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We study in detail the Ramanujan smooth expansions, for arithmetic functions; we start with the most general ones, for which we supply the "$P-$local expansions", for arguments with all prime-factors $p\le P$ (namely, $P-$smooth arguments), that are also square-free; then, we supply general results for interesting subsets of arithmetic functions, regarding both their $P-$local and (global) Ramanujan smooth expansions., Comment: Plain TeX, 20 pages, plus 1 p.bib

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is studied. The Kronecker lemma is generalized to the case when the corresponding Dirichlet series diverges. Several assertions are proven and illustrative examples are given., Comment: 13 pages

The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet series. Several assertions have been proven and numerous examples have been considered., Comment: 14 pages

Annotated bibliography of 18th, 19th, and early 20th century works involving Lambert series. A tour of 19th and early 20th century analytic number theory., Comment: 42 pages

We prove that $|\sum_{n \leq x} \mu(n) | \leq x/160\,383$ for all $x\geq 8.4 \times 10^9$ thus improving the record for this type of estimates which was $|\sum_{n \leq x} \mu(n) | \leq x/4345$ for all $x\geq 2\,160\,535$ and was due to Cohen, Dress and El Marraki. We also prove that $|\sum_{n \leq x} \mu(n) | \leq x/180\,194$ for all $x\geq 10^{19}$., Comment: 11 pages, In French

We provide numerical bounds for $\Sigma(X)=\sum_{\substack{d_1,d_2\le X}}\frac{\mu(d_1)\mu(d_2)}{[d_1,d_2]}$. We show in particular that $0\le \Sigma(X)\le 17/25$ for every $X\ge2$.

The paper considers estimates for the asymptotics of summation functions of bounded multiplicative arithmetic functions. Several assertions on this subject are proved and examples are considered., Comment: 13 pages

In this paper, we establish a mean value result of arithmetic functions over shorter intervals with the Selberg-Delange method using the Hooley-Huxley contour.

The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic functions. Several assertions on this topic are proved and examples are considered., Comment: 16 pages

The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and examples are considered., Comment: 16 pages

In this paper, we investigate the asymptotics of a class of weighted sums over multiplicative functions and apply our results to deduce a stronger asymptotic form of Yitang Zhang's smoothened GPY sieve with coefficient expressions that are friendly to numerical computations.

We define the asymptotic behavior "almost everywhere" of additive and multiplicative arithmetic functions in the paper. Classes of additive and multiplicative arithmetic functions are singled out for which the asymptotics coincides "almost everywhere" with the asymptotics of the corresponding strongly additive and multiplicative arithmetic functions. Several assertions are proved and examples are considered., Comment: 20 pages

Let $\zeta^k(s) = \sum_{n=1}^\infty \tau_k(n) n^{-s}, \Re s > 1$. We present three conditional results on the ternary additive correlation sum $$\sum_{n\le X} \tau_3(n) \tau_3(n+h),\quad (h\ge 1),$$ and give numerical verifications of our method. The first is a conditional proof for the full main term of the above correlation sum for any composite shift $1 \le h \le X^{2/3}$, on assuming an averaged level of distribution for the three-fold divisor function $\tau_3(n)$ in arithmetic progressions to level two-thirds. The second is a conditional derivation for the leading order main term asymptotics of this correlation sum, also valid for any composite shift $1 \le h \le X^{2/3}$. The third result gives a complete expansion of the polynomial for the full main term for the special case $h=1$ from both our method and from the delta-method, showing that our answers match. Our method is essentially elementary, especially for the $h=1$ case, uses congruences, and, as alluded to earlier, gives the same answer as in prior prediction of Conrey and Gonek [Duke Math. J. 107 (3) 2002], previously computed by Ng and Thom [Funct. Approx. Comment. Math. 60(1) 2019], and unpublished heuristic probabilistic arguments of Tao. Our procedure is general and works to give the full main term with a power-saving error term for any correlations of the form $\sum_{n\le X} \tau_k(n) f(n+h)$, to any composite shift $h$, and for a wide class of arithmetic function $f(n)$., Comment: details on contours shifting added; presentation improved. 46 pages, 2 tables, 4 figures

Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivi\'{c} proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.

It is conjectured that the sum $$ S_r(n)=\sum_{k=1}^{n} \frac{k}{k+r}\binom{n}{k} $$ for positive integers $r,n$ is never integral. This has been shown for $r\le 22$. In this note we study the problem in the ``$n$ aspect" showing that the set of $n$ such that $S_r(n)\in {\mathbb Z}$ for some $r\ge 1$ has asymptotic density $0$. Our principal tools are some deep results on the distribution of primes in short intervals.

We consider a method for estimating asymptotics of arithmetic functions on a arithmetic progression in the paper. This question is closely related with the distribution of primes. Problems often arise even with the definition of asymptotic of the mean value of arithmetic functions, and even more with the determination of asymptotics of moments of higher orders. Therefore, the paper proposes (a different from the traditional) probabilistic approach based on the limiting distribution of arithmetic functions on an arithmetic progression. We consider arithmetic functions with limiting normal distribution. Several assertions are proved about the estimation of asymptotics of moments of strongly additive arithmetic functions on a arithmetic progression, as well as additive functions of the class and arithmetic functions of the class having the same limit distribution. Versions 1 and 2 consider the asymptotics of the moments of arithmetic functions with normal limit distribution in natural series., Comment: 18 pages

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$., Comment: 10 pages

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's result corresponded to Huxley's $7/12$ exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramar\'e's identity to extract a small prime factor from the $n$-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the M\"obius function as well as some results on multiplicative functions and almost primes in short intervals., Comment: 18 pages; referee comments incorporated

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., Comment: 13 pages; comments are welcome

Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved., Comment: 11 pages

A conjecture due to Cilleruelo states that for an irreducible polynomial $f$ with integer coefficients of degree $d\geq 2$, the least common multiple $L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log L_f(N)\sim (d-1)N\log N$ as $N\to \infty$. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of $N$ depending on the range of shifts., Comment: minor changes

The paper compares the asymptotic of the expressions $\frac {1} {x} \sum\limits_{n \leq x} {f(n)}$ and $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$, $\frac {1} {x} \sum\limits_{p \leq x} {f(p)}$ and $\sum\limits_{p \leq x} {\frac {f(p)} {p}}$. The asymptotic of sums $\sum\limits_{n \leq x} {\frac {f(n)} {n}}$ and $\sum\limits_{p \leq x} {\frac {f(p)} {p}}$ ($n,p$ - respectively, positive and prime numbers) are determined if the asymptotic of sums are known, respectively: $\sum\limits_{n \leq x} {f(n)}$,$\sum\limits_{p \leq x} {f(p)}$., Comment: 9 pages

Let $P(x)$ be an irreducible quadratic polynomial in $\mathbb{Z}[x]$. We show that for almost all $n$, $P(n)$ does not lie in the range of Euler's totient function., Comment: 18 pages

The paper compares probabilistic and exact methods for estimating the asymptotic behavior of summation arithmetic functions, and estimates of the results are obtained by precise methods. Conditions for stationarity in the broad sense are investigated for summation arithmetic functions. A lemma and theorems about the estimation of the standard deviation for the summation arithmetic Mertens and Lowville functions completely satisfying the stationarity conditions in the broad sense are proved., Comment: 9 pages

A famous conjecture of Chowla states that the Liouville function $\lambda(n)$ has negligible correlations with its shifts. Recently, the authors established a weak form of the logarithmically averaged Elliott conjecture on correlations of multiplicative functions, which in turn implied all the odd order cases of the logarithmically averaged Chowla conjecture. In this note, we give a new and shorter proof of the odd order cases of the logarithmically averaged Chowla conjecture. In particular, this proof avoids all mention of ergodic theory, which had an important role in the previous proof., Comment: 15 pages, no figures, submitted, J. Numb. Thy. Bordeaux

Following Wigert, various authors, including Ramanujan, Gronwall, Erd\H{o}s, Ivi\'{c}, 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 functions $f$. In particular, this class contains functions counting ideals of given norm in the ring of integers of an arbitrary, fixed quadratic number field. As a consequence, we determine such maximal orders for several multiplicative $f$ arising as a normalized function counting representations by certain binary quadratic forms. Incidentally, for the non-multiplicative function $r_2$ which counts how often a positive integer is represented as a sum of two squares, this entails the asymptotic formula $$ \max_{n\leq x} \log r_2(r_2(n))= \frac{\sqrt{\log x}}{\log \log x} (c/\sqrt{2}+o(1))$$ with some explicitly given constant $c>0$., Comment: To appear in Mathematika

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that $X^{8/33+\varepsilon} \leq H \leq X^{1-\varepsilon}$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H \geq X^{1/3+\varepsilon}$. We also obtain an analogous result for $\sum_n \Lambda(n) \Lambda(N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying H\"older's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates., Comment: 80 pages, no figures. updated references

Let $\tau(n)$ be the number of divisors of $n$. We give an elementary proof of the fact that $$ \sum_{n\le x} \tau(n)^r =xC_{r} (\log x)^{2^r-1}+O(x(\log x)^{2^r-2}), $$ for any integer $r\ge 2$. Here, $$ C_{r}=\frac{1}{(2^r-1)!} \prod_{p\ge 2}\left( \left(1-\frac{1}{p}\right)^{2^r} \left(\sum_{\alpha\ge 0} \frac{(\alpha+1)^r}{p^{\alpha}}\right)\right). $$, Comment: 8 pages, revised

An arithmetic function $f$ is called a $sieve$ $function$ of $range$ $Q$ if its Eratosthenes transform $g=f\ast\mu$ has support in $[1,Q]$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). We continue our study of the distribution of such functions over short $arithmetic$ $bands$, $n\equiv ar+b\, (\bmod\,q)$, with $1\le a\le H=o(N)$ and $r,b$ integers such that g.c.d.$(r,q)=1$. In particular, we discuss the optimality of some results., Comment: 5 pages, plain TeX

Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} \tau_k(n) \tau_{\ell}(n+h)$ where $\tau_k$ and $\tau_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main result is a lower bound of the correct order of magnitude for $D_{k,\ell}(x,h)$, uniform in $h$. In addition, the conjectural asymptotic formula for $D_{k,\ell}(x,h)$ is studied. Using an argument of Ivi\'{c} and Conrey-Gonek the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao., Comment: This is the second version of the article. The article now considers the more general sums $D_{k,\ell}(x,h) = \sum_{n \le x} \tau_k(n) \tau_{\ell}(n+h)$. We also show that our probabilistic argument and Terry Tao's probabilistic argument give the same conjecture for $D_{k,\ell}(x,h)$

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to 0 modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_1(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_1(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_1(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the other hand however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_1(n)$ tends to infinity on the integers outside this set. Finally we consider the asymptotic behaviour of the difference between these two functions and find that $\sum_{n\leq x}\big( a(n)-P_1(n)\big) \sim \left(1-\frac{\pi}{4}\right)\sum_{n\leq x} P_2(n)$, where $P_2(n)$ is the second largest divisor of $n$.

We study the arithmetic function sopfr$(n)$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$. In particular we obtain the asymptotic formula $$ \sum_{n \leq x} \rm{sopfr}(n) \sim \frac{\pi^2}{12} \frac{x^2}{\log x},$$ which holds as well for the function sopf$(n)$ (OEIS A008472) that just gives the sum of distinct prime factors of $n$. This asymptotic formula was already stated by R. Jakimcyuk \cite{rj12} which was brought to our attention after the completion of the first version of this manuscript., Comment: 4 pages

For pedagogical purposes (inclusion in lecture notes) we review the proof of the theorem stated in the title. At the end we state a problem., Comment: 6 pages, just for fun of reading proofs