function". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 5 (2): 173–174. doi:10.1007/BF00536652. MR 0197656. S2CID 122126180.
^ a b c d e f g h i j k l m n o p q Arias de Reyna, Juan (1982). "Definición y estudio de una función indefinidamente diferenciable de soporte compacto" [Definition and study of an infinitely differentiable function of compact support]. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales (in Spanish). 76. Madrid: 21–38. arXiv:1702.05442.
^ a b c d e f g h i j k l m n o p q Arias de Reyna, Juan (2018). "Arithmetic of the Fabius function" (PDF). Integers. 18. A51. arXiv:1702.06487.
^ a b c d Jessen, Børge; Wintner, Aurel (1935). "Distribution functions and the Riemann zeta function". Transactions of the American Mathematical Society. 38 (1): 48–88. doi:10.1090/S0002-9947-1935-1501802-5. MR 1501802.
^ a b c d e f g h i Reshetnikov, Vladimir. "Fabius function: asymptotic completion audit". ProveIt. GitHub. Retrieved 24 August 2026.
^ Sloane, N. J. A. (ed.). "Sequence A288163". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ a b Rvachev, V. A. (1990). "Compactly supported solutions of functional-differential equations and their applications". Russian Mathematical Surveys. 45 (1): 87–120. doi:10.1070/RM1990v045n01ABEH002324. MR 1050928.
^ Reshetnikov, Vladimir. "Fabius function formalization". ProveIt. GitHub. Retrieved 24 August 2026.
^ Alkauskas, Giedrius (2001), Dirichlet series associated with Thue–Morse sequence (PDF) (preprint)
^ Sloane, N. J. A. (ed.). "Sequence A272755 (Numerators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A272757 (Denominators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Bolnez, Pierrot (9 July 2016). "How do I numerically evaluate and plot the Fabius function?". Mathematica Stack Exchange. Retrieved 24 August 2026. Answered by Vladimir Reshetnikov, who attributes the algorithm to V. A. Rvachev.
^ a b Haugland, Jan Kristian (2016). "Evaluating the Fabius function". arXiv:1609.07999 [math.GM].
^ Reshetnikov, Vladimir (7 February 2017). "A conjecture about certain values of the Fabius function". MathOverflow. Retrieved 24 August 2026.

External links#edit

Lean 4 formalization and exact evaluator
FabiusF at the Wolfram Function Repository
OEIS: A272755 (numerators of ${\displaystyle F(2^{-n})}$ ) and OEIS: A272757 (denominators)
MathOverflow question on the reciprocal-power values

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