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Let f(z) be a holomorphic or Maass cusp form for the full modular group S L (2 , Z) . In this paper, we consider the asymptotic behavior of the sums U w (x) = ∑ n ≤ x λ w , f × f (n) , which can be regarded as the general divisor problems considered by Kanemitsu et al. (Monatsh Math 136(1):17–34, 2002), and get some new results which improve previous results. [ABSTRACT FROM AUTHOR]

Let f and g be two distinct normalized primitive Hecke cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f ⊗ f ⊗ f ⊗ g (n) and λ sym 2 f ⊗ f ⊗ g (n) the nth normalized coefficients of the automorphic L-functions L (f ⊗ f ⊗ f ⊗ g , s) and L (sym 2 f ⊗ f ⊗ g , s) , respectively. In this paper, we are interested in the average behavior of the coefficients λ f ⊗ f ⊗ f ⊗ g (n) and λ sym 2 f ⊗ f ⊗ g (n) on a primitive integral binary quadratic form with negative discriminant whose class number is 1, and we also provide the asymptotic formulae of these summatory functions. As an application, we also consider the number of sign changes of the sequences { λ f ⊗ f ⊗ f ⊗ g (n) } n ⩾ 1 and { λ sym 2 f ⊗ f ⊗ g (n) } n ⩾ 1 on the same binary quadratic form in short intervals. [ABSTRACT FROM AUTHOR]

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let σ o mex (n) (resp., σ e mex (n) ) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we find Hardy-Ramanujan type asymptotic formulae for both σ o mex (n) and σ e mex (n) . We also prove some infinite families of congruences for σ o mex (n) and σ e mex (n) modulo 4 and 8 [ABSTRACT FROM AUTHOR]

Let (n, q) denote the greatest common divisor of positive integers n and q, and let f r denote the characteristic function of r-full numbers. We consider several asymptotic formulas for sums of the modified square-full ( r = 2 ) and cube-full numbers ( r = 3 ), which is ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f r q d log x q with any positive real numbers x and y. Moreover, we derive the asymptotic formula of the above with r = 2 under the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]

The near orthgonality of certain k-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average 1 k s (r + 1) ∑ j = 1 k s j r c k (s) (j) remains positive for all r ≥ 1 . Further, we give a lower bound for max N ∑ j = 1 N s c k (s) (j) . [ABSTRACT FROM AUTHOR]

Let R (1 , 1) (n) denote the coefficients of the Dirichlet series ζ ′ (s) L ′ (s , χ 4) = Σ n = 1 ∞ R (1 , 1) (n) n - s for R e s > 1 and P (1) (x) the error term of Σ n ≤ x R (1 , 1) (n). A representation of the Chowla–Walum type formula for P (1) (x) is derived. As a direct application, we shall give a new order estimate for P (1) (x) , which constitutes an improvement over the evaluation originating from Furuya et al. Furthermore, the asymptotic formula of the integral ∫ 1 X P (1) k (x) d x is established for k = 3 , 4 . [ABSTRACT FROM AUTHOR]

Let u = (u n) n ≥ 0 be a Lucas sequence, that is, a sequence of integers satisfying u 0 = 0 , u 1 = 1 , and u n = a 1 u n - 1 + a 2 u n - 2 for every integer n ≥ 2 , where a 1 and a 2 are fixed nonzero integers. For each prime number p with p ∤ 2 a 2 D u , where D u : = a 1 2 + 4 a 2 , let ρ u (p) be the rank of appearance of p in u , that is, the smallest positive integer k such that p ∣ u k . It is well known that ρ u (p) exists and that p ≡ (D u ∣ p) (mod ρ u (p)) , where (D u ∣ p) is the Legendre symbol. Define the index of appearance of p in u as ι u (p) : = p - (D u ∣ p) / ρ u (p) . For each positive integer t and for every x > 0 , let P u (t , x) be the set of prime numbers p such that p ≤ x , p ∤ 2 a 2 D u , and ι u (p) = t . Under the Generalized Riemann Hypothesis, and under some mild assumptions on u , we prove that # P u (t , x) = A F u (t) G u (t) x log x + O u x (log x) 2 + x log log (3 x) φ (t) (log x) 2 , for all positive integers t and for all x > t 3 , where A is the Artin constant, F u (·) is a multiplicative function, and G u (·) is a periodic function (both these functions are effectively computable in terms of u ). Furthermore, we provide some explicit examples and numerical data. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. We consider sums of type ∑ n 1 ⋯ n k ≤ x F (n 1 , ... , n k) , taken over the hyperbolic region { (n 1 , ... , n k) ∈ N k : n 1 ⋯ n k ≤ x } , where F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ∑ n 1 ⋯ n k ≤ x f ((n 1 , ... , n k)) and ∑ n 1 ⋯ n k ≤ x f ([ n 1 , ... , n k ]) , involving the GCD and LCM of the integers n 1 , ... , n k , where f : N → C belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k = 2 . [ABSTRACT FROM AUTHOR]

For a number field K , and integral ideals I and J in its number ring O K , Nowak studied the asymptotic behavior of the average of Ramanujan sums C J (I) over both ideals I and J . In this article, we extend this investigation by establishing asymptotic formulas for the second moment of averages of Ramanujan sums over quadratic and cubic number fields, thereby generalizing previous works of Chen, Kumchev, Robles, and Roy on moments of averages of Ramanujan sums over rationals. Additionally, using a special property of certain integral domains, we obtain second moment results for Ramanujan sums over some other number fields. [ABSTRACT FROM AUTHOR]

In this short paper, we study sums of the shape ∑ n ⩽ x f ([ x / n ]) / [ x / n ] , where f is the Euler totient function φ , the Dedekind function Ψ , the sum-of-divisors function σ or the alternating sum-of-divisors function β. We improve previous results when f = φ and derive new estimates when f = Ψ , f = σ and f = β. [ABSTRACT FROM AUTHOR]

Let p ¯ (n) denote the overpartition function. In this paper, we obtain an inequality for the sequence Δ 2 log p ¯ (n - 1) / (n - 1) α n - 1 which states that log (1 + 3 π 4 n 5 / 2 - 11 + 5 α n 11 / 4 ) < Δ 2 log p ¯ (n - 1) / (n - 1) α n - 1 < log (1 + 3 π 4 n 5 / 2 ) for n ≥ N (α) , where α is a non-negative real number, N (α) is a positive integer depending on α , and Δ is the difference operator with respect to n. This inequality consequently implies log -convexity of { p ¯ (n) / n n } n ≥ 19 and { p ¯ (n) n } n ≥ 4 . Moreover, it also establishes the asymptotic growth of Δ 2 log p ¯ (n - 1) / (n - 1) α n - 1 by showing lim n → ∞ Δ 2 log p ¯ (n) / n α n = 3 π 4 n 5 / 2 . [ABSTRACT FROM AUTHOR]

We provide examples of multiplicative functions f supported on the k-free integers such that at primes f (p) = ± 1 and such that the partial sums of f up to x are o (x 1 / k) . Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent 1/k: There are examples such that the partial sums up to x are o (x 1 / (k + 1 2) + ϵ) , for all ϵ > 0 . This generalizes to the k-free integers the results of Aymone (J. Number Theory, 212:113-121, 2020). [ABSTRACT FROM AUTHOR]

Let f 3 denote the characteristic function of cube-full numbers, and let (n, q) be the greatest common divisor of positive integers n and q. For any positive real numbers x and y, we shall consider several asymptotic formulas for sums of sums of modified cube-full numbers, which is ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d f 3 q / d k with k = 1 , 2 . [ABSTRACT FROM AUTHOR]

Let Z n be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group Z m × Z n , where m and n are arbitrary positive integers. Let k ≥ 2 be a fixed integer, we consider the k-th power sequence. For x ≥ 1 , define S k (x) : = ∑ m , n ≤ x s (m k , n k). In this paper, we obtain an asymptotic formula of S k (x) by using a multidimensional Perron formula and the complex integration method. [ABSTRACT FROM AUTHOR]

Let k ⩾ 2 be a fixed natural number and d k (n) denote the number of ways n can be written as a product of k positive integers. We use Δ k (x) to denote the error term in the asymptotic formula of the summatory function of d k (n). The aim of this paper is to study the hybrid moments of Δ 2 (x) and Δ 3 (x) . The higher power moments of Δ 2 (x) are also studied. [ABSTRACT FROM AUTHOR]

Let us denote by d(n) the number of divisors of n, by li (t) the logarithmic integral of t, by β 2 the number log 3 / 2 log 2 = 0.584 ... and by R(t) the function t ↦ 2 t + ∑ ρ t ρ / ρ 2 log 2 t , where ρ runs over the non-trivial zeros of the Riemann ζ function. In his PHD thesis about highly composite numbers, Ramanujan proved, under the Riemann hypothesis, that log d (n) log 2 ⩽ li (log n) + β 2 li (log β 2 n) - log β 2 n log log n - R (log n) + O log n (log log n) 3 holds when n tends to infinity. The aim of this paper is to give an effective form to the above asymptotic result of Ramanujan. [ABSTRACT FROM AUTHOR]

We define the k-dimensional generalized Euler function φ k (n) as the number of ordered k-tuples (a 1 , ... , a k) ∈ N k such that 1 ≤ a 1 , ... , a k ≤ n and both the product a 1 ⋯ a k and the sum a 1 + ⋯ + a k are prime to n. We investigate some of the properties of the function φ k (n) , and obtain a corresponding Menon-type identity. [ABSTRACT FROM AUTHOR]

In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three L-functions which can be factorized as a product of a degree one and a degree two L-functions. We emphasize that we can break the 1/2-barrier in the error term, and we get an explicit exponent. Our proof relies on methods from the theory of exponential sums. [ABSTRACT FROM AUTHOR]

In this paper, we study Ramanujan sums c J (I) , where I and J are integral ideals in an arbitrary quadratic number field. In particular, the asymptotic behaviour of sums of c J (I) over both I and J is investigated. [ABSTRACT FROM AUTHOR]

Let n , q , a ∈ N , d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over n ≤ x and n ≡ a (mod q) . Previously we obtained an asymptotic formula for the second moment V (x , Q) = ∑ q ≤ Q ∑ a = 1 q A (x , q , a) - M (x , q , a) 2 , where Q ≤ x and M(x, q, a) is a usual expected value for A(x, q, a). In this article, we compute the second moment similar to the previous one but M(x, q, a) is replaced by a better approximating function ρ R (x , q , a) given by ρ R (x , q , a) = ∑ n ≤ x n ≡ a (mod q) ∑ r ≤ R c r (n) r log n r 2 + 2 γ , where c r (n) is Ramanujan's sum. The advantage of using ρ R (x , q , a) instead of M(x, q, a) is that the new second moment, for a suitable R, gives us practically the same main term and the error term of smaller or the same order of magnitude with much simpler and shorter calculation than that of V(x, Q). [ABSTRACT FROM AUTHOR]

We give explicit bounds on sums of d (n) 2 and d 4 (n) , where d(n) is the number of divisors of n and d 4 (n) is the number of ways of writing n as a product of four positive integers. In doing so, we make a slight improvement on the upper bound for class numbers of quartic number fields. [ABSTRACT FROM AUTHOR]

Let F (x) ∈ Z [ x 1 , x 2 , ... , x n ] , n ≥ 3 , be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum Σ k , F (X ; B) = ∑ x ∈ X B ∩ Z n τ k F (x) ; for X tending to infinity, where B ⊂ R n is an n-dimensional box such that min x ∈ X B F (x) ≥ 0 for all sufficiently large X, and τ k (·) is the kth divisor function for any integer k ≥ 2 . [ABSTRACT FROM AUTHOR]

Let f be an arithmetic function and let S # denote the extended Selberg class. We denote by L (s) = ∑ n = 1 ∞ f (n) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of L (s) , which belongs to S # , are the coefficients of the Laurent expansion of L at its pole s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Elliott's weighted Erdős–Kac theorem to the case of short intervals. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the normalized Fourier coefficients of a holomorphic Hecke cusp form of full level. We study a generalized divisor problem with λ f (n) over some special sequences. More precisely, for any fixed integer k ≥ 2 and j ∈ { 1 , 2 , 3 , 4 } , we are interested in the following sums S k (x , j) : = ∑ n ≤ x λ k , f (n j) = ∑ n ≤ x ∑ n = n 1 n 2 ⋯ n k λ f (n 1 j) λ f (n 2 j) ⋯ λ f (n k j). [ABSTRACT FROM AUTHOR]

We introduce a shifted convolution sum that is parametrized by the squarefree natural number t. The asymptotic growth of this series depends explicitly on whether or not t is a congruent number, an integer that is the area of a rational right triangle. This series presents a new avenue of inquiry for the congruent number problem. [ABSTRACT FROM AUTHOR]

We obtain uniform estimates for N k (x , y) , the number of positive integers n up to x for which ω y (n) = k , where ω y (n) is the number of distinct prime factors of n which are < y . The motivation for this problem is an observation due to the first author in 2015 that for certain ranges of y, the asymptotic behavior of N k (x , y) is different from the classical situation concerning N k (x , x) studied by Sathe and Selberg. We demonstrate this variation of the classical theme; to estimate N k (x , y) we study the sum S z (x , y) = ∑ n ≤ x z ω y (n) for R e (z) > 0 by the Buchstab–de Bruijn method. We also utilize a certain recent result of Tenenbaum to complete our asymptotic analysis. [ABSTRACT FROM AUTHOR]

Investigating a question of Alladi, we describe the local distribution of small prime factors of integers, with emphasis on the transition phase occurring for certain values of the parameters. [ABSTRACT FROM AUTHOR]

We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of x k in the reversion of the power series ∑ k ≥ 1 μ (k) x k , where μ (k) is the usual number-theoretic Möbius function. Using this result, we deduce an asymptotic expression for the number of such systems. [ABSTRACT FROM AUTHOR]

The Landau–Selberg–Delange method gives an asymptotic formula for the partial sums of a multiplicative function f whose prime values are α on average. In the literature, the average is usually taken to be α with a very strong error term, leading to an asymptotic formula for the partial sums with a very strong error term. In practice, the average at the prime values may only be known with a fairly weak error term, and so we explore here how good an estimate this will imply for the partial sums of f, developing new techniques to do so. [ABSTRACT FROM AUTHOR]

Let Λ = { λ j } j = 1 ∞ be a set of positive integers, and let p Λ (n) be the number of ways of writing n as a sum of positive integers λ ∈ Λ such that χ (λ) = 1 and f (λ) ≡ j (mod m) , where m ≥ 1 and j ≥ 0 are fixed integers. Here, χ and f are a certain multiplicative function and an additive function, respectively. In this paper, we obtain the asymptotic formula for ln ∑ n ≤ x p Λ (n) ∼ ln p Λ ([ x ]) as x → ∞ . [ABSTRACT FROM AUTHOR]

Hafner and Stopple proved a conjecture of Zagier on the asymptotic expansion of a Lambert series involving Ramanujan's tau function with the main term involving the nontrivial zeros of the Riemann zeta function. Recently, Chakraborty et. al. have extended this result to any cusp form over the full modular group and also over any congruence subgroup. The aim here is to study the asymptotic behaviour of a similar Lambert series involving the coefficients of Maass cusp forms over the full modular group. [ABSTRACT FROM AUTHOR]

Let L^ be the operator given by L^{an}n≥0={an+12-anan+2}n≥0. A sequence {an}n≥0 is called asymptotically r-log-concave if L^k{an}n≥N are non-negative sequences for 1≤k≤r and some integer N. Let p(n) be the number of integer partitions of n. We prove that the sequence {p(n)}n≥1 is asymptotically r-log-concave for any positive integer r. Moreover, we give a method to compute the explicit N such that {p(n)}n≥N is r-log-concave. [ABSTRACT FROM AUTHOR]

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel. [ABSTRACT FROM AUTHOR]

We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms ∑H≤h≤2HWhH∑X≤n≤2Xλ1(n-h)λ2(n)λ3(n+h), which is nontrivial provided that H≥X2/3+ε. The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions. [ABSTRACT FROM AUTHOR]

We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Halász. Several applications are derived, including estimates for the difference of mean-values of so-called pretentious functions, local laws for the distribution of prime factors in an arbitrary set, and weighted distribution of additive functions. [ABSTRACT FROM AUTHOR]

We study the typical behaviour of the size of the ratio set A / A for a random subset $$A\subset \{1,\dots , n\}$$ . For example, we prove that $$|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2 $$ for almost all subsets $$A\subset \{1,\dots ,n\}$$ . We also prove that the proportion of visible lattice points in the lattice $$A_1\times \cdots \times A_d$$ , where $$A_i$$ is taken at random in [1, n] with $$\mathbb P(m\in A_i)=\alpha _i$$ for any $$m\in [1,n]$$ , is asymptotic to a constant $$\mu (\alpha _1,\dots ,\alpha _d)$$ that involves the polylogarithm of order d. [ABSTRACT FROM AUTHOR]

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In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative. We also prove results concerning the non-vanishing in short intervals and strong cancellation among the coefficients evaluated at powers of a fixed prime. Our results rest on a Serre's type density result established by Kowalski and Saha in the Appendix. [ABSTRACT FROM AUTHOR]

We study the parity of the prime-counting function $$\pi (x)$$ , and more generally the distribution of its values in residue classes modulo $$q$$ . We prove that for any integer $$q\ge 2$$ and $$0\le a\le q-1$$ , the function $$\pi (n)$$ lies in the residue class $$a \pmod q$$ for a positive proportion of integers $$n\ge 1$$ . Based on the numerical evidence, we conjecture that $$\pi (n)$$ should be equidistributed among the residue classes modulo $$q$$ , and we prove an average version of this conjecture using the large sieve. [ABSTRACT FROM AUTHOR]

We prove that the partition function $$p(n)$$ is log-concave for all $$n>25$$ . We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer's estimates on the remainders of the Hardy-Ramanujan and the Rademacher series for $$p(n)$$ . [ABSTRACT FROM AUTHOR]

In 2008, Deza and Varukhina established asymptotic formula for the mean value of the arithmetic function $$\tau _{k_1}^K(n)\tau _{k_2}^K(n)\cdots \tau _{k_l}^K(n)$$ , where $$K$$ is a quadratic or cyclotomic field, and $$\tau _{k}^K(n)$$ is the $$k$$ -dimensional divisor function in the number field $$K$$ . Recently, Lü generalized their results to any Galois extension $$K$$ of the rational field. It seems interesting to deal with similar problems which involve different number fields. In this paper, we are concerned with the mean value of the arithmetic function $$\tau _{k_1}^{K_1}(n)\tau _{k_2}^{K_2}(n)\cdots \tau _{k_l}^{K_l}(n)$$ , where $$K_j$$ are number fields whose discriminants are relatively prime. [ABSTRACT FROM AUTHOR]

For an arithmetic multiplicative function $$f\ge 0\ $$ , the generalized divisor function $$ \fancyscript{T}(m, v):= \sum \nolimits _{d|m,\; d\le v}f(d), \ m\in \mathbb {N}, \ v\in \mathbb {R},\ $$ and the corresponding arithmetically defined random process $$ X(m,t):= \fancyscript{T}(m,m^t)/\fancyscript{T}(m,m),\ 0\le t\le 1,$$ are considered. An asymptotic formula for the second moment of $$X(m,t)$$ is obtained. The remainder term is estimated as well. [ABSTRACT FROM AUTHOR]

An abstract Central Limit Theorem is applied to give weighted Central Limit Theorems for classical holomorphic and Maass wave forms. Examples and conjectural related results are considered. [ABSTRACT FROM AUTHOR]

A correction is made to the renormalising constants in an example relating the classical theorem of Erdős-Kac to the Fourier coefficients of automorphic forms. [ABSTRACT FROM AUTHOR]