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Distributions of additive functions

on shifted primes
Gediminas Stepanauskas

Institute of Mathematics, Vilnius University
1

Introduction

A function f defined on positive integers n is called additive, if

f(mn)=f(m)+f(n)for every coprime pair (m. n)= ı.
In probabilistic number theory, a central problem is to decide when an additive func‐tion f renormalised or not possesses a ıimit distribution. First, we can study the frequency\nu_{x}(f(n)<u) :=\frac{1}{[x]}\#\{n\leq x : f(n)<v\}.
More generally, for additive function f it is natural to consider when functions \alpha(x) and\beta(x)>0 may be found such that frequenciesv_{x}(\frac{f(n)-\mathfrak{a}(x)}{\beta(\tau)}<u)
possess a limiting distribution.

Erdós and Wintner [7] solved completely the case when a(x)=0, \beta(x)= ı. Theyobtained that\nu_{x}(f(71)<?/)\Rightarrow F(ll)asxarrow\infty

if and only if three series\sum_{|f(p)|>1}\frac{1}{p}, \sum_{|f(p)|\underline{<}1}\frac{f(p)}{p}, \sum_{|f(p)|\underline{<}1}\frac{f^{2}(p)}{p}(1)

converge.

Here and further \Rightarrow is the sign of a weak convergence and F(u) means some limit
distribution function.

The case which corresponds to the choice / 3(T)=1 was compıetely solved by Elliottand Ryavec [2] and by Levin and Timofeev [17]. They got that there exists a real function\alpha(x) for which\nu_{x}(f(n)-a(x)<u)\Rightarrow F(u)(2)

asx:arrow\infty

if and only if there is a constant ( so that f (it)=clog n+h(,\iota) , where theseries\sum_{|h(p)|>1}\frac{ \imath} {p}s \sum_{|h(p)|\leq]}\frac{h^{2}(p)}{p}

(3)

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are convergent.

See also the paper of Erdó s [9].
The common case was partially solved by Erdós and Kac [8] and by Kubilius [15].
\acute{}They proved that for some class of additive functions\nu_{x}(\frac{J(7\downarrow)-A(x)}{B(x)}<u)\Rightar ow F(u)as\tauarrow\infty

(4)

with

A(x)= \sum_{p\leq x}\frac{f(p)}{p}, B(x)=(\sum_{p\leq x}\frac{f^{2}(p)}{p}) ^{1/2}

The distributions of additive functions are interesting also on subsets of the set of
positive integers. In the literature you can find papers devoted to the behaviour of additive
functions on arithmetically interesting subsequences like\{an+b:\gamma 1\in \mathbb{N}\} ,\{[n^{\alpha}] : n\in \mathbb{N}\} ,{ G(\prime\iota) : Il\in \mathbb{N}, G is a polinomial},\{[\mathfrak{a}n] : n\in \mathbb{N}\} ,{ n:n\in \mathbb{N}, n is squarefree},{ n:n\in \mathbb{N}, n has no large prime factors},\{p+a:p\in \mathbb{P}\}. \{op+b:p\in \mathbb{P}\},and others.

The methods used in the investigation of asymptotic behaviour of additive functions
on the set of positive integers can be applied to other sequences which are well distributed
in most residue classes to moduli which are not to large.

2

Distributions on shifted primes

In the present paper I will give a survey on distributions of additive functions on the setof shifted prime numbers.

The (črse of shifted 1 ) rin1(^{\backslash },b^{\backslash } was conside.red 1 ) y\Gamma 3_{r}\iota r\cdot t)_{C}'\iota n , Vinogra.dov, Levin [1] K_{\subset}'\prime\iota t_{\dot{r}) i[12,13], Halberstam, Hildebrand [11], Timofeev [33, 34, 35, 37], Elliott [3], and others.
A. Hildebrand [ı1] and N.M. Timofeev [34] proved that\nu_{x}(f(p+1)<u)\Rightarrow F(u)if and only if three series (1) converge.
The conditions for the convergence in this result are the same as in the classicalErdós‐Wintner theorem, although the limit distributions may be different. Therefore. thedistributions of f(n) and f(p+1) can only converge simultaneously.
The case (2) for shifted primes was solved by Timofeev [34] as well. He got that there
exist a real function a(x) for which\nu_{x}(f(n)-\alpha(x)<u)\Rightarrow F(u)

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as xarrow\infty if and only if there is a constant c so that f(71)=c\log n+h(n) and the series(3) are convergent. In this case \alpha(x) can be choseno( \tau)=r\log r+|h(p)1\leq 1\sum_{1)\leq x}\frac{h(p)}{p}Barban, Vinogradov, Levin [1] and Hildebrand [11] considered the analogues of (4).
They proved that for some class of additive functions with suitable \alpha(x) and \beta(x)lJ_{x}( \frac{f(p+1)-\alpha(x\cdot)}{\beta(x)}<u)\Rightar ow F(u) .
Elliott [4] investigated the case\nu_{x}(\frac{f([x]-p)-\alpha(x)}{\beta(x)}<u)\Rightar ow F(u) .
Let \omega(n) mean the number of prime divisors of n . And let \tau_{k}(n) denote the number ofways of expressing n as the product of k divisors. \tau_{2}(n)=\tau(n) is the number of divisorsof n . From the results above, can be deduced thatlノx ( \frac{\omega(p+1)-\log\log x}{\sqrt{\log\log x} <u)\Rightar ow\frac{ \imath} {\sqrt{2\pi} \int_{-\infty}^{u}e^{-v^{2}/2}dv=:\Phi(u) ,\frac{1}{\pi(x)}\sum_{p\leq x}\omega(p+1)\sim\log\log x,\nu_{x}(\frac{\log_{k}\tau_{k}(p+1)-\log\log x}{\sqrt{\log\log x} <v) \Rightar ow\Phi(v).

Investigations of the limit behaviour of additive functions on shifted primes can begeneralized to the sum of additive functions with different shifts. The idea to consider

the sums of shifted additive functions is not new. The first result in this direction belongs

to LeVeqv . [ı6]. hIore general results later were established 1 ) y Kul)iıius [15], K_{c!t}'ai[14],Hildebrand [10], Elliott [3, 4, 5, 6], Timofeev and Usmanov [36], Stepanauskas [28, 29,
30, 3ı], and others. There the cases when the values of additive functions can be taken(^{\backslash }on different arithmetic progressions, on shifted primes, and when the number of additive
functions as summands may slowly increase together with x were examined. All theseresults were given by using elementary methods, the method of characteristic functions,or the Kubilius model of probability spaces.

The results (general enough) for shifted primes are given by Stepanauskas [28, 30]:
1. The distributions\nu_{x}(f_{i}(a_{1}p+b_{1})+ +f_{s}(a_{s}p+b_{s})<u)
converge weakly as xarrow\infty to some limit distribution if the tree series converge:|f_{1}.(p.)| \leq1\sum_{|f_{S}(p)1\leq1}\frac{f_{1}(p)+\cdots+f_{s}(p)}{p} <\infty.

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\sum_{|f_{i}(p)|>1}\frac{1}{p}<\infty_{:}\sum_{|f_{?}(p)|\leq 1}\frac{f_{i} ^{2}(p)}{p}<\infty.
The conditions for the convergence in this result are the same as in the result for arith‐
metic progressions for positive integers, although the limit distributions may be different.

Therefore, the distributions of the sum for f_{i}(a_{i}n+b_{i}) and the sum for f_{i}(a_{i}p+b_{i}) canonly converge simultaneously.

2. Let \phi be the Euler totient function, \sigma(Ib) be the sum of positive divisors of n . Then\nu_{x}(\frac{\phi(o_{1}p+b_{1}). \cdot\phi(o_{s}p+b_{s}) {(c\iota_{1}p+b_{1} ). (a_{s}p+b_{s}) <u)\Rightar ow F(?1) .
\nu_{x}(\frac{\sigma(o_{1}p+b_{ \imath} ). \cdot\sigma(a_{s}p+b_{s}) {(c\iota_{1}p+b_{1}). (a_{s}p+b_{s}) <u)\Rightar ow F(u)_{:}
u_{x}(\frac{\phi(o_{1}p+b_{1})\ldots\phi(o_{k}p+b_{k})\sigma(o_{k+1}p+b_{k+1} \ldots\phi(a_{s}p+b_{s})}{(a_{1}p+b_{1})\ldots(a_{s}p+b_{s})}<u)\Rightar ow F(u)asXarrow\infty.

3. Let for

i=1 , 2, . . . , sf_{i}(p)arrow 0, parrow\infty,

and for at least one of f_{k}\sum_{\log\log x<p\leq x}\frac{f_{i}^{2}(p)}{p}ar ow 0, xar ow\infty,\sum_{p\leq x}\frac{f_{k}^{2}(I,)}{p}ar ow\infty, xar ow\infty.

Then the sumf_{1}(a_{1}p+b_{1})+\cdots+.f_{s}(a_{s}q+b_{s})is asymptotically uniformly distributed mod 1 on the set of primes.
4. Let the integer‐valued additive functions f_{i} , i=1 ,, s , be such that the series\sum_{|f_{t}(p)|\neq 0}\frac{1}{])}<\infty.
converge. Then\nu_{x}(f_{1}(a_{1}p+b_{1})+\cdots+f_{s}(a_{s}p+b_{s})=k)arrow\lambda_{k} :for every k\in \mathbb{Z}.

xarrow\infty,

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3

Distributions of sets of additive functions

on shifted primes
Elliott [4] showed that every stable law F can occur as a limit distribution for\iota\prime_{x}(\frac{f(n)-\mathfrak{a}(x)}{\beta(x)}<Z1)(5)with suitable chosen a and \beta and that there are uncountably many distributions F whichcannot occur as the limit distributions for (5) no matter how they are centred and nor‐malised. The Poisson distribution is among them.
It is clear that the set of possible limit distributions can be expanded if we consider
additive functions which values vary together with x . i.e. the sets of additive functions

f_{x}(n) : \tau\geq 2 . In the books [15. 3, 4] there were considered (at most) additive functions
h_{x} having a special expression:\cdot

h_{x}(n)= \frac{f(7\iota)}{\beta(x)},where \beta is some normalising function. The more common sets were investigated by Šiaulys[18, 19, 20, 21, 22].
The question which appear here is what happens with the case of shifted primes.

In [23, 24] by Šiaulys and Stepanauskas the case of the Poisson limit distribution wasconsidered. I will present here several consequences.
Let f_{x} : x\geq 2_{:} be a set of strongly additive functions such that f_{x}(p)\in\{0,1\}\mathb {P}, \forall x\geq 2.
1. For every parameter \lambda>0 the Poisson distribution\Pi(u,\lambda):=\sum_{k=0,1k<v}.\frac{\lambda^{k} {A\cdot!}e^{-k}can occur (accordingly choosing f_{x} ) as limit distribution for\nu_{x}(f_{x}(p+1)<u) , \nu_{x}(f_{x}(p+1)+g_{x}(p+2)<u) .
2. Letandf_{x}(p)=\begin{ar ay}{l} 1 if logz: <p\leq(\log x:)^{\alpha}. 0 otherwise; \end{ar ay}g_{x}(p)=\begin{ar ay}{l} 1 if \log\log x<p\leq(\log\log a^{\pi})^{\beta}, 0 otherwise, \end{ar ay}

with some 0 , \beta>1 . Then\lim_{xar ow\infty}\nu_{\lambda}(f_{x}(p+1)=k)=\frac{(\log \mathfrak{a})^{k} { \cap k!},\forall p\in

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\lim_{xar ow\infty}\nu_{x}(f_{x}(p+1)+g_{x}(p+2)=k)=\frac{(\log o\beta)^{k} {\alpha\beta k!}.
3. Let \prime\sqrt{}) and \chi be unboundedly increasing functions such that \log V'(x)/\log xarrow 0
and \log\lambda(x)/\log xarrow 0 as xarrow\infty . Thent xactly
ntervalkprime( \psi l(x\cdot), \psi^{o}(x\cdot)]\}\sim\frac{(\log\cap)^{k} {\alpha K!} \frac{T}{\log x}\dot{}\#\{p\leq x,factorsfrom
p+lhasehei\#{

p \underline{<}x_{:}(p+1)(p+2)

has exactly prime factors } \sim\underline{(2\log\alpha)^{k} \underline{J^{\cdot} .
kfrom the interval (\psi(x)_{:}\psi^{o}(x) ]\alpha^{2}k!

\log xfactorsfrom
wherehr( \tau/,(\gam a\cdot)\tau/f^{\alpha}(x)],\cup(\chi.(x)\chi^{\thounteta}(\tau)] ed f\# {p\leqx,t_{heinterva1(\psi,(x)\psi^{\mathfrak{ }(x)] andthepr\dot{\imath}me}(p+{\imath})(p+,2)hasexact1yk,pr\dot{\imath}rome\,} sim\frac{(\log\alpha\beta)^{k} \alpha\betak!}\frac{I}\logx}
imeforSp+arec

factors of p+2 from ()_{\backslash }'(x) , x^{\beta}(x) ]Discrete uniform distributionu(u, L):=\frac{1}{L}\sum_{A<u}k=0,1,\ldots,L-1
for shifted primes was considered by Šiaulys, Stepanauskas and Žvinyte [32, 27]. I give

only some consequences from these investigations.

1. The discrete uniform distribution U(\iota\nu, L) can occur as a limit distribution forlJ_{x}(f_{x} ( p+ ı) <v) ,\nu_{x}(f_{x}(p+1)+g_{x}(p+2)<u)with some set of additive functions if L=2 and cannot occur if L=3.4.5\ldots..
2. Let.
Thenf_{x}(p)=\begin{ar ay}{l } 1 if p=3, 0 otherwise. \end{ar ay}\nu_{x}(f_{x}(p+1)<v)\Rightarrow \mathcal{U}(u, 2) .
3. Let.
Thenf_{x}(p)+.q_{x}(q)=\begin{ar ay}{l } 1 if p=2, 0 otherwise. \end{ar ay}\nu_{x} (f_{x}(n)+g_{x}(n+ {\imath}) <v)\Rightarrow \mathcal{U}(u, 2) .
4. Let either

f_{x}(p)=g_{x}(p)=\begin{ar ay}{l } 1 if p=5, 0 otherwise, \end{ar ay}

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or

.
Then

f_{x}(P)+q_{x}(J) =\begin{ar ay}{l } 1 if p=3. 0 otherwise. \end{ar ay}\nu_{x}(f_{x}(p+1)+g_{x}(p+2)<u)\Rightarrow u(u_{:}2) .
In the proofs of the main results for the Poisson and discrete limit distributions, theauthors combined different methods. It was used elementary ınethods, the method ofcharacteristic functions, and the Kubilius method of probability spaces. But the methodof factorial moments played here the crucial role.
It would be interesting to examine the binomial, the Bernoulli, the shifted Poisson,
the geometrical, and other distributions for shifted primes. Can they occur or not? It is

known [25_{:}26] that some of them can occur as limit distributions even for 1J_{x}(f(n)<u) .
but some of them can not.

ACKNOWLEDGEMENT. The author expresses his gratitude to professor Kohji Mat‐sumoto for the possibility to take part in the symposium at RIMS.

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Institute of Mathematics

Vilnius University
Viınius 03225

LITHUANIA

E‐mail address: gediminas.stepanauskas@mif.vu.lt