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Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046

Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121

Reviewed by Pieter Moree

These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1. In s = 1 it has a true if and only if for every e > 0, we have det()RC# n12/ + e, simple pole. In 1859 Bernhard Riemann published ‘Über die Anzahl n e where Rr= () is the nn# matrix with r = 1 if j = 1 or i divides der Primzahlen unter einer gegebenen Grösse’. This only 9-pages- nij ij j, and r = 0 otherwise. long paper, the only published work of Riemann on number theory, ij In Chapter 4, after some chapters on history, basic properties is without doubt the most important paper ever written in analyt- of g()s and one with derivations of some basic estimates involving ic number theory; indeed it is foundational, as Riemann makes prime numbers, Schoenfeld’s criterion is proved. It says that the RH essential use of s being a complex variable (allowing methods of is equivalent to the inequality that |(} xx)|- # xx()log 2 /(8r) for complex analysis to be applied), whereas a century earlier Leon- x $ 74, where }()xp= / n # log . The sum is over all prime pow- hard Euler only considered g()s for real values of s. px ers pxn # and each of those chips in a weight log p. The function The uniqueness of prime factorization of the integers finds its }()x turns out to be easier to study than r()x , on the other hand analytic counterpart in the identity g()sp=-% ()1 --s 1 valid for p usually results on }()x can be easily translated to results on r()x . v > 1 (where the product extends over all the primes p). Given this Schoenfeld’s criterion is important, but not very surprising. identity it is perhaps not so surprising that the behaviour of g()s is More surprising is Robin’s criterion which states that, for n > 5040, very closely related to the distributional properties of the primes.

The Riemann Hypothesis (formulated in the 1859 paper) states 1 c / d < enloglog , that all the zeros in the critical strip 01< 9 satisfy (2) (thus to wit: we solved pletely explicit. George Pólya considered the family of deforma- tions half of RH!). In addition we showed that all 5-free integers > 5040 3 th satisfy (2), where an integer is said to be k-free if no k power of mui2 zu HUm ()ze|= # ()ue du. an integer > 1 divides it (otherwise it is said to be k-full). It then -3 continues to discuss various improvements of our work (the author RH is equivalent with H ()z having only real zeros. It can be shown and Trudgian showed for example that the inequality holds for 11- 0 there is a unique finite real numberK such that H ()z has only free integers). Not discussed, but fresh on the arXiv is the result m real zeros if and only if m $ K. The constant K is now called the de of Morrill and Platt that RH is true if and only if (2) holds for all Bruijn–Newman constant. RH is equivalent with K # 0. De Bruijn 20-full integers. 1 showed in 1950 that K # 2 . Newman showed that K exists, i.e. We say that N is a colossally abundant number if for some e > 0 that K > - 3. The lower bound was improved many times over the the function v()nn/ 1 + e attains its maximum at N. It is not difficult -11 1 years and the best we know currently is that -10 < 5040, Chapter 6 concerns a criterion of 2006 from Cardon and Robert there exists a colossally abundant counterexample. Chapter 9 is which in essence is about approximation of g()1 + iz by a particu- devoted to a study of these numbers and some variations. 2 lar sequence of orthogonal polynomials. An inequality closely related to that of Robin is Amoroso showed that if AxN()= %nN# Un()x is the product of n c the firstN cyclotomic polynomials, then its so-called height, for < enloglog . me+ {()n (3) any e > 0, is bounded above by CNe (with Ce a constant) if and only if g()s has no zero with real part exceeding m. This result This is now called the Nicolas inequality. The intriguing work of is proved in Chapter 7. mostly Nicolas related to his inequality and refinements thereof are Chapter 9 concerns Weil’s version of the explicit formula. For a discussed in Chapter 5. large class of test functions this relates a sum involving Riemann About five chapters in total are devoted to thisand closely zeros to a sum involving primes and some remaining terms that related material, thus making it one of the main topics discussed can be regarded as associated with the so-called prime at infinity. in Volume 1. By a judicious choice of test functions this allows one to prove I will now discuss Volume 2, which draws on a rather broader many results. spectrum of mathematical methods and ideas than Volume 1. Weil’s work on explicit formulae has been very influential and Marcel Riesz showed in 1916 that RH is equivalent to also was an input for a proof of the RH for curves over finite fields (in the case of elliptic curves a proof is given here). It has been 3 n + 1 ()- 1 n 14/ + e the goal of many to use this approach for RH and generalizations / xC# e x . thereof. Even though Weil provided a bridge between the two cas- n = 1 ()nn- 12!(g ) es and meanwhile vast and deep new mathematical theories have This implies that the values of g()s at all even integers determine been developed, Weil’s bridge remains to be crossed ... the truth of RH. Hardy and Littlewood proved a variation where The book concludes with several appendices, giving more back- all the values of g()s at odd integers > 1 appear. In 2005 Luis ground material, for example on the Fourier, Laplace and Mellin Báez-Duarte unified and generalized the existing series equivalenc- transform. There is also a manual for a set of functions written to es, giving rise to a very large family. Chapter 2 provides the details. assist the reader to reproduce and possibly extend, calculations mentioned in the book. One of the most beautiful and surprising equivalences to the Riemann Hypothesis is related to Banach and Hilbert space meth- The first volume falls mostly in the realm of computational num- ods. By r we denote the entire part of a real number r. Bertil Ny- ber theory and the author rederived some results, for example man in 19505? in his PhD thesis (!) proved that RH is true if and only some of the classical ones of Rosser and Schoenfeld. The second if the linear span of the functions kxa()|=-a//xxa 1 is dense volume draws on much more areas of mathematics and everybody, 66@@ in the Hilbert space L2 (,01) of square integrable functions on the I think, will be in for some surprises there. For the results stated interval (,01), where 01< a # . Arne Beurling, the PhD supervisor mostly proofs are given. This requires a broad range of mathemat- of Nyman, extended the criterion to Lp-spaces. These intriguing ical tools to be used. Sometimes the author cleaned up some of results are discussed in Chapter 3. the original proofs. Put All in all these books serve as a good introduction to a wide range of mathematics related to the Riemann Hypothesis and 1 k make for a valuable contribution to the literature. They are truly mk|= / 1 - 1 - , t d cmt n encyclopedic and I am sure will entice many a reader to consult some literature quoted and who knows, eventually make an own where the sum is over the non-trivial Riemann zeta zeros. Li’s cel- contribution to the area. ebrated criterion says that RH is true if and only if mk > 0 for every I thank Alexandre Kosyak for helpful comments on several ear- k $ 1. Note that mt1 = /t 1/ . It was already known to Riemann lier versions of this review.