NUMBER THEORY: SAILING ON THE SEA OF NUMBER THEORY

By EVELYN SANCHEZ

CONTENTS CONVOLUTIONS OF THE VON MANGOLDT FUNCTION AND RELATED DIRICHLET SERIES CONSTRUCTING NEW NON-CONGRUENT NUMBERS BY GRAPH THEORY DISTRIBUTION OF UNITS OF AN ALGEBRAIC NUMBER FIELD MODULO AN IDEAL SIGN CHANGES OF FOURIER COE?CIENTS AND EIGENVALUES OF CUSP FORMS SHIFTED CONVOLUTION SUMS OF FOURIER COE?CIENTS OF CUSP FORMS TWO EXPOSITIONS ON ARITHMETIC OF CUBICS DISTRIBUTION OF POINTS ON MODULAR HYPERBOLAS A SURVEY OF PROBLEMS AND RESULTS ON RESTRICTED SUMSETS A GENERAL MODULAR RELATION IN ANALYTIC NUMBER THEORY L-FUNCTIONS OF FUNCTION FIELDS P(M1, , MR)?S (11) WAS ?RST STUDIED BY MELLIN [29,30], AND INDEPENDENTLY BY BARNES [5,6] FOR P A LINEAR FORM, AT THE BEGINNING OF THE 20TH CENTURY MELLIN PROVED THE MEROMORPHIC CONTINUATION OF (11) TO THE WHOLE COMPLEX PLANE C IF ALL THE COE?CIENTS OF P HAVE POSITIVE REAL PARTS SEVERAL MATHEMATICIANS AFTER MELLIN PROVED THE MEROMORPHIC CONTINUATION OF (11) UNDER WEAKER ASSUMP- TIONS AT PRESENT, THE ASSUMPTION (H0S) INTRODUCED BY ESSOUABRI [12] IS THE WEAKEST ESSOUABRI [11] ALSO POINTED OUT THAT THE MULTI-VARIABLE GENERALIZA × · · · × (M1 + · · · + MR)?SR, (13) WHICH IS CALLED THE EULER-ZAGIER R-FOLD SUM, HAS BEEN STUDIED EXTENSIVELY IN RECENT YEARS THE MEROMORPHIC CONTINUATION OF (13) TO CR IS INCLUDED IN THE ABOVE THEOREM OF ESSOUABRI [11], BUT [11] IS UNPUBLISHED VARIOUS DI?ERENT PROOFS OF THE CONTINUATION WERE PUBLISHED BY ARAKAWA AND KANEKO [3], ZHAO [37], AKIYAMA, EGAMI AND TANIGAWA [1], AND THE SECOND-NAMED AUTHOR [27] THE METHOD IN [27] IS BASED ON THE MELLIN-BARNES INTEGRAL FORMULA (1 + ?)?S = 1 · · · MSR R , (16) NUMBER THEORY: SAILING ON THE SEA OF NUMBER THEORY OLUTIONS OF THE VON MANGOLDT FUNCTION × · · · × AR(MR) (M1 + · · · + MR)SR (18) ALSO BEHAVES NICELY IN FACT, THE FOLLOWING THEOREM WAS PROVED IN MATSUMOTO AND TANIGAWA [28] THEOREM 11 ([28]) ASSUME THAT ?K(S) (1 ?K ?R) ARE ABSOLUTELY CON- VERGENT FOR ? > ?K(> 0), CAN BE CONTINUED MEROMORPHICALLY TO THE WHOLE PLANE C, HOLOMORPHIC EXCEPT FOR A POSSIBLE POLE (OF ORDER AT MOST 1) AT S = ?K, AND OF POLYNOMIAL ORDER IN ANY ?XED STRIP ?1 ?? ??2 THEN ?R(S1, , SR; ?1, , ?R) CAN BE CONTINUED MEROMORPHICALLY TO THE WHOLE SPACE CR, AND THE LOCATION OF ITS POSSIBLE SINGULARITIES CAN BE DESCRIBED EX- PLICITLY IN PARTICULAR, IF ALL ?K(S) ARE ENTIRE, THEN ?R(S1, , SR; ?1, , ?R) IS ALSO ENTIRE THE PROOF OF THE ABOVE THEOREM IS AN ANALOGUE OF THE SECOND-NAMED AUTHOR’S PROOF OF THE MEROMORPHIC CONTINUATION OF (13) GIVEN IN [27], WHOSE BASIC TOOL IS THE MELLIN-BARNES FORMULA (14) THE IDEA OF APPLYING FORMULA (14) IN SUCH A SITUATION HAD BEEN ALREADY MENTIONED BY THE ?RST-NAMED AUTHOR [10] IN THE ONE-VARIABLE CASE THE AUTHORS EXPRESS THEIR SINCERE GRATITUDE TO PROFESSOR GAUTAMI BHOWMIK FOR POINTING OUT AN ERROR IN THE ORIGINAL MANUSCRIPT, AND USE- FUL SUGGESTIONS IN PARTICULAR, THE FORM OF CONJECTURE (B) BELOW WAS ?RST SUGGESTED BY HER NUMBER THEORY: SAILING ON THE SEA OF NUMBER THEORY LOG K · LOG(N ?K) ?N(LOG N)2 (26) IN THE PRESENT PAPER WE WILL SHOW, UNDER THE ASSUMPTION OF CERTAIN CON- JECTURES, THAT ?2(S) HAS THE NATURAL BOUNDARY ON THE LINE ?S = 1 (THEOREM NUMBER THEORY: SAILING ON THE SEA OF NUMBER THEORY