An average Brun-Titchmarsh theorem and shifted primes with a large prime factor
Runbo Li
TL;DR
The work develops an average version of Brun–Titchmarsh for large moduli by combining Harman-type sieve methods with Maynard’s large-moduli input, refining Baker–Harman’s prior bounds. The central mechanism is to compare sieve sums over admissible residue classes $\mathcal{A}^q$ and the full interval counts $\mathcal{B}^q$, decomposing the sums via Buchstab’s identity and high-dimensional sieves to control negative contributions and bound the averaged constant $C(\theta)$. By exploiting Maynard’s Proposition 8.3 and performing delicate numerical evaluation of sieve integrals, the paper shows that $C(\theta)$ remains small enough on $[1/2,0.679]$ to deduce an improved bound for almost all moduli, yielding that there are infinitely many primes $p$ with $P^{+}(p+a) > p^{0.679-\varepsilon}$. As an application to shifted primes with a large prime factor, the author obtains new exponent bounds on $P^{+}(p-1)$, improving upon Baker–Harman’s 1998 results; the Appendix discusses conjectural distributions of $P^{+}(p+a)$ and historical progress toward bounds from both large and small prime factors. The combination of advanced sieve techniques, multi-dimensional information, and numerical sieve-integral evaluation marks a notable advancement in average prime distributions for large moduli and their arithmetic applications.
Abstract
The author studies an average version of Brun-Titchmarsh theorem with large moduli. Using Maynard's recent breakthrough on the Bombieri-Friedlander-Iwaniec type triple convolution estimates, we refine the previous result of Baker and Harman (1996). As an application, we improve a result of Baker and Harman (1998) on shifted primes with a large prime factor, showing that the largest prime factor of $p - 1$ is larger than $p^{0.679}$ for infinitely many primes $p$.