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Let λf(n), σ(n) and ϕ(n) denote nth Fourier coefficient of the primitive holomorphic cusp form f, the sum-of-divisors function, and the Euler totient function, respectively. In this paper, we first refine related previous results on the power sum ∑ n ≤ x λ f j (n) σ b (n) ϕ c (n) and then investigate the average behavior of the power sum ∑ n = a 2 + b 2 + c 2 + d 2 ≤ x , (a , b , c , d) ∈ Z 4 λ f j (n) σ b (n) ϕ c (n). [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the monomial-prime numbers, which are of the form p n k for primes p and integers k ≥ 2 . First, we give an asymptotic estimate on the number of numbers of a general form pf(n) for arithmetic functions f satisfying certain growth conditions, which generalizes Bhat's recent result on the Square-Prime Numbers. Then, we prove three Mertens-type theorems related to numbers of a more general form, partially extending the recent work of Qi-Hu, Popa and Tenenbaum on the Mertens sum evaluations. At the end, we evaluate the average and variance of the number of distinct monomial-prime factors of positive integers by applying our Mertens-type theorems. [ABSTRACT FROM AUTHOR]

Let Λ denote the von Mangoldt function, and (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 involving the von Mangoldt function; S k (x , y) : = ∑ n ≤ y ∑ q ≤ x ∑ d | (n , q) d Λ q d k for k = 1 , 2 . [ABSTRACT FROM AUTHOR]

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 be any arithmetic function and define S f (x) := ∑ n ≤ x f ([ x / n ]) . If the function f is small, namely, f(n) ≪ nε, then the error term Ef(x) in the asymptotic formula of Sf(x) has the form O(x1/2+ε). In this paper, we shall study the mean square of Ef(x) and establish some new results of Ef(x) for some special functions. [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 ⩾ 3 r\geqslant 3 be an integer and 푄 any positive definite quadratic form in 푟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 퐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 . [ABSTRACT FROM AUTHOR]

Subsets of rational numbers are specified as preimages of values of arithmetical functions. The uniformity of distribution of elements of these sets is proved and interpreted in the context of Diophantine approximation to real numbers. [ABSTRACT FROM AUTHOR]

We characterize the limiting behavior of partial sums of multiplicative functions f : F q [ t ] → S 1 . In contrast to the number field setting, the characterization depends crucially on whether the notion of discrepancy is defined using long intervals, short intervals, or lexicographic intervals. Concerning the notion of short interval discrepancy, we show that a completely multiplicative f : F q [ t ] → { - 1 , + 1 } with q odd has bounded short interval sums if and only if f coincides with a "modified" Dirichlet character to a prime power modulus. This confirms the function field version of a conjecture over Z that such modified characters are extremal with respect to partial sums. Regarding the lexicographic discrepancy, we prove that the discrepancy of a completely multiplicative sequence is always infinite if we define it using a natural lexicographic ordering of F q [ t ] . This answers a question of Liu and Wooley. Concerning the long sum discrepancy, it was observed by the Polymath 5 collaboration that the Erdős discrepancy problem admits infinitely many completely multiplicative counterexamples on F q [ t ] . Nevertheless, we are able to classify the counterexamples if we restrict to the class of modified Dirichlet characters. In this setting, we determine the precise growth rate of the discrepancy, which is still unknown for the analogous problem over the integers. [ABSTRACT FROM AUTHOR]

Let [t] be the integral part of a real number t, and let f be the arithmetic function satisfying some simple condition. We establish a new asymptotical formula for the sum S f (x) = ∑ n ≤ x f ([ x / n ]) , which improves the recent result of J. Stucky (2022). [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]

Asymptotic formulae are established for the number of natural numbers mm with largest square-free divisor not exceeding mϑ{m}^{{\vartheta }}, for any fixed positive parameter ϑ{\vartheta }. Related counting functions are also considered.

We give asymptotic analysis of power series associated with lacunary partition functions. New partition theoretic interpretations of some basic hypergeometric series are offered as examples. [ABSTRACT FROM AUTHOR]

This paper concerns the visibility properties of lattice points in multiple random walks on N k , where k ≥ 2 is an integer. We study two aspects of the visibility: simultaneous visibility in multiple random walkers; and that only some of these walkers are visible. Combining tools from number theory and probability theory, we prove the corresponding densities of the above two parts. [ABSTRACT FROM AUTHOR]

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences. [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]

Finding elliptic curves with high ranks has been the focus of much research. Recently, with the goal of generating elliptic curves with a large rank, some authors used large integers n which have many divisors, amongst which one can find divisors d such that d + n / d is a perfect square. This strategy is in itself a motivation for studying the function τ □ (n) which counts the number of divisors d of an integer n for which d + n / d is a perfect square. We show that ∑ n ≤ x τ □ (n) = c □ x 3 / 4 + O (x) for some explicit constant c □ . Moreover, letting ρ 1 (n) : = max { d ∣ n : d ≤ n } and ρ 2 (n) : = min { d ∣ n : d ≥ n } stand for the middle divisors of n, we show that the order of magnitude of the number of positive integers n ≤ x for which ρ 1 (n) + ρ 2 (n) is a perfect square is x 3 / 4 / log x . [ABSTRACT FROM AUTHOR]

In this article, we study the distribution of index of Farey fractions by providing asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with B -free denominators. Additionally, we reconsider the square-free case earlier done in Alkan et al. (Ramanujan J 16(2):131–161, 2008), and obtain new results for moments of indices with square-free denominators. We also obtain higher level correlation measures of the index function, generalizing earlier known results on two level correlations. [ABSTRACT FROM AUTHOR]

We study the correlations of pairs of complex logarithms of Z -lattice points in C at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighbourhood to itself in the Bianchi orbifold PSL 2 (Z [ i ]) \ H R 3 . [ABSTRACT FROM AUTHOR]

We apply the character sums method of Lenstra, Moree, and Stevenhagen to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form ∑ n = 1 ∞ g (n) # G (n) in which g(n) is a multiplicative arithmetic function and { G (n) } is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants. [ABSTRACT FROM AUTHOR]

A multiplicative function f is said to be resembling the Möbius function if f is supported on the square-free integers, and f (p)= ±1 for each prime p. We prove O- and Ω-results for the summatory function ∑n≤xf(n) for a class of these f, and the point is that these O-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is O(x1/3+ε) under the Riemann Hypothesis. On the other hand it is proved to be Ω(x1/4) unconditionally. It is interesting to compare these with the corresponding results for the Möbius function. [ABSTRACT FROM AUTHOR]

For integer n ⩾ 1 and real u, let Δ(n, u):= ∣{d: d ∣ n, eu < d ⩽ eu+1}∣. The Erdős-Hooley Delta-function is then defined by Δ(n):=maxu∈ℝ Δ(n, u). We improve the current upper bounds for the average and normal orders of this arithmetic function. [ABSTRACT FROM AUTHOR]

Let P be a fixed set of primes possessing a limiting frequency ν , as detected by the weight 1/p. We show that for any fixed α ∈ (0 , 1) , the ⌈ α Ω (n) ⌉ -th smallest prime factor of n, denoted P (α) (n) , belongs to P on a set of n with natural density ν . We prove a similar result for the largest prime factor P ≤ y (n) of n not exceeding y, whenever y → ∞ . As corollaries, P (α) (n) and P ≤ y (n) conform to Benford's leading digit law. Finally, we establish the equidistribution of P (α) (n) in coprime residue classes, in an essentially optimal range of uniformity in the modulus. [ABSTRACT FROM AUTHOR]

Using hyperbola method, Dirichlet, in 1849, proved that the error term in the study of the summatory function of the divisor function d(n) is O (x) . Then in 1904, Voronoï used the method of contour integration to improve the error term as O (x 1 3 + ϵ) , for any positive ϵ . Recently, Gupta and Maji (J. Math. Anal. Appl.507 (2022) 125738) studied the following generalized divisor function: for any k ∈ N , r ∈ Z , D k , r (n) = ∑ d k | n n d k r. In this paper, we obtain a Voronoï error bound for the summatory function of D k , r (n) . [ABSTRACT FROM AUTHOR]