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 encompasses a choice of papers in Analytic and straightforward quantity conception in reminiscence of Professor Paul Erdös, one of many maximum mathematicians of this century. Written by way of many prime researchers, the papers take care of the latest advances in a large choice of themes, together with arithmetical services, best numbers, the Riemann zeta functionality, probabilistic quantity concept, homes of integer sequences, modular kinds, walls, and q-series. *Audience:* Researchers and scholars of quantity conception, research, combinatorics and modular varieties will locate this quantity to be stimulating.

**Read or Download Analytic and Elementary Number Theory: A Tribute to Mathematical Legend Paul Erdös PDF**

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

**Example text**

Erdos [2] conjectured that for any finite set A of positive integers, ( 1) where In other words, no set A can have simultaneously few sums and few products. Notice that trivially (2) 60 FORD Our chief interest here is the behavior of the function fh(k) = min{IEh(A)I : IAI = k, A c N}. Erdos and Szemeredi [3] proved the nontrivial bounds klH « Jz(k) « k2-cjlog2k, (3) where c and 8 are positive constants an logk x denotes the kth iterate of the logarithm. Nathanson [7] showed that 8 = 1/31 is admissible, and we note that the argument works for any finite set of positive real numbers.

More generally, if h :=:: 2 define hA = {a 1 + · · · + ah :a; E A}, Ah = {a 1 · · · ah :a; E A}. Erdos [2] conjectured that for any finite set A of positive integers, ( 1) where In other words, no set A can have simultaneously few sums and few products. Notice that trivially (2) 60 FORD Our chief interest here is the behavior of the function fh(k) = min{IEh(A)I : IAI = k, A c N}. Erdos and Szemeredi [3] proved the nontrivial bounds klH « Jz(k) « k2-cjlog2k, (3) where c and 8 are positive constants an logk x denotes the kth iterate of the logarithm.

4 j2P; fori E ]z, hence by (10), 1 1 I:-::::-. P; 2 (12) . I lE 2 LetM, = III I, M2 inequality, = 1/zl and H = M 1 +M2 • By(8), (11), (12) and theCauchy-Schwarz 12BI + IB 2 1::::: ~I 413 M, + L P; iEh > ~M I [413 + lM22 - l = ~z 4 1\H- Mz) + 2Mi. 2 The right side is minimized at M 2 = ~[ 4 1 3 . Since H ::::: ~ [k fl] :::;: ~ - ~, we obtain 12BI + IB 2 1::::: > ~Ht 4 1 3 - : 2 ! _ 1s;3 _ ~z413. - 2 32 (13) 2 Ignoring the last term, the optimal value of l is The lemma now follows from (13), since k::: 107 and l :::;: (~k) 3 1 7 - 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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