An arithmetic function arising from Carmichael’s conjecture

نویسنده

  • Paul POLLACK
چکیده

Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n, the equation φ(n) = φ(m) has a solution m 6= n. This suggests defining F (n) as the number of solutions m to the equation φ(n) = φ(m). (So Carmichael’s conjecture asserts that F (n) ≥ 2 always.) Results on F are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of F contains every natural number k ≥ 2. Also, the maximal order of F has been investigated by Erdős and Pomerance. In this paper we study the normal behavior of F . Let K(x) := (log x)(log log x)(log log log x). We prove that for every fixed > 0, K(n)1/2− < F (n) < K(n)3/2+ Manuscrit reçu le 15 mai 2010.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Gross–Zagier formula and arithmetic fundamental lemma

The recent work [YZZ] completes a general Gross–Zagier formula for Shimura curves. Meanwhile an arithmetic version of Gan–Gross–Prasad conjecture proposes a vast generalization to higher dimensional Shimura varieties. We will discuss the arithmetic fundamental lemma arising from the author’s approach using relative trace formulae to this conjecture.

متن کامل

On the range of Carmichael’s universal-exponent function

Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds x/(log x).36 for all large x, while for φ it is equal to x/(log x)1+o(1), an old result of Erdős. We also improve on an ...

متن کامل

Heegner points, Stark-Heegner points, and values of L-series

Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and Swinnerton-Dyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1...

متن کامل

On the Growth of Torsion in the Cohomology of Arithmetic Groups

Let G be a semisimple Lie group with associated symmetric space D, and let Γ ⊂ G be a cocompact arithmetic group. Let L be a lattice inside a ZΓ-module arising from a rational finite-dimensional complex representation of G. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup Hi(Γk;L )tors as Γk ranges over a tower of congruence subgrou...

متن کامل

On the Euler Function of the Catalan Numbers

We study the solutions of the equation φ(Cm)/φ(Cn) = r, where r is a fixed rational number, Ck is the kth Catalan number and φ is the Euler function. We note that the number r = 4 is special for this problem and for it we construct solutions (m,n) to the above equation which are related to primes p such that 2p− 1 or 4p− 3 is also prime. 1. An observation concerning φ(Cn+1)/φ(Cn) For a positive...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2011