The History of Titchmarsh Divisor Problem

THE HISTORY OF TITCHMARSH DIVISOR PROBLEM Let τ (n) = P 1 be the divisor function, a 6= 0 be fixed integer. We define the following constants, where d|n γ is the Euler-Mascheroni constant.   ζ(2)ζ(3) Y p C1 (a) = 1− 2 ζ(6) p −p+1 p|a   2 log p X log p X p  C2 (a) = C1 (a) γ − + p2 − p + 1 (p − 1)(p2 − p + 1) p p|a Theorem 1 (1931). [T] Under GRH for Dirichlet L-functions,   X x log log x . (1) τ (p + a) = C1 (a)x + O log x p≤x Theorem 2 (1963). [L] Unconditionally by dispersion method,   X x log log x . (2) τ (p + a) = C1 (a)x + O log x p≤x Halberstam(1967) [H] gave a simpler unconditional proof using Bombieri-Vinogradov theorem and BrunTitchmarsh inequality. Bombieri, Friedlander, and Iwaniec(1986) [BFI], independently by Fouvry(1984) [F] obtained more precise formula Theorem 3. [BFI] Let A > 0 be fixed.  X (3) Λ(n)τ (n + a) = C1 (a)x log x + (2C2 (a) − C1 (a))x + O n≤x x logA x  . Using partial summation to above, we have Corollary 1. (4) X  τ (p + a) = C1 (a)x + 2C2 (a)Li(x) + O p≤x x logA x  . This result heavily relies on Bombieri-Vinogradov type result without having absolute value in the sum. Theorem 4. [BFI] Let A > 0, then there is B > 0 depending on A such that   X x x a,A . (5) ψ(x; q, a) − φ(q) logA x −B q≤x(log x) (q,a)=1 By partial summation, we have Corollary 2. Let A > 0, then there is B > 0 depending on A such that   X π(x) x (6) π(x; q, a) − a,A . φ(q) logA x −B q≤x(log x) (q,a)=1 1 2 In view of this corollary, it looks like the moduli q came almost close to x. However, up to the full moduli q ≤ x, the estimate is very different. In fact, from the following lemma and Corollary 1: Lemma 1. X n≤x (n,a)=1 1 = C1 (a) log x + C2 (a) + O φ(n)  log x x  . We obtain the following asymptotic for the full moduli. Corollary 3. X  (7) q≤x (q,a)=1 π(x) π(x; q, a) − φ(q)   = (C2 (a) − C1 (a))Li(x) + O x log2 x  . For the primes in arithmetic progressions, A. T. Felix (2011) [Fe] proved that Theorem 5. [Fe] Fix integers a 6= 0 and k ≥ 1. Then     X p−a ck x τ (8) = x+O , k k log x p≤x p≡a mod k where Y ck = C1 (a) 1+ p|k p−1 p2 − p + 1  . Q Let q 0 = p|q p = rad(q). D. Fiorilli (2012) [Fi] obtained more precise formula. As a special case of [Fe, Theorem 2.4], we have Theorem 6. [Fi] Fix integers a 6= 0 and q ≥ 1. Then X (9) Λ(qm + a)τ (m) − M.T  |a|/q 0, we have   X (11) Λ(n)τ (n − 1) = C1 x log x + (2C2 − C1 )x + O x1−δ . n≤x 3 It would be natural to consider similar problems for k-divisor functions τk (n). Also in the P paper [D], it was mentioned that current methods are not sufficient to obtain asymptotic formulas of p≤x τk (p − 1) for k ≥ 3. On the other hand, in an expository note by D. Koukoulopoulos (2015) [K, Exercise 4.3.2], Theorem 8. Unconditionally, we have  Y X 1 2 1− τ3 (p + a)  x log x . (12) p p≤x p|a Assuming Elliott-Halberstam Conjecture (EH), there is an absolute constant C(a) such that X τ3 (p + a) = C(a)x log x + O(x). (13) p≤x