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On the mean value of functions related to the divisors function in the ring of polynomials over a finite field
V. V. Iudelevich Lomonosov Moscow State University
(Moscow)
Abstract:
Let g:Fq[T]→R be a multiplicative function which values at the degrees of the irreducible polynomial, depends only on the exponent, such that g(Pk)=dk polynomial P and for some arbitrary sequence of reals {dk}∞k=1. This paper regards the sum T(N)=∑degF=NF is monicg(F), where F ranges over polynomials of degree N with leading coefficient equal to 1 (unitary polynomials). For the sum T(N), an exact formula is found, and various asymptotics are calculated in cases of q→∞; q→∞, N→∞; qN→∞. In particular, the following asymptotic formulas are obtained ∑degF=NF is monicτ(Fk)=(k+NN)qN+ON,k(qN−1), N⩾1, q→∞; ∑degF=NF is monic1τ(F)=qN4N((2NN)−23(2N−4N−2)q−1+O( 4N√Nq−2)), N→∞, q→∞; ∑degF=NF is monic1τ(F)=C1⋅(2NN)4NqN+O(qN−0.5N1.5), C1=+∞∏l=1(√q2l−qllnqlql−1)πq(l), qN→∞; where τ(F) is a number of monic divisors of F, and πq(l) is a number of monic irreducible polynomials of degree l. The second and third equalities are analogous for polynomials over a finite field of one of Ramanujan's results ∑n≤x1d(n)=x√lnx(a0+a1lnx+…+aN(lnx)N+ON(1(lnx)N+1)), where d(n) is a classical divisor function, and ai are some constants. In particular, a0=1√π∏plnpp−1√p(p−1).
Keywords:
the ring of polynomials over a finite field, divisor function.
Received: 03.02.2020 Accepted: 22.10.2020
Citation:
V. V. Iudelevich, “On the mean value of functions related to the divisors function in the ring of polynomials over a finite field”, Chebyshevskii Sb., 21:3 (2020), 196–214
Linking options:
https://www.mathnet.ru/eng/cheb935 https://www.mathnet.ru/eng/cheb/v21/i3/p196
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Abstract page: | 237 | Full-text PDF : | 69 | References: | 34 |
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