By Larry Joel Goldstein
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Over three hundred sequences and plenty of unsolved difficulties and conjectures concerning them are provided herein. The booklet includes definitions, unsolved difficulties, questions, theorems corollaries, formulae, conjectures, examples, mathematical standards, and so forth. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, nearly primes, cellular periodicals, services, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, and so forth.
Extra resources for Analytic Number Theory
Let a > 0. If for real s > 0 F(s) =A f(l +a) (1 + r(s)), s0t where lr(s)I :::; Cose and A> 0, Co > 0, and€> 0 are constants, then fox f(t) dr(t) where Ip( x) I :::; Ci/ ln x, = AxOt(l + p(x)), C 1 > 0 being a constant. We will show how this theorem contains the Hardy-Littlewood theorem (with remainder term) for a power series. Suppose that as z -+ 1 a series with nonnegative coefficients an satisfies a relation ~ anzn = where e > 0. Then as s -+ (l ! z)0t (1+0((1 - zt)), 0 L 00 1 ane-ns = 0 (1 + O(sc)), 8 n=O where€> 0.
This proves the lemma. LEMMA 11. Suppose 11",,(x) = E~ akxk is a polynomial of degree v with real coefficients such that 17r,,(x)I $ M on the interval [O, l]. Then its coefficients satisfy the estimate where C is an absolute constant. PROOF. In this lemma the letter C will stand for absolute constants, not necessarily equal to each other. Consider the v + 1 points Xi = i / v, 48 1. FACTS FROM ANALYSIS i = 0, 1, ... , v, in [O, l]. Since a polynomial of degree vis determined by its values at L' + 1 points, it follows from the Lagrange interpolation formula that 11",,,(x) L 11",,,(xi) (xi = i=O (x - xo) · · · (x - Xi-1)(x - Xi+i) · · · (x - x,,,) ( · xo) ···(xi - Xi-1)(xi - Xi+i) · · · Xi - x,,,) v Let
TAUBERIAN THEOREMS FOR POWER SERIES 29 Postnikov  proved a complex analogue of the Hardy-Littlewood Tauberian theorem. Suppose a power series f(z) = E;;" anzn has radius of convergence equal to l. Let z = reie (r = lzl). Assume that f(z) = 1/(1- z) + 0(1) for IOI :::; c < 1r. Then Lan= N + O(lnN). THEOREM. n5,N Comparing Postnikov's theorem with Fatou's, we see that imposing conditions in the complex domain leads (in the given situation) to a significant improvement in the remainder term. There is a complex Tauberian theorem for Dirichlet integrals, namely Ikehara's theorem, that is useful in analytic number theory.