Generalizing the Titchmarsh Divisor Problem
نویسنده
چکیده
Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem ∑ p≤x d(p− a) = cx + O ( x log log x log x ) where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: ∑ p≤x p≡a mod k d ( p− a k ) = Ckx + O ( x log x ) where Ck is a constant dependent on k and a and the implied constant is dependent on k. We also apply it a question related to Artin’s conjecture for primitive roots.
منابع مشابه
A Geometric Variant of Titchmarsh Divisor Problem
We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of abelian varieties. For any abelian variety A defined over Q, we study the asymptotic distribution of the primes of Z which split completely in the division fields of A. For all abelian varieties which contain an elliptic curve we establish an asymptotic formula for such primes under the assumption of GRH. We e...
متن کاملShifted convolution and the Titchmarsh divisor problem overFq[t]
Author for correspondence: J. C. Andrade e-mail: [email protected] Shifted convolution and the Titchmarsh divisor problem overFq[t] J. C. Andrade1, L. Bary-Soroker2 and Z. Rudnick2 1Institut des Hautes Études Scientifiques (IHÉS), Le Bois-Marie, 35 Route de Chartres, Bures-sur-Yvette 91440, France 2Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Av...
متن کاملShifted convolution and the Titchmarsh divisor problem over 𝔽q[t].
In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.
متن کاملFailure to respond as a problem in generalizing the results of research projects
This article has no abstract.
متن کاملذخیره 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