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We deduce asymptotic formulas for the sums ∑ n 1 , … , n r ≤ x f ( n 1 ⋯ n r ) and ∑ n 1 , … , n r ≤ x f ( [ n 1 , … , n r ] ) , where r ≥ 2 is a fixed integer, [ n 1 , … , n r ] stands for the least common multiple of the integers n 1 , … , n r and f is one of the divisor functions τ 1 , k ( n ) ( k ≥ 1 ), τ ( e ) ( n ) and τ ⁎ ( n ) . Our formulas refine and generalize a result of Lelechenko (2014). A new generalization of the Busche–Ramanujan identity is also pointed out. [ABSTRACT FROM AUTHOR]

Text For each positive integer n , let L ( n ) be the length of longest arithmetic progressions in the least positive reduced residue system modulo n . In this paper, we give an asymptotic formula for ∑ n ≤ x L ( n ) . This solves a problem of Pongsriiam (2018) [4] . Video For a video summary of this paper, please visit https://youtu.be/xJEpSBYoZJ0 . [ABSTRACT FROM AUTHOR]

Résumé Let α ∈ ] 0 , 1 ] and let Q j ( 1 ⩽ j ⩽ r ) denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the Q j ( n ) for x < n ⩽ x + x α is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the Q j . [ABSTRACT FROM AUTHOR]

We continue our study of convolution sums of two arithmetical functions f and g , of the form ∑ n ≤ N f ( n ) g ( n + h ) , in the context of heuristic asymptotic formulæ. Here, the integer h ≥ 0 is called, as usual, the shift of the convolution sum. We deepen the study of finite Ramanujan expansions of general f , g for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of f and g , namely in terms of its shift h and we compare this ‘‘shift Ramanujan expansion’’, with our previous finite expansions in terms of the f and g arguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formulæ. [ABSTRACT FROM AUTHOR]

We study the difference between the number of primitive roots modulo p and modulo p + k for prime pairs p , p + k . Assuming the Bateman–Horn conjecture, we prove the existence of strong sign biases for such pairs. More importantly, we prove that for a small positive proportion of prime pairs p , p + k , the dominant inequality is reversed. [ABSTRACT FROM AUTHOR]

We consider a quadratic analogue of the classical Titchmarsh divisor problem and prove an upper bound for the average of τ ( p 2 + 1 ) over p ⩽ X . [ABSTRACT FROM AUTHOR]

We obtain an asymptotic formula for the average value of the divisor function over the integers n ≤ x in an arithmetic progression n ≡ a mod q , where q = p k for a prime p ≥ 3 and a sufficiently large integer k . In particular, we break the classical barrier q ≤ x 2 / 3 − ε (with an arbitrary ε > 0 ) for such formulas, and, using some new arguments, generalise and strengthen a recent result of R. Khan (2015), making it uniform in k . [ABSTRACT FROM AUTHOR]

In this article, we completely determine the length of longest arithmetic progressions in the least positive reduced residue system and in all reduced residue systems modulo n for every positive integer n . [ABSTRACT FROM AUTHOR]

Assume that α , β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation ∑ n ∈ ( X , 2 X ] α ( n ) β ( n − h ) γ ( n + h ) averaging over a short variable h . More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh. [ABSTRACT FROM AUTHOR]

We study number theoretic properties of the map x ↦ x x ( mod p ) , where x ∈ { 1 , 2 , … , p − 1 } , and improve on some recent upper bounds, due to Kurlberg, Luca, and Shparlinski, on the number of primes p < N for which the map only has the trivial fixed point x = 1 . A key technical result, possibly of independent interest, is the existence of subsets N q ⊂ { 2 , 3 , … , q − 1 } such that almost all k -tuples of distinct integers n 1 , n 2 , … , n k ∈ N q are multiplicatively independent (if k is not too large), and | N q | = q ⋅ ( 1 + o ( 1 ) ) as q → ∞ . For q a large prime, this is used to show that the number of solutions to a certain large and sparse system of F q -linear forms { L n } n = 2 q − 1 “behaves randomly” in the sense that | { v ∈ F q d : L n ( v ) = 1 , n = 2 , 3 , … , q − 1 } | ∼ q d ( 1 − 1 / q ) q ∼ q d / e . (Here d = π ( q − 1 ) and the coefficients of L n are given by the exponents in the prime power factorisation of n .) [ABSTRACT FROM AUTHOR]

For any positive integers k and j , we consider some asymptotic formulas for weighted averages of the Cohen–Ramanujan sums s k ( s ) ( j ) = ∑ d | k , d s | j f ( d ) g ( k d ) with any fixed positive integer s ≥ 1 , and any arithmetical functions f and g . [ABSTRACT FROM AUTHOR]

We study a weighted divisor function ∑ ′ m n ≤ x cos ⁡ ( 2 π m θ 1 ) sin ⁡ ( 2 π n θ 2 ) , where θ i ( 0 < θ i < 1 ) is a rational number. By connecting it with the divisor problem with congruence conditions, we establish an upper bound, mean-value, mean-square and some power-moments. [ABSTRACT FROM AUTHOR]

Using bounds for certain sums of fractional parts over r -full numbers coming from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions in short intervals. [ABSTRACT FROM AUTHOR]

Let gcd ⁡ ( k , j ) be the greatest common divisor of the integers k and j . We establish some asymptotic formulas for weighted averages of the gcd-sum functions, that is ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ⁡ ( k , j ) ) with f = id , ϕ , ϕ s , ψ and ψ s for any fixed positive integers r and s , where ϕ , ϕ s , ψ and ψ s are the Euler, the Jordan, the Dedekind and the generalized Dedekind function, respectively, and also prove the mean square formulas of the error term of the gcd-sum functions ∑ k ≤ x 1 k r + 1 ⋅ ∑ j = 1 k j r ϕ ( gcd ⁡ ( k , j ) ) and ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r ψ ( gcd ⁡ ( k , j ) ) . [ABSTRACT FROM AUTHOR]

For two arithmetical functions f and g , we study the convolution sum of the form ∑ n ≤ N f ( n ) g ( n + h ) in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup. [ABSTRACT FROM AUTHOR]

We deduce an asymptotic formula with error term for the sum ∑ n 1 , … , n k ≤ x f ( [ n 1 , … , n k ] ) , where [ n 1 , … , n k ] stands for the least common multiple of the positive integers n 1 , … , n k ( k ≥ 2 ) and f belongs to a large class of multiplicative arithmetic functions, including, among others, the functions f ( n ) = n r , φ ( n ) r , σ ( n ) r ( r > − 1 real), where φ is Euler's totient function and σ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function f , the function of k variables f ( [ n 1 , … , n k ] ) is multiplicative. [ABSTRACT FROM AUTHOR]

In a recent work, Luca and Stănică examined quotients of the form φ ( C m ) φ ( C n ) , where φ is Euler's totient function and C 0 , C 1 , C 2 … is the sequence of the Catalan numbers. They observed that the number 4 (and analogously 1 4 ) appears noticeably often as a value of these quotients. We give an explanation of this phenomenon, based on Dickson's conjecture. It turns out not only that the value 4 is (in a certain sense) special in relation to the quotients φ ( C n + 1 ) φ ( C n ) , but also that the value 4 k has similar “special” properties with respect to the quotients φ ( C n + k ) φ ( C n ) , and in particular we show that Dickson's conjecture implies that, for each k , the number 4 k appears infinitely often as a value of the quotients φ ( C n + k ) φ ( C n ) . [ABSTRACT FROM AUTHOR]

Let r ≥ k ≥ 2 be fixed positive integers. Let ϱ r , k denote the characteristic function of the set of r -tuples of positive integers with k -wise relatively prime components, that is any k of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum ∑ n 1 , … , n r ≤ x ϱ r , k ( n 1 , … , n r ) by elementary arguments. Our result improves the error term obtained by J. Hu [5] . [ABSTRACT FROM AUTHOR]

This paper deals with function field analogues of the famous theorem of Landau which gives the asymptotic density of sums of two squares in Z . We define the analogue of a sum of two squares in F q [ T ] , q odd and estimate the number B q ( n ) of such polynomials of degree n in two cases. The first case is when q is large and n fixed and the second case is when n is large and q is fixed. Although the methods used and main terms computed in each of the two cases differ, the two iterated limits of (a normalization of) B q ( n ) turn out to be exactly the same. [ABSTRACT FROM AUTHOR]

Given several completely q j -additive integer functions with pairwise prime q j 's, we consider their joint behavior modulo some integer vector. Under several natural conditions, it is possible to show that this behavior displays some “independence” properties with respect to the various moduli. In this paper we deal with this phenomenon along arithmetic sequences, as well as on short intervals. [ABSTRACT FROM AUTHOR]

For two arithmetical functions f and g with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum ∑ n ≤ N f ( n ) g ( n + h ) under some suitable conditions. In this follow up article we improve these results with a weakened hypothesis which is in some sense minimal. [ABSTRACT FROM AUTHOR]

Given two arithmetical functions f , g we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums ∑ n ≤ N f ( n ) g ( n + h ) , building on an earlier work of Gadiyar, Murty and Padma. A key role in our method is played by the theory of Ramanujan expansions for arithmetical functions. [ABSTRACT FROM AUTHOR]

Let μ ( A ) be the Möbius function defined in a polynomial ring F q [ T ] with coefficients in the finite field F q of q elements ( q is odd). In this paper, we present a function field version of partial progress toward a conjecture of Good and Churchhouse. We calculate the mean and the large q limit of the variance of partial sums of the Möbius function on short intervals. Our calculation closely follows the framework of a recent work of Keating and Rudnick, where they consider the distribution of the von Mangoldt function in function fields. [ABSTRACT FROM AUTHOR]