By Carl Pomerance, Michael Th. Rassias (eds.)

This quantity features a selection of learn and survey papers written by means of the most eminent mathematicians within the overseas neighborhood and is devoted to Helmut Maier, whose personal examine has been groundbreaking and deeply influential to the sphere. particular emphasis is given to issues relating to exponential and trigonometric sums and their habit in brief periods, anatomy of integers and cyclotomic polynomials, small gaps in sequences of sifted leading numbers, oscillation theorems for primes in mathematics progressions, inequalities relating to the distribution of primes in brief periods, the Möbius functionality, Euler’s totient functionality, the Riemann zeta functionality and the Riemann speculation. Graduate scholars, examine mathematicians, in addition to machine scientists and engineers who're drawn to natural and interdisciplinary examine, will locate this quantity an invaluable resource.

*Contributors to this volume:*

Bill Allombert, Levent Alpoge, Nadine Amersi, Yuri Bilu, Régis de los angeles Bretèche, Christian Elsholtz, John B. Friedlander, Kevin Ford, Daniel A. Goldston, Steven M. Gonek, Andrew Granville, Adam J. Harper, Glyn Harman, D. R. Heath-Brown, Aleksandar Ivić, Geoffrey Iyer, Jerzy Kaczorowski, Daniel M. Kane, Sergei Konyagin, Dimitris Koukoulopoulos, Michel L. Lapidus, Oleg Lazarev, Andrew H. Ledoan, Robert J. Lemke Oliver, Florian Luca, James Maynard, Steven J. Miller, Hugh L. Montgomery, Melvyn B. Nathanson, Ashkan Nikeghbali, Alberto Perelli, Amalia Pizarro-Madariaga, János Pintz, Paul Pollack, Carl Pomerance, Michael Th. Rassias, Maksym Radziwiłł, Joël Rivat, András Sárközy, Jeffrey Shallit, Terence Tao, Gérald Tenenbaum, László Tóth, Tamar Ziegler, Liyang Zhang.

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**Example text**

4 Before proceeding it bears repeating that the same limiting support as K gets large (namely, . 1; 1/) can be achieved by just trivially bounding as above (that is, without exploiting cancelation in sums of Kloosterman sums), but we present here an argument connecting the Kuznetsov formula to the Petersson formula as studied in [27] instead. It suffices to study S WD 2 X X `D1 p Â log p O `=2 p log T X . p` ; 1I c/ 2 log T c 1 c ! 2k C 1/HT c 2 (73) and bound S by something growing strictly slower than T 2 .

R/ R Ã Â X . 1/m x2m X 2k C 1 x2kC1 D c8 . 2k C 1// 2 m 0 k 0 2 3 X . X/ D m 0 . mCn/Š gives us the claimed calculation. 1 C 1 y2 /n (see [1], p. 362) and bound trivially. Appendix 2: An Exponential Sum Identity The following proposition and proof are also used in [2]. 1. Suppose X Ä T. X/ WD T X hQQ . x/. 48 L. Alpoge et al. Proof. Observe that k 7! sin . k=2/ is supported only on the odd integers, and maps 2k C 1 to . 1/k . X/hQQ k 0 k622TZ Â k 2T Ã k 2 k 2T sin sin : (99) Since sin sin k 2 k 2T D e ik 2 e ik 2T e ik 2 e ik 2T 1 T 2 X D ˛D .

Goldfeld, The class number of quadratic fields and the conjectures of Birch and SwinnertonDyer. Ann. Scuola Norm. Sup. Pisa 3(4), 623–663 (1976) 19. B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986) 20. A. Gülo˘glu, Low-lying zeros of symmetric power L-functions. Int. Math. Res. Not. 9, 517–550 (2005) 21. B. Hayes, The spectrum of Riemannium. Am. Sci. 91(4), 296–300 (2003) 22. D. Hejhal, On the triple correlation of zeros of the zeta function. Int.