By Krishnaswami Alladi (auth.), K. Alladi, P. D. T. A. Elliott, A. Granville, G. Tenebaum (eds.)

ISBN-10: 1441950583

ISBN-13: 9781441950581

ISBN-10: 1475745079

ISBN-13: 9781475745078

This quantity includes a choice of papers in Analytic and ordinary quantity idea in reminiscence of Professor Paul Erdös, one of many maximum mathematicians of this century. Written through many top researchers, the papers take care of the latest advances in a wide selection of issues, together with arithmetical features, major numbers, the Riemann zeta functionality, probabilistic quantity concept, homes of integer sequences, modular varieties, walls, and q-series. *Audience:* Researchers and scholars of quantity conception, research, combinatorics and modular types will locate this quantity to be stimulating.

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**Extra info for Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös**

**Example text**

49 MODULO SMALL PRIMES Proof: The result follows from Proposition I and the following well-known q-series identity: oo LC(2,N)qN N=O = n oo (1 2n)2 1-q n n=l ( - q ) oo = Lq"2i". n=O D Theorem 2. 3_) q3n+l. D Theorem 3. If N < 25'+1, then L = p(N) (mod 5), as(ao)as(aJ) · · · as(as) ao+25a 1+···+25'a,=N where as(n) := (n Proof: + 1) Ldln+l d. The result follows from Proposition 1 and the identity (see [3, 4]) 5 (5 ) oo oo (d) n ~ = LC(5,N)qN+I = L L S . d ·qn. 7J (z) N=O n=l din D Theorem 4. If N < 49s+l, then p(N) Proof: = It is well known that [3] 50 ONO Since TJ\Z)TJ 3(7z) = L~t r:(n)qn (mod 7) where r(n) is Ramanujan's tau-function, the result now follows by Proposition 1 and the Lehmer congruence [21] r:(n) =nL (mod 7).

Whenr(m) > O,denoteby(bj,bj) (1 :s j :s r(m))thedistinctpairsofnumber sbj E B*, bj E B; with product m. Notice that bj1 + b}2 E B* + B; for each of the r(m ) 2 pairs (h, jz). ]4 b'-]4 = m. ) in B* with bj1 < bj3 , Eq. ]4 and thus i, m and n are uniquely determined. If we let N; be the number of quadruples corresponding to each i, then by (7) and the Cauchy-Schwarz inequality, N; 2: LL:Sm(n) -1 m n Also, N; :;:: 0 for each i. If bj1 < bj3 , then (9) implies bj2 < bj4 and hence 14 L N· < -1. 4 . (10) I- ' Define It = { i = 1 (mod 3) : S; P/ :;:: lz = { i = 1 (mod 3) : S; P/ < ~1 4 }, ~1 4 }.

The proof, which seems to be new, may be used to derive other identities similar to (1). /Xft) 2- 2 wdw, 2 where (2) 1 l = (c) c+iT lim T--+oo c-iT For a proof of (2) see, for example, [6, p. 87]. Note that in (2) we may shift the line of integration to ffie w = c, 0 < c < 1, and that the integral is absolutely convergent for 0 < c < 1/2, since Ieos wl::::: cosh v, f(w) « lvlu-lf 2 e-rrlvlf2 for w = u + iv. If 1 = L[f(x)] 00 f(x)e-sx dx is the (one-sided) Laplace transform of f(x), then for ffies > 0 L[~(x)] = = foo Jo (L'd(n)- x(logx n::'Ox ~ loo n=l n ~d(n) sx e- dx + 2y- 1)- ~) e-sx dx + logs-y 2 1 -- 4s S 1~ =- ~d(n)e s n=l -sn + logs - y 1 1 ~ (w)f(w)s-w s2 1 -4s 1 = --.

### Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös by Krishnaswami Alladi (auth.), K. Alladi, P. D. T. A. Elliott, A. Granville, G. Tenebaum (eds.)

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