The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/φ(q))c_q(a)$, given by Hardy ($φ:=$ Euler's totient function). Both converge pointwise (not absolutely) in N. A $G:$N $\rightarrow $C is called a Ramanujan coefficient, abbrev. R.c., iff (if and only if) $\sum_{q=1}^{\infty}G(q)c_q(a)$ converges in all $a\in $N; given $F:$N $\rightarrow $C, we call $<F>$, the set of its R.c.s, the Ramanujan cloud of $F$. Our Main Theorem in arxiv:1910.14640, for Ramanujan expansions and finite Euler products, implies a complete Classification for multiplicative Ramanujan coefficients of $0$. Ramanujan's $G_R(q):=1/q$ is a normal arithmetic function $G$, i.e., multiplicative with $G(p)\neq 1$ on all primes $p$; while Hardy's $G_H(q):=1/φ(q)$ is a sporadic $G$, namely multiplicative, $G(p)=1$ for a finite set of $p$, but there's no $p$ with $G(p^K)=1$ on all integers $K\ge 0$ (Hardy's has $G_H(p)=1$ iff $p=2$). The $G:$N $\rightarrow $C multiplicative, such that there's at least a prime $p$ with $G(p^K)=1$, on all $K\ge 0$, are defined to be exotic. This definition completes the cases for multiplicative $0-$Ramanujan coefficients. The exotic ones are a kind of new phenomenon in the $0-$cloud (i.e., $<0>$): exotic Ramanujan coefficients represent $0$ only with a convergence hypothesis. The not exotic, apart from the convergence hypothesis, require in addition $\sum_{q=1}^{\infty}G(q)μ(q)=0$ for normal $G\in <0>$, while sporadic $G\in <0>$ need $\sum_{(q,P(G))=1}G(q)μ(q)=0$, $P(G):=$product of all $p$ making $G(p)=1$. We give many examples of R.c.s $G\in <0>$; we also prove that the only $G\in <0>$ with absolute convergence are the exotic ones; actually, these generalize to the weakly exotic, not necessarily multiplicative.
We call $R_G(a):=\sum_{q=1}^{\infty}G(q)c_q(a)$ the 'Ramanujan series', of coefficient $G:$N$\to$C, where $c_q(a)$ is the well-known Ramanujan sum. We study the convergence of this series (a preliminary step, to study Ramanujan expansions and define $G$ a 'Ramanujan coefficient' when $R_G(a)$ converges pointwise, in all natural $a$. Then, $R_G:$N$\to$C is well defined ('w-d'). The 'Ramanujan cloud' of a fixed $F:$N$\to$C is $<F>:=${$G:N\to C|R_G \; w-d, F=R_G$}. (See the Appendix.) We study in detail the multiplicative Ramanujan coefficients $G$ : their $<F>$ subset is called the 'multiplicati…
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number theory.
In this paper, we consider a general form of the analogue of Ramanujan's sum in the ring of polynomials over a finite field. We first prove some multiplicative properties of such functions before considering their finite Fourier series and some specific examples. In the end, we also prove a result about the Dirichlet series of such functions.
All the $F:$N$\rightarrow $C having Ramanujan expansion $F(a)=\sum_{q=1}^{\infty}G(q)c_q(a)$ (here $c_q(a)$ is the Ramanujan sum) pointwise converging in $a\in $N, with $G:$N$\rightarrow $C a multiplicative function, may be factored into two Ramanujan expansions, one of which is a finite Euler product : details in our Main Theorem. This is a general result, with unexpected and useful consequences, esp., for the Ramanujan expansion of null-function, say 0. The Main Theorem doesn't require other analytic assumptions, as pointwise convergence suffices; this depends on a general property of Euler …
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of $\log f(f(n))$ for a class of multiplicative fun…
We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. Moreover, we deduce from the criterion that the collection of multiplicative functions which are not strong Benford forms a group under pointwise multiplication. In contrast to earlier work, our approach is based on Halász's Theorem.
Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization
We consider a wide class of summatory functions F{f;N,p^m}= \sum_{k\leq N} f(p^m k), m \in \mathbb Z_+ \cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k \in \mathbb Z_+, where p is a prime number . Assuming an asymptotic behavior of summatory function, F{f;N,1} \stackrel{N\to \infty}{=} G_1(N) [1+ { \cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2 \geq 0, - \infty < b_1, b_2< \infty , we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\inf…
We extend the Matomäki-Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function $f$ in typical intervals of length $h(\log X)^c$, with $h = h(X) \rightarrow \infty$ and where $c = c_f \geq 0$ is determined by the distribution of $\{|f(p)|\}_p$ in an explicit way. We give three applications. First, we show that the classical Rankin-Selberg-type asymptotic formula for partial sums of $|λ_f(n)|^2$, where $\{λ_f(n)\}_n$ is the sequence of no…
Let $τ(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ Δ(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}τ(n) q^n \,,q=e^{2πi z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $τ(n)\neq 0$ for all $n\geq 1$; since $τ(n)$ is multiplicative, it suffices to study primes $p$ for which $τ(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $τ$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq…
We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal weight, (possibly different) squarefree level and trivial nebentypus. As a first application, we show that for the Ramanujan $τ$ function and any admissible $k$-tuple of distinct non-negative integers $a_1,\ldots,a_k$ the set $$ \{n \in \mathbb{N} : |τ(n+a_1)| < \cdots < |τ(n+a_k)|\} $$ has positive natural density. This result improves upon recent work of Bilu, …
