Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

Abstract

We show that the expected asymptotic for the sums ${\sum }_{X<n\le 2X}\Lambda (n)\Lambda (n+h)$, ${\sum }_{X<n\le 2X}{d}_k(n)d_l(n+h)$, and ${\sum }_{X<n\le 2X}\Lambda (n)d_k(n+h)$ hold for almost all $h\in [-H,H]$, provided that $X^{8/33+\epsilon }\le H\le X^{1-\epsilon }$, with an error term saving on average an arbitrary power of the logarithm over the trivial bound. Previous work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao covered the range $H\ge X^{1/3+\epsilon }$. We also obtain an analogous result for ${\sum }_n\Lambda (n)\Lambda (N-n)$. Our proof uses the circle method and some oscillatory integral estimates (following a paper of Zhan) to reduce matters to establishing some mean-value estimates for certain Dirichlet polynomials associated to "Type $d_3$" and "Type $d_4$" sums (as well as some other sums that are easier to treat). After applying Hölder's inequality to the Type $d_3$ sum, one is left with two expressions, one of which we can control using a short interval mean value theorem of Jutila, and the other we can control using exponential sum estimates of Robert and Sargos. The Type $d_4$ sum is treated similarly using the classical $L^2$ mean value theorem and the classical van der Corput exponential sum estimates.

80 pages, no figures. updated references

References in corpus (4)

• Linear correlations of multiplicative functions
• A note on the Liouville function in short intervals
• The shifted convolution of generalized divisor functions
• Correlations of the von Mangoldt and higher divisor functions II. Divisor correlations in short ranges

Cited by in corpus (6)

• Linear correlations of multiplicative functions
• Higher uniformity of arithmetic functions in short intervals I. All intervals
• Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions
• On the Hardy-Littlewood-Chowla conjecture on average
• Averages of the Möbius function on shifted primes
• Singmaster's conjecture in the interior of Pascal's triangle