research on number theoretic concepts attached formal group

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(1) We extend a well-known distribution relation for ordinary Bernoulli polynomials to that of Bernoulli polynomials attached to formal group.(2) Let N be an elementary extension of N and n ∈ N-N. We prove that PTC (n) has no proper endextension of Δ^b_1-LLIND and consider conditions that a model of bounded arithmetic has a proper endextension.(3) We study the structure of uniform random binary recursive circuits. We show that a suitably normalized version of the number of outputs converges in distribution to a normal random variate. We also discuss the connection of the number of outputs to a non-classical urn model, and our investigation provides a first solved instance of this new class of urns.(4) Let ζ(s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form Σ^q_<a=1> ζ(s, a/q)^h or Σ^q_<a=1> |ζ(s, a/q)|^<2h> are studied, where q and h are positive integers. These mean values can be written as linear combinations of Σ^q_<a=1> ζ_r(s1, ・・・, sr, a/q), where ζ_r(s1, ・・・, sr;α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of Σ^q_<a=1> ζ_r(s1, ・・・, sr;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for Σ^q_<a=1> ζ(s, a/q)h and Σ^q_<a=1> |ζ(s, a/q)|^<2h> is obtained. In particular, the asymptotic expansion of Σ^q_<a=1> ζ(1/2, a/q)^3 with respect to q is written down.