Authors establish the Erdős-Graham conjecture for large k and provide GRH-conditional counterexamples for small k using sieves and exponential sums.
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Resolves Erdős-Straus, Erdős, and Erdős-Pomerance-Sárközy conjectures on prime factors of consecutive integers via probabilistic method and quantitative correlations.
Establishes best known uniform bounds on newform correlation and decorrelation integrals, yielding effective holomorphic QUE and extending prior decorrelation results via refined weak subconvexity for Rankin-Selberg L-functions.
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Binomial coefficients with divisors avoiding an interval
Authors establish the Erdős-Graham conjecture for large k and provide GRH-conditional counterexamples for small k using sieves and exponential sums.
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Quantitative correlations and some problems on prime factors of consecutive integers
Resolves Erdős-Straus, Erdős, and Erdős-Pomerance-Sárközy conjectures on prime factors of consecutive integers via probabilistic method and quantitative correlations.
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Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions
Establishes best known uniform bounds on newform correlation and decorrelation integrals, yielding effective holomorphic QUE and extending prior decorrelation results via refined weak subconvexity for Rankin-Selberg L-functions.