By the end, he was reading the lyrics from a ring of autocue screens placed among the footlights, a small, silver-toupeed figure no longer able to rely on the memory that had once held the entire library of Broadway theatre song, from All the Things You Are to Zing! Went the Strings of My Heart. The great voice had been steadily deteriorating, its range contracted and its intonation unsteady. But, even then, there was enough left to evoke more than just the ghost of the finest male interpreter of songs written by composers and lyricists who mostly saw themselves as craftsmen, but produced masterpieces of seemingly imperishable sophistication and sensitivity.

As late as 1990, when Frank Sinatra was in his 75th year, a few thousand people believed it worth missing England’s World Cup semi-final against Germany in order to trek out to an unprepossessing shed in east London, a venue then known as the Docklands Arena. Instead of Gazza’s tears, we got Sinatra celebrating the Fourth of July with a repertoire including My Heart Stood Still, a song written in 1927 by Richard Rodgers and Lorenz Hart for the hit musical A Connecticut Yankee. As ever, and not yet needing the services of an electronic prompter, he meticulously credited not just the composers but Nelson Riddle’s swooning arrangement for strings, brass and woodwind before intoning the opening lines: “I laughed at sweethearts I met at school / All indiscreet hearts seemed romantic fools … ”

That’s what Broadway people would call the verse – the bit tacked on at the beginning, a kind of prologue that made sense in the context of a musical but usually got left off by subsequent interpreters. Like Mabel Mercer, whose phrasing he admired and studied, Sinatra preferred to perform such songs with all their component parts and nuances intact. And his insistence on such a formality symbolised the reverence with which he approached the material that formed the core of the performances during his last years, a guarantee of dignity even as he grew infirm. On what would have been his 100th birthday – Sinatra was born in Hoboken, New Jersey, on 12 December 1915 – it’s worth remembering that for all that he is fixed in the public mind as the Chairman of the Board, or the boorish leader of the Rat Pack, Sinatra was at his most remarkable as a singer and, even as his voice faltered as his career neared its close, it was as a singer that he sought to cement his legacy.

He had explicitly foreshadowed his own decline. He was not quite 50 years old when he recorded September of My Years, a song of bittersweet elegance tailor-made by his buddies Jimmy Van Heusen and Sammy Cahn to express the encroaching sense of his prime drawing to a close. This was 1965, and that prime had begun barely only a dozen years earlier, with his first recordings for the Capitol label: the start of a string of albums that included Songs for Swingin’ Lovers! and In the Wee Small Hours. He would never again sound as much at one with his voice, with the material, or with the times.

The tide was turning against him. The Rat Pack shtick had worn threadbare at just the moment when an investigation into his liaisons with known mobsters – from Willie Moretti in New Jersey to Sam Giancana in Chicago – cost him the right to a casino operator’s licence, and thus to own shares in the hotels in Las Vegas and on Lake Tahoe where he performed and acted as a frontman. An association that once imbued his public image with a frisson of dark glamour had become tainted by sleaze.

The beat of the music had changed, too. Like most entertainers of his generation, he tried at first to ignore the tidal wave generated when the Beatles dropped their pebble into the pond of popular entertainment. Ten years earlier he had taken the same approach to Elvis, and emerged more or less unscathed. But this was different.

An attempt to confront the new age on its own terms – a fascinating song cycle called Watertown, written by Bob Gaudio and Jake Holmes, and depicting the melancholy existence of a suburban husband who could have come from the pages of Cheever or Updike – was met with indifference; so was a less creditable attempt to reach the new audience with an album of songs by Rod McKuen. It was possible, of course, that Sinatra himself could not discern the difference between the two.

Six years after proclaiming that he had come to his September, he announced his retirement. It didn’t last, of course, and after a mere 16 months of silence the billboards and music paper ads proclaimed a single message: Ol’ Blue Eyes Is Back, with a new album and a TV special of that title, followed by a world tour at the start of 1974.

The hiatus had not noticeably blunted the edge of his temper. During a press conference in Australia, he memorably described the journalists who followed his every move as “bums, parasites, fags and buck-and-a-half hookers”. But eventually the ageing process – and perhaps a fourth marriage, to Barbara Marx – did the trick. The thousand concerts he undertook between 1973 and his final show in January 1995 were mostly reflective affairs in which, having dabbled in but renounced the work of George Harrison, Stevie Wonder and Jim Croce, he returned to the songs of his burnished yesterdays.

The valedictory air of his concerts deepened as the years passed, carrying friends and associates away with them. At Carnegie Hall in 1984, 10 years after his comeback, he told the audience: “Tonight and for the rest of this year we are dedicating each performance to three men who were very instrumental in my career and also as friends. I’m speaking of Count Basie, Gordon Jenkins and Don Costa. We miss them.” The dedicatees were one great bandleader, with whom he recorded an ebullient album on stage in Las Vegas in 1966, and two great arrangers – one of whom, Jenkins, had also composed a song called Goodbye, which may have inspired his most profoundly accomplished and affecting recorded performance, back in 1958.

In his final decade, he continued to tour widely while returning time and again to serenade the high-rollers in Atlantic City and Las Vegas, his spiritual homes. But by the time he pitched up at an indoor sports arena in Atlanta, Georgia in January 1994, entering what would be the last 12 months of his performing career, his appeal was sufficient to fill barely half of the Omni Coliseum’s 15,000 seats.

That night, fronting a 50-piece orchestra, he made his way haltingly through the classics, including I’ve Got the World on a String, My Funny Valentine, Come Rain or Come Shine and I Guess I’ll Hang My Tears Out to Dry. He fumbled the introductions, often repeating himself, and ended with My Way, a song he had once been unafraid to introduce to a Carnegie Hall audience as “a pain in the you-know-where”. Afterwards he stayed on to shake hands with fans who clustered at the front of the stage. Only a handful were left by the time an aide shepherded him back to the dressing room.

Just over a year later, on 25 February 1995, at the Marriott Hotel in Palm Desert, not far from his California home, he provided the climax to the Frank Sinatra golf tournament with a short six-song set. It was the very last public performance of his career. He sang You Make Me Feel So Young and finished with The Best is Yet to Come.

At the end of that year, silent now, he was given an 80th birthday gala celebration at the Shrine Auditorium in Los Angeles, where he listened with an expression of benign vagueness as Bruce Springsteen ironed the chromatic kinks out of Angel Eyes, a song by Matt Dennis and Earl Brent that was so often a highlight of Sinatra’s concerts. Bob Dylan followed Springsteen to sing his own Restless Farewell, road-testing the gentle, careful style that would so strikingly re-emerge 20 years later with Shadows in the Night, his album of Sinatra-linked standards.

“Happy birthday, Mr Frank,” Dylan said, but the next two-and-a-half years would find Sinatra afflicted by heart problems, bladder cancer and dementia. He died on 14 May 1998, in a Los Angeles hospital, aged 82. But he had said his goodbyes every time he sang Angel Eyes, as he did to close his not-quite-farewell show in 1971 and many more times during those last thousand concerts, always investing its final line with the air of a man draining the last drop of bourbon, stubbing out his cigarette, pulling on his coat, adjusting the angle of his fedora and heading through the saloon door into the lonely, rainswept night: “’Scuse me while I disappear … ”