Does there exist an asymptotic formula for $\sum_{p\le x}\tau(p−1)^n$ ? Here $n$ is an arbitrary positive integer and $\tau$ is the divisor function. The case of $n=1$ was done by Linnik, but when $n$ is big the question seems to be open.
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$\begingroup$ You probably wanted $n$ to be an exponent over $\tau(p-1)$. I updated the text accordingly (along with some linguistic and grammatical improvements). The question is good, by the way. $\endgroup$– GH from MOCommented Aug 20, 2016 at 13:51
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$\begingroup$ Do you have a reference to the Linnik result? $\endgroup$– Igor RivinCommented Aug 21, 2016 at 4:27
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1$\begingroup$ @IgorRivin: Linnik Ju. V. New versions and new uses of the dispersion methods in binary additive problems. (Russian) Dokl. Akad. Nauk SSSR 137 1961 1299–1302. See also Iwaniec's Analytic Number Theory page420 $\endgroup$– user97495Commented Aug 21, 2016 at 8:26
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1$\begingroup$ For each fixed n, one can get the correct order of magnitude of such sums (but not an asymptotic formula) using a method of Erdos, described elegantly on Terry Tao's blog: terrytao.wordpress.com/2011/07/23/erdos-divisor-bound A similar problem is tackled in this recent paper (but with $\tau_n(p-1)$ instead of $\tau(p-1)^n$): link.springer.com/article/10.1007/s11139-014-9669-1 $\endgroup$– so-called friend DonCommented Aug 21, 2016 at 15:57
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