Paul Erdos and Carl Pomerance have proofs on an asymptotic upper bound on the number of preimages of Euler's totient function ϕ and the sum-of-divisors functions σ. In this paper, we will extend the ...upper bound to the number of preimages of iterates of ϕ and σ. Using these new asymptotic upper bounds, a conjecture in de Koninck and Kátai's paper, “On the uniform distribution of certain sequences involving the Euler totient function and the sum of divisors function” is now proven and many corollaries follow from their proven conjecture.
We consider two sequences a(n) and b(n), 1\leq n<\infty, generated by Dirichlet series of the forms \begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad \sum ..._{n=1}^{\infty }\dfrac {b(n)}{\mu _n^{s}}, \end{equation*} satisfying a familiar functional equation involving the gamma function \Gamma (s). A general identity is established. Appearing on one side is an infinite series involving a(n) and modified Bessel functions K_{\nu }, wherein on the other side is an infinite series involving b(n) that is an analogue of the Hurwitz zeta function. Six special cases, including a(n)=\tau (n) and a(n)=r_k(n), are examined, where \tau (n) is Ramanujan’s arithmetical function and r_k(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new.
In this paper, we prove that the Hecke eigenvalue square for a holomorphic cusp form and the Piltz divisor functions are good weighting functions for the pointwise ergodic theorem. Additionally, we ...prove similar results for various other arithmetical functions in the last section.
The sum of the reciprocals of all prime numbers diverges but the divergence is very slow. We propose effective lower and upper bounds for partial sums under the Riemann hypothesis. We give an ...explicit error term for all Mertens’ theorems.
Fake mu's Martin, Greg; Mossinghoff, Michael; Trudgian, Timothy
Proceedings of the American Mathematical Society,
08/2023, Letnik:
151, Številka:
8
Journal Article
Recenzirano
Let \digamma (n) denote a multiplicative function with range \{-1,0,1\}, and let F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n). Then F(x)/\sqrt {x} = a\sqrt {x} + b + E(x), where a ...and b are constants and E(x) is an error term that either tends to 0 in the limit or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of \sqrt {x}) in the first case, and apparent bias b in the latter. For example, if \digamma (n)=\mu (n), the Möbius function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n) has apparent bias 0, while if \digamma (n)=\lambda (n), the Liouville function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n) has apparent bias 1/\zeta (1/2). We study the bias when \digamma (p^k) is independent of the prime p, and call such functions fake \mu ’s . We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function F(x) with apparent bias b, we also show that F(x)/\sqrt {x}-a\sqrt {x}-b changes sign infinitely often.
The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard factorization of n! into primes. We show that for each θ > 0 the ...primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!θ =∏p⩽nθ pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.
We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov ...range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on GL₃. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially ℓ-adic cohomology and the Riemann Hypothesis.
For integer n and real u, define Δ(n,u)≔|{d:d∣n,eu<d⩽eu+1}|. Then, the Erdős–Hooley Delta function is defined as Δ(n)≔maxu∈RΔ(n,u). We provide uniform upper and lower bounds for the mean-value of ...Δ(n) over friable integers, i.e. integers free of large prime factors.
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