Knowledge (XXG)

Barnes zeta function

Source πŸ“

515: 245: 54: 556: 580: 240:{\displaystyle \zeta _{N}(s,w\mid a_{1},\ldots ,a_{N})=\sum _{n_{1},\dots ,n_{N}\geq 0}{\frac {1}{(w+n_{1}a_{1}+\cdots +n_{N}a_{N})^{s}}}} 549: 365:
Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character
575: 542: 283: 275: 468: 424: 37: 522: 25: 372: 398: 352: 344: 422:
Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions",
487: 443: 390: 336: 526: 477: 433: 380: 328: 499: 455: 495: 451: 376: 569: 356: 29: 17: 438: 491: 447: 394: 340: 514: 482: 385: 332: 409:
Barnes, E. W. (1904), "On the theory of the multiple gamma function",
348: 402: 319:
Barnes, E. W. (1899), "The Theory of the Double Gamma Function. ",
363:
Barnes, E. W. (1901), "The Theory of the Double Gamma Function",
463: 530: 57: 239: 309: = 1 it is the Riemann zeta function. 464:"On Barnes' multiple zeta and gamma functions" 550: 8: 557: 543: 321:Proceedings of the Royal Society of London 481: 437: 384: 228: 218: 208: 189: 179: 160: 146: 127: 122: 106: 87: 62: 56: 371:(274–286), The Royal Society: 265–387, 48:The Barnes zeta function is defined by 33: 7: 511: 509: 36:). It is further generalized by the 529:. You can help Knowledge (XXG) by 14: 513: 267:has real part greater than  462:Ruijsenaars, S. N. M. (2000), 327:, The Royal Society: 265–268, 225: 166: 112: 68: 1: 263:have positive real part and 581:Mathematical analysis stubs 24:is a generalization of the 597: 508: 439:10.1016/j.aim.2003.07.020 411:Trans. Camb. Philos. Soc. 276:meromorphic continuation 469:Advances in Mathematics 425:Advances in Mathematics 525:–related article is a 483:10.1006/aima.2000.1946 386:10.1098/rsta.1901.0006 333:10.1098/rspl.1899.0101 241: 38:Shintani zeta function 523:mathematical analysis 242: 26:Riemann zeta function 576:Zeta and L-functions 286:are simple poles at 55: 22:Barnes zeta function 377:1901RSPTA.196..265B 237: 159: 538: 537: 235: 118: 588: 559: 552: 545: 517: 510: 502: 485: 458: 441: 418: 405: 388: 359: 246: 244: 243: 238: 236: 234: 233: 232: 223: 222: 213: 212: 194: 193: 184: 183: 161: 158: 151: 150: 132: 131: 111: 110: 92: 91: 67: 66: 30:E. W. Barnes 596: 595: 591: 590: 589: 587: 586: 585: 566: 565: 564: 563: 506: 461: 421: 408: 362: 318: 315: 308: 278:to all complex 262: 224: 214: 204: 185: 175: 165: 142: 123: 102: 83: 58: 53: 52: 46: 12: 11: 5: 594: 592: 584: 583: 578: 568: 567: 562: 561: 554: 547: 539: 536: 535: 518: 504: 503: 476:(1): 107–132, 459: 432:(2): 362–395, 419: 406: 360: 314: 311: 306: 258: 248: 247: 231: 227: 221: 217: 211: 207: 203: 200: 197: 192: 188: 182: 178: 174: 171: 168: 164: 157: 154: 149: 145: 141: 138: 135: 130: 126: 121: 117: 114: 109: 105: 101: 98: 95: 90: 86: 82: 79: 76: 73: 70: 65: 61: 45: 42: 28:introduced by 13: 10: 9: 6: 4: 3: 2: 593: 582: 579: 577: 574: 573: 571: 560: 555: 553: 548: 546: 541: 540: 534: 532: 528: 524: 519: 516: 512: 507: 501: 497: 493: 489: 484: 479: 475: 471: 470: 465: 460: 457: 453: 449: 445: 440: 435: 431: 427: 426: 420: 416: 412: 407: 404: 400: 396: 392: 387: 382: 378: 374: 370: 366: 361: 358: 354: 350: 346: 342: 338: 334: 330: 326: 322: 317: 316: 312: 310: 305: 302: =  301: 298: =  297: 293: 290:= 1, 2, ..., 289: 285: 284:singularities 282:, whose only 281: 277: 272: 270: 266: 261: 257: 253: 229: 219: 215: 209: 205: 201: 198: 195: 190: 186: 180: 176: 172: 169: 162: 155: 152: 147: 143: 139: 136: 133: 128: 124: 119: 115: 107: 103: 99: 96: 93: 88: 84: 80: 77: 74: 71: 63: 59: 51: 50: 49: 43: 41: 39: 35: 31: 27: 23: 19: 531:expanding it 520: 505: 473: 467: 429: 423: 414: 410: 368: 364: 324: 320: 303: 299: 295: 291: 287: 279: 273: 268: 264: 259: 255: 251: 249: 47: 21: 15: 18:mathematics 570:Categories 313:References 44:Definition 492:0001-8708 448:0001-8708 417:: 374–425 395:0264-3952 357:186213903 341:0370-1662 274:It has a 199:⋯ 153:≥ 137:… 120:∑ 97:… 81:∣ 60:ζ 500:1800255 456:2078341 373:Bibcode 294:. For 32: ( 498:  490:  454:  446:  401:  393:  355:  349:116064 347:  339:  250:where 521:This 403:90809 399:JSTOR 353:S2CID 345:JSTOR 527:stub 488:ISSN 444:ISSN 391:ISSN 337:ISSN 254:and 34:1901 20:, a 478:doi 474:156 434:doi 430:187 381:doi 369:196 329:doi 16:In 572:: 496:MR 494:, 486:, 472:, 466:, 452:MR 450:, 442:, 428:, 415:19 413:, 397:, 389:, 379:, 367:, 351:, 343:, 335:, 325:66 323:, 271:. 40:. 558:e 551:t 544:v 533:. 480:: 436:: 383:: 375:: 331:: 307:1 304:a 300:w 296:N 292:N 288:s 280:s 269:N 265:s 260:j 256:a 252:w 230:s 226:) 220:N 216:a 210:N 206:n 202:+ 196:+ 191:1 187:a 181:1 177:n 173:+ 170:w 167:( 163:1 156:0 148:N 144:n 140:, 134:, 129:1 125:n 116:= 113:) 108:N 104:a 100:, 94:, 89:1 85:a 78:w 75:, 72:s 69:( 64:N

Index

mathematics
Riemann zeta function
E. W. Barnes
1901
Shintani zeta function
meromorphic continuation
singularities
doi
10.1098/rspl.1899.0101
ISSN
0370-1662
JSTOR
116064
S2CID
186213903
Bibcode
1901RSPTA.196..265B
doi
10.1098/rsta.1901.0006
ISSN
0264-3952
JSTOR
90809
Advances in Mathematics
doi
10.1016/j.aim.2003.07.020
ISSN
0001-8708
MR
2078341

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

↑