Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, k)$. For $k=0$, $F(s, k)$ seems to be the negative of the log derivative of the Riemann zeta function plus some entire function. What about for general real $k$ ?
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$\begingroup$ If k is large, then it seems to be approximately F(s+k-1/k, 0) and for k small, its approximately F(s, 0). If you differentiate with respect to k, you get some weird sum + F(s-2,k)? Just spitballing here. $\endgroup$– AsvinCommented Apr 20, 2020 at 11:50
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